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O set 0 implies that on X there is a simple TG-path in B(q',250) from q' to Pi rH1 that avoids 8D( extrH11')' Since the number of edges traversed by these paths is bounded, it is easy to see that the probability that S follows the latter path and X holds given 0 is chosen sufficiently small. The latter is equivalent to showing that the random variable P[ST=zTIS'I T ] is bounded away from zero on A by a function of 8 and A. Suppose that A holds, and that ZTEa'.f. Let 6 and 6 be the two connected components of ar\{zr}. The construction of Y n in the proof (3.48) shows that there is a path Yo connecting 6 and 6 in B(zT,8r)\B(zT' !8r) that separates ZT from a!. in DT such that Yo is an (s, diam Yo)-barrier for (D, IT) for every sE (0,8'], where 8' E (0,1) is a universal constant. (The assumption that Al holds is used here.) If ST never visits a vertex adjacent to Yo, then Yo separates ST from a!.. In this situation, if I \ IT does not hit Yo, then ST does not visit a vertex adjacent to it, and therefore ZT=ST' Thus, Theorem 3.11 (or Remark 3.12) applies to give the needed lower bound on P[ST=zTIS'IT ] when ST does not visit a vertex adjacent to Yo. 1109 . Now, "(
1la :=fn{iE{5 and
1La :=£n{iE{51 and
5.5. Let A>O. Then there are sequences
'\to and
Xn
Yn'\to, and an a>O,
such that (5.20) (5.21)
923
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G. F. LAWLER. O. SCHRAMM AND W. WERNER
Actually (see, e.g., [30]), much stronger statements hold: In the above, :=::: may be replaced by:::::, and these statements hold for every sequence tending to 00. But the present statement will be sufficient to determine the values of the Brownian intersection exponents, and it can be easily proved as follows. Proof. We will only give the detailed proof of (5.21). The proof of (5.20) is easier,
follows exactly the same lines, and is safely left to the reader. Because of (5.19), it suffices to find the lower bound for the left-hand side of (5.21). Let us first introduce some notation. Let rE (0. i), and consider the measure JL~ XJL~ on the space of pairs (B, B'). Let
Let B* be the path B stopped when it hits C l / 4 ' if it does, and B*=B, if it does not hit C l / 4 . Similarly, define B'* from B'. Let Q3 and ~' be the traces of Band B', respectively. Let £* be the event that the traces of B* and B'* are disjoint. On £*, let 0;' and O2 be the domains defined for B* and B'*, as 0 1 and O2 were defined for B and B'. Then 0ie OJ on £, j=l, 2. For j=l, 2, on £*, let be the 7r-extremal distance between CT and C l / 4 in 0;. Otherwise, set =00. For a>O, let 'Va be the event that the distance between ~nA(~, 1) and ~'nA(~, 1) is at least a. Suppose that aE (0, i). Observe that for (B, B')~Va, there is for j=l or j =2 a path of length at most a in OJ nA(!. 1) which separates C r from C l in OJ. It then follows that £j~£;+cl10g(1/a), for some constant Cl>O. Consequently,
.c;
.c;
(5.22) =
a Cl min{>.,I} f(4r),
since the image of JL~ under the map Bt---+4B* is JL~r' By (5.19), we have f(r):=:::r W ,U..l) (and f is non-decreasing). Consequently, if a is chosen sufficiently small, there is a sequence Yn,\-0 (for instance a subsequence of 4 -n) such that f(Yn) ~2aCl min{A,l} f(4Yn). For these Yn, (5.22) gives (5.23) Fix such an a and such a Yn' Let Ia be the event that iEOl and the distance from i to Q3 U~' is at least 110 a. Observe that if we apply an independent random uniform rotation 924
VALUES OF BROWNIAN INTERSECTION EXPONENTS, II: PLANE EXPONENTS
303
about 0 to a pair (B, B')EDant, then with probability at least a/1OIT the rotated pair is in La. Since the integrand in (5.23) is invariant under rotations, (5.23) and (5.19) give
It therefore suffices to show that when 0:>0 is small, we have LantcHo.. To prove this, consider a pair (B,B')EIa . Let A be the subarc of DInGI that has i as one endpoint and the other endpoint is in IB, and let A' be the subarc of DI nG1 that has i as one endpoint and the other endpoint is in IB'. Then the extremal distance from A to IB' in 0 1 is bounded from below by a positive constant depending only on a, as is the extremal distance from A' to IB in 0 1 . Conformal invariance of extremal distance therefore shows that the distance from 4>(i) to {O, iIT} is bounded from below by a constant depending 0 only on a, which proves that (B,B')E1lo:, where a>O depends only on a.
6. The universality argument We are now ready to combine the results derived so far to prove our main theorems. As in [27], we follow the universality ideas presented in [32]. First, Theorem 5.6 (~(1, 1)=~) will be proved, followed by Theorem 5.7 (giving ~(1, 1, 1, A)). As we have seen, Theorems 1.1 and 1.2 are immediate consequences.
6.1. Proof of e(1, 1)= ~ Let r>O, let K t be a radial SLE 6-process in U starting at i, and let T=T(r) be the least t such that Kt nGr =/=0. Set Ji:=KT , let P r denote the law of J't, and let Er denote expectation with respect to this measure. As before, we let JL~ denote the Brownian excursion measure in A(r, 1) started from Gn and let IB denote the trace of the excursion B. We are interested in the event £* in which JinIB=0 and IB crosses A(r, 1) (i.e., IBnG1 =/=0). The proof will proceed by computing (Pr x JL~)[£*] in two different ways. In the first computation, we begin by conditioning on Ji and then taking the expectation, while the second computation begins by conditioning on B. When IB does not separate Gr from G in U, let OB denote the connected component of A(r, l)\IB that touches both circles Gr and G, and .cB the corresponding IT-extremal distance.
When Ji does not disconnect Or from a in V, let OJ( denote the connected component of A(r, 1) \Ji that touches both circles, and .cj( the corresponding IT-extremal distance. 925
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Suppose first that oR is given and that £.R
where
is as in Lemma 5.4. On the other hand, we know from Theorem 3.1, with b=l, 1\;=6, that fL
Combining these two facts implies that (6.1) Suppose now that B is given and that it does cross the annulus without separating the disk Cr from C. The second part of Lemma 4.3 shows that there exists a constant c> 0 such that
Hence, combining this with (5.18) shows that
(Pr xJL~)[f*l:::;; rE(l,l)+o(l).
(6.2)
On the other hand, suppose now that B is given and that BE1I.0., where 11.0. is as in Lemma 5.5. Then by the first part of Lemma 4.3, there exists a constant d such that
Combining this with Lemma 5.5, we find that
Comparing with (6.2) gives
Now, by (6.1),
when n~oo, which proves ~(1, 1)=~. 926
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305
6.2. The determination of e(1, 1, 1, A) The goal now is to prove (5.7). As oXH€(l, 1, 1, oX)=€(l, 1, t(1, oX» is continuous on [0,00) (see [31]), we can restrict ourselves to the case where OX>O. The proof goes along similar lines as the proof of €(1, 1)=~. This time, we will consider two independent Brownian excursions B and B' in the annulus A(r, 1), and one SLE6 it as before. We are interested in the event that B, B' and it all cross the annulus, that they remain disjoint, and that B, it and B' are in clockwise order. Let us call this event l. Note that in this case, it crosses the annulus in 0 1 , where we use the notations of §5.3. We shall compute in two different ways the quantity
On the one hand, we know from conformal invariance and the restriction property of Brownian excursions and from (5.17) that when it crosses the annulus without separating Cr from C,
But we know from Theorem 3.1 that
(6.4) where
v = v(b) = H4b+l+V1+24b). Note that v is a continuous increasing function of b on (0,00). Consequently, (6.3) and (6.4) give
i
exp( -oX£2) d/-L;(B) d/-L;(B') dPr(it)
~ r.,(W,I,A)).
Combining this with (5.4) shows that
(6.5) where -
a = a(oX) := v(€(l, 1, oX») =
418
2
((11 + v'24 oX + 1) -4).
Suppose now that Band B' are given and that E is satisfied; i.e., that B and B' cross the annulus without intersecting each other. Then Lemma 4.3 shows that
927
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
for some constant c. Hence, combining this with (5.19) shows that (6.6) Suppose now that B and B' are given and that (B, B')Eiio:. Then Lemma 4.3 shows that there exists a constant c' > 0 such that
Combining this with Lemma 5.5 gives
Comparing with (6.6) implies that
when n-+oo. Consequently, by (6.5),
~(1,
1, 1, A)=a(A).
928
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VALUES OF BROWNIAN INTERSECTION EXPONENTS, II: PLANE EXPONENTS
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References [1] AHLFORS, L. V., Conformal Invariants: Topics in Geometric FUnction Theory. McGrawHill, New York, 1973. [2] AIZENMAN, M., DUPLANTIER, B. & AHARONY, A., Path crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Lett., 83 (1999), 1359-1362. [3] AZENCOTT, R., Behaviour of diffusion semi-groups at infinity. Bull. Soc. Math. Prance, 102 (1974), 193-240. [4] BISHOP, C. J., JONES, P. W., PEMANTLE, R. & PERES, Y., The dimension of the Brownian frontier is greater than 1. J. Funct. Anal., 143 (1997), 309-336. [5] BURDZY, K. & LAWLER, G. F., Non-intersection exponents for Brownian paths. Part I: Existence and an invariance principle. Probab. Theory Related Fields, 84 (1990), 393-410. [6] - Non-intersection exponents for Brownian paths. Part II: Estimates and applications to a random fractal. Ann. Probab., 18 (1990),981-1009. [7] CARDY, J. L., Conformal invariance and surface critical behavior. Nuclear Phys. B, 240 (1984), 514-532. [8] - Critical percolation in finite geometries. J. Phys. A, 25 (1992), L201-L206. [9] - The number of incipient spanning clusters in two-dimensional percolation. J. Phys. A, 31 (1998), L105. [10] CRANSTON, M. & MOUNTFORD, T., An extension of a result of Burdzy and Lawler. Probab. Theory Related Fields, 89 (1991), 487-502. [11] DUPLANTIER, B., Random walks and quantum gravity in two dimensions. Phys. Rev. Lett., 81 (1998), 5489-5492. [12] - Two-dimensional copolymers and exact conformal multifractality. Phys. Rev. Lett., 82 (1999), 880-883. [13] - Harmonic measure exponents for two-dimensional percolation. Phys. Rev. Lett., 82 (1999), 3940-3943. [14] DUPLANTIER, B. & KWON, K.-H., Conformal invariance and intersection ofrandom walks. Phys. Rev. Lett., 61 (1988),2514-2517. [15] DUPLANTIER, B. & SALEUR, H., Exact determination of the percolation hull exponent in two dimensions. Phys. Rev. Lett., 58 (1987), 2325-2328. [16] KENYON, R., Conformal invariance of domino tiling. Ann. Probab., 28 (2000), 759-795. [17] - Long-range properties of spanning trees. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys., 41 (2000), 1338-1363. [18] - The asymptotic determinant ofthe discrete Laplacian. Acta Math., 185 (2000), 239-286. [19] LAWLER, G. F., Intersections of Random Walks. Birkhiiuser Boston, Boston, MA, 1991. [20] - Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab., 1:2 (1996), 1-20 (electronic). [21] - Cut times for simple random walk. Electron. J. Probab., 1:13 (1996), 1-24 (electronic). [22] - The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab., 1:5 (1996), 29-47 (electronic). [23] - The frontier of a Brownian path is multifractal. Preprint, 1997. [24] - Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J., 4:5 (1998), 1-67 (electronic). [25] - Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions, in Random Walks (Budapest, 1998), pp. 219-258. Bolyai Soc. Math. Stud., 9. Janos Bolyai Math. Soc., Budapest, 1999. [26] LAWLER, G. F. & PUCKETTE, E. E., The intersection exponent for simple random walk. Combin. Probab. Comput.,9 (2000),441-464. 929
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[27J LAWLER, G. F., SCHRAMM, O. & WERNER, W., Values of Brownian intersection exponents, I: Half-plane exponents. Acta Math., 187 (2001), 237-273. [28J - Values of Brownian intersection exponents, III: Two-sided exponents. To appear in Ann. Inst. H. Poincare Probab. Statist. http://arxiv.org/abs/math.PR/0005294. [29J - Analyticity of intersection exponents for planar Brownian motion. To appear in Acta Math., 188 (2002). http://arxiv.org/abs/math.PR/0005295. [30] - Sharp estimates for Brownian non-intersection probabilities. To appear in In and Out of Equilibrium. Probability with a Physics Flavor. Progr. Probab. Birkhauser Boston, Boston, MA. [31] LAWLER, G. F. & WERNER, W., Intersection exponents for planar Brownian motion. Ann. Probab., 27 (1999),1601-1642. [32] - Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. (JEMS),2 (2000),291-328. [33] L6wNER, K., Untersuchungen liber schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann., 89 (1923), 103-121. [34] MADRAS, N. & SLADE, G., The Self-Avoiding Walk. Birkhauser Boston, Boston, MA, 1993. [35J MANDELBROT, B. B., The Fractal Geometry of Nature. Freeman, San Francisco, CA, 1982. [36J NIENHUIS, B., Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Statist. Phys., 34 (1984), 731-761. . [37] POMMERENKE, CH., On the Loewner differential equation. Michigan Math. J., 13 (1966), 435-443. [38J REVUZ, D. & YOR, M., Continuous Martingales and Brownian Motion. Grundlehren Math. Wiss., 293. Springer-Verlag, Berlin, 1991. [39] SCHRAMM, 0., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221-288. [40] WERNER, W., Bounds for disconnection exponents. Electron. Comm. Probab., 1:4 (1996), 19--28 (electronic). GREGORY F. LAWLER Department of Mathematics Duke University Box 90320 Durham, NC 27708-0320 U.s.A. [email protected]
ODED SCHRAMM Microsoft Research 1, Microsoft Way Redmond, WA 98052 U.S.A. [email protected]
WENDELIN WERNER Departement de Mathematiques Universite Paris-Sud Bat. 425 FR-91405 Orsay Cedex France wendelin. [email protected]
Received May 18, 2000
930
The Annals of Probability 2004, Vol. 32, No. lB, 939--995 © Institute of Mathematical Statistics, 2004
CONFORMAL INVARIANCE OF PLANAR LOOP-ERASED RANDOM WALKS AND UNIFORM SPANNING TREES By GREGORY F. LAWLER,! ODED SCHRAMM AND WENDELIN WERNER
Cornell University, Microsoft Corporation and Universite Paris-Sud This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that aD is a C 1-simp1e closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc A c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
1. Introduction. 1.1. Motivation from statistical physics. One of the main goals of both probability theory and statistical physics is to understand the asymptotic behavior of random systems when the number of microscopic random inputs goes to 00. These random inputs can be independent, such as a sequence of independent random variables, or dependent, as in the Ising model. Often, one wishes to understand these systems via some relevant "observables" that can be of a geometric or analytic nature. In order to understand this asymptotic behavior, one can attempt to prove convergence toward a suitable continuous model. The simplest and most important example of such random continuous models is Brownian motion, which is the scaling limit of random walks. In particular, a simple random walk on any lattice in ]Rd converges to (a linear image of) Brownian motion in the scaling limit. Physicists and chemists have observed that critical systems (i.e., systems at their phase transition point) can exhibit macroscopic randomness. Hence, various quantities related to the corresponding lattice models should converge as the mesh is refined. In fact, one of the important starting points for theoretical physicists working on two-dimensional critical models is the assumption that the continuous limit is independent of the lattice and, furthermore, displays conformal invariance. This assumption has enabled them to develop and use techniques from conformal field theory to predict the exact values of certain critical exponents. Until very Received February 2002; revised March 2003. 1Supported in part by the NSF and the Mittag-Leffler Institute. AMS 2000 subject classifications. 82B41. Key words and phrases. Loop-erased random walk, uniform spanning trees, stochastic Loewner evolution.
939 I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_30, 931
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recently, the existence of the limit, its conformal invariance and the derivation of the exponents assuming conformal invariance remained beyond mathematical justification for the basic lattice models in critical phenomena, such as percolation, the Ising model and random cluster measures. Although there are many interesting questions about higher-dimensional systems, we will limit our discussion to two dimensions where conformal invariance plays an essential role. 1.2. Recent progress. In [38], a one-parameter family of random growth processes (loosely speaking, random curves) in two dimensions was introduced. The growth process is based on Loewner's differential equation, where the driving term is time-scaled one-dimensional Brownian motion, and is therefore called stochastic Loewner evolution, or SLEK • The parameter K :::: 0 of SLE is the time scaling constant for the driving Brownian motion. It was conjectured that the scaling limit of the loop-erased random walk (LERW) is SLE2, and this conjecture was proved to be equivalent to the conformal invariance of the LERW scaling limit (see [38]). The argument given was quite general and shows that a conformally invariant random path satisfying a mild Markovian property, which will be described below, must be SLE. On this basis, it was also conjectured there that the scaling limits of the critical percolation interface and the uniform spanning tree Peano curve are the paths of SLE6 and SLEg, respectively, and it was claimed that conformal invariance is sufficient to establish these conjectures. (For additional conjectures regarding curves tending to SLE, including the interfaces in critical random cluster models-also called FK percolation modelsfor q E [0,4], see [36].) At some values of the parameter K, SLE has some remarkable properties. For instance, SLE6 has a locality property (see [27]) that makes it possible to relate its outer boundary to that of planar Brownian motion. This has led to the proof of conjectures concerning planar Brownian motion and simple random walks (see [27]-[29]).
In [40] and [41], Smimov recently proved the existence and conformal invariance of the scaling limit of critical site percolation on the two-dimensional triangular lattice: he managed to prove Cardy's formula (see [5]) which is a formula for the limit of the probability of a percolation crossing between two arcs on the boundary of the domain. Combining this information with the independence properties of percolation, Smirnov then showed that the scaling limit of the percolation interface is SLE6. This has led to the rigorous determination of critical exponents for this percolation model (see [30] and [42]). 1.3. LERWand UST defined. The uniform spanning tree (UST), which can be interpreted as the q = 0 critical random cluster model (see [15]), is a dependent model that has many remarkable features. In particular, it is very closely related to the loop-erased random walk, whose definition (see [24]) we now briefly recall. 932
CONFOR M AL I NVARI ANCE
941
Consider any fini te or rec urrent connected graph G , a vertex a and a set of vertices V . A loop-erased random walk (LERW) from a to V is a random simple curve joining a to V obtained by erasing the loops in chronol og ical order from a sim ple random walk started at a and stopped upon hitting V. In other words, if (f (n) , 0 :5 11 :5 T) is a simple random walk on G started from a and stopped at its first hitting time T of V , the loop erasure f3 = (f3o , ... , f3e) is defin ed inductive ly as foll ows: f30 = a ; if f3n E V , then 11 = i; and otherwi se f3n +l = r (k ), wherek = I + max {m :5 T: r (m) =f3n }' A spanning tree T of a connected grap h G is a subgraph of G such that fo r every pair of vertices lJ ,1/ in G there is a unique sim ple path (i.e., self-avoiding) in T with these vertices as endpoi nts. A ullifo rm spallll ing tree (UST) in a finite, connected graph G is a sample from the uniform probability measure on span ning trees of G. It has been shown in [34] that the law of the self-avoidin g path wit h endpoint s a and b in the UST is the same as that of the LERW from a to fb i. See Figure I . Wilson [44] establi shed an even stronger connection between LERW and UST by giving an algorithm to ge nerate UST's using LERW. Wilson 's algorithm run s as follows. Pick an arbitrary ordering vo, VI, ... 'VII! for the verti ces in G. Let To = {vol. Inducti ve ly, for 11 = 1, 2, ... , 111 , defin e Tn to be the uni on of Tn- l and a (conditiona ll y independent) LERW path from Vn to Tn_ I. (If Vn E Tn_ I, then Tn = Tn_ I.) Then, regardless of the chosen order of the vertices, T,/J is a UST on G. Wilson 's algorithm gives a natural extension of the definiti on of UST to infinite recurrent graphs. In fact, for transient graph s, there are two natural definiti ons which often coinc ide, but this interesting theory is somewhat re moved from the topic of this paper. Many striking properties of UST and LERW have been
FI G. 1.
The LERIV ill the UST. 933
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
discovered. See [32] for a survey of UST's and [26] for a survey of properties ofLERWin Zd, d > 2. Exploiting a link with domino tilings and deriving discrete analogs of CauchyRiemann equations, Kenyon (see [20] and [21]) rigorously established the values of various critical exponents predicted for the LERW (see [11], [14] and [32]) in two dimensions. In particular, he showed that the expected number of steps of a LERW joining two comers of the N x N square in the square grid ZZ is of the order of magnitude of N 5/ 4 . He also showed conformal invariance for the leading term in the asymptotics of the probability that the LERW contains a given edge. This was the first mathematical evidence for full conformal invariance of the LERW scaling limit. In [3] and [1] subsequential scaling limits of the UST measures in Zd were shown to exist, using a compactness argument. Moreover, these papers prove that all the paths in the scaling limit that intersect a fixed bounded region are uniformly Holder continuous. In [38] the topology of subsequential scaling limits of the UST on ZZ was determined. In particular, it was shown that every subsequential scaling limit of the LERW is a simple path. 1.4. A short description of SLE. We now briefly describe SLE; precise definitions are deferred to Section 2.1. Chordal SLE is a random growing family of compact sets K t , t E [0, (0), in the closure IHr of the upper half plane IHr. The evolution of K t is given by the Loewner differential equation with "driving function" Brownian motion. From [36], it is known that when K i= 8 the process is described by a random curve y : [0, (0) ---+ IHr, in the sense that, for every t ~ 0, IHr \ K t is the unbounded component of IHr \ y [0, t]. A corollary of our results is that this holds for K = 8 as well. The curve y satisfies y (0) = and limt-HXJ y (t) = 00. If K :s 4, then y is a simple curve and K t = y[O, t]. There is another version of SLE called radial SLE. Radial SLE also satisfies the description above, except that the upper half plane IHr is replaced by the unit disk 1U, yeO) is on the unit circle a1U and limt-+oo yet) = 0. Both radial and chordal versions of SLE may be defined in an arbitrary simply connected domain D ~ C by mapping over to D using a fixed conformal map 4> from IHr or 1U to D.
°
°
1.5. The main results of the paper. Let D ~ C be a simply connected domain with ED. For 8 > 0, let J-L8 be the law of the loop erasure of a simple random walk on the grid 8Z z, started at and stopped when it hits aD. See Figure 2. Let v be the law of the image of the radial SLEz path under a conformal map from the unit disk 1U to D fixing 0. When the boundary of D is very rough, the conformal map from 1U to D might not extend continuously to the boundary, but the proof of the following theorem in fact shows that even in this case the image of the SLEz path has a unique endpoint on aD.
°
934
CONFORMALINVAR~CE
FIG. 2.
943
A sample of the loop-erased random walk; proved to converge to radial SLE2.
On the space of unparameterized paths in C, consider the metric p (f3, y) = infsuPtE[O,I] [S(t) - y(t)[, where the infimum is over all choices ofparameterizations Sand y in [0, 1] of f3 and y. THEOREM 1.1 (LERW scaling limit). The measures J.L8 converge weakly to v as 8 ---+ 0 with respect to the metric p on the space of curves. Since SLE is conformally invariant by definition, this theorem implies conformal invariance of the LERW. The theorem and proof apply also to some other walks on lattices in the plane where the scaling limit of the walk is isotropic Brownian motion. It even applies in the nonreversible setting. See Section 6 for further details. There are two distinct definitions for the UST corresponding to a domain D ~ C, as follows. Let G F(D) denote the subgraph of Z2 consisting of all the edges and vertices which are contained in D. If G F(D) is connected, then we refer to the UST on G F(D) as the UST on D with free boundary conditions. Let Gw(D) denote the graph obtained from Z2 by contracting all the vertices outside of D to a single vertex (and removing edges which become loops). Then the UST on Gw(D) is the UST on D with wired boundary conditions. Since the UST is built from the LERW via Wilson's algorithm, it is not surprising that conformal invariance of the UST scaling limit should follow from that of the LERW scaling limit. In fact, [38], Theorem 11.3, says just that. COROLLARY 1.2 (UST scaling limit). The wired andfree UST scaling limits (as defined in [38]) in a simply connected domain Dee whose boundary is a C I-smooth simple closed curve exist and are conformally invariant. 935
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One can eas ily show, using [38] , Theore m Il.l (i), that the wired tree depends continuously on the domain, a nd he nce for that case D may be an arbitrary simpl y connected domain. However, some regularity assumption is needed for the free UST scaling limit: conformal invariance fa ils for the domain whose boundary conta ins the topologist's sine curve (the closure of {x + i sin(l /x ):x E (0, I'I}). T he UST Peano curve is an entirely different curve derived from the UST in two dime nsions . The c urve is rathe r re markable, as it is a natural random path visiting every vertex in an appropri ate graph or lattice . We now roughly describe two natura l definiti ons of thi s c urve; furth e r detai ls appea r in Sect ion 4. Let C be a fin ite planar graph , with a particular e mbedding in the plane, and let C t de note its planar dual, agai n with a partic ular e mbedding. Then the re is a bijecti on e
FI G. 3. The graph . dllal gra"h. lice. dllal Iree mul Pem/O clm'e. The j'erlex oj II/e dllal 8ra"h corresponding 10 Ifle unbOlllule(1 face is drawn as a cycle.
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CONFORMAL INVARIANCE
945
x-coordinate is odd. Now rescale the resulting oriented graph by 1/2 and translate it by (1/4, 1/4). It is easy to check that a Hamiltonian path (a path visiting every vertex exactly once) respecting the orientation on the resulting oriented graph is the same as the UST Peano path of '1.2 . It should be expected that the uniform measure on Hamiltonian paths in '1.2 has the same scaling limit as that of the UST Peanopath. Given a domain D, one can consider the UST Peano curve for the wired or for the free UST (which is essentially the same as the wired, by duality). However, the conjecture from [38] regarding the convergence to chordal SLE pertains to the UST Peano curve associated with the tree with mixed wired and free conditions. Let D C
°
Again, the convergence is weak convergence of measures with respect to the metric p. Figure 4 shows a sample of the UST Peano path on a fine grid. As explained above, it was proved in [36] that each SLEK is generated by a path, except for K = 8. In Section 4.4, the remaining case K = 8 is proved, using the convergence of the Peano curve.
FIG. 4.
An arc from a sample of the UST Peano path; proved to converge to chordal SLES.
937
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
Corollary 1.2 and Theorem 1.3 (and their proofs) apply to other reversible walks on planar lattices (the self-duality of Z2 does not play an important role); see Section 6. To add perspective, we note that the convergence to SLE of the LERW and the UST Peano curve are two boundary cases of the conjectured convergence in [36] of the critical FK random cluster measures with parameter q E (0,4). For these parameter values, the scaling limit of the interface of a critical cluster with mixed boundary values is conjectured to converge to chordal SLEK (q), where K(q) = 4njcos- 1(-,J(ij2). The boundary case K(O) = 8 corresponds to the convergence of the UST Peano path to SLE8. The outer boundary of the scaling limit of a macroscopic critical cluster is not the same as the scaling limit of a critical cluster outer boundary, because of "fjords" which are pinched off in the limit. The former is conjectured to "look like" SLE16/K(q), but a precise form of this conjecture is not yet known. In the case q = 0, however, such a correspondence is easy to explain. In Z2, an arc of the Peano curve is surrounded on one side by a simple path in the tree, and on the other side by a simple path in the dual tree. Both these paths are LERWs. Similar correspondences exist for the UST in a subdomain of ffi.2, but one has to set appropriate boundary conditions. Thus, the convergence ofLERW to SLE2 also corresponds to the case q = 0, as 16jK(0) = 2. Suppose that 0 E D and a, f3 CaD, as before. Consider the simple random walk on 8Z2 which is reflected off f3 and stopped when it hits a. Using an analogous method to that of the present paper, one could handle the scaling limit of the loop erasure of this walk. It is described by a variant of SLE2 where the driving term is Brownian motion with time scaled by 2, but having an additional drift. The drift is not constant, but can be explicitly computed. The identification of the scaling limit as one of the SLEs should facilitate the derivation of critical exponents and also the asymptotic probabilities of various events, including some results which have not been predicted by arguments from physics. This was the case for critical site percolation on the triangular grid; see [30], [39], [40] and [42].
1.6. Some comments about the proof Since a loop-erased random walk is obtained in a deterministic way from a simple random walk (by erasing its loops) and since a simple random walk converges to Brownian motion in the scaling limit, it is natural to think that the scaling limit of the LERW should simply be the process obtained by erasing the loops from a planar Brownian motion. The problem with this approach is that planar Brownian motion has loops at every scale, so that there is no simple algorithm to erase loops. In particular, there is no "first" loop. Our proof does use the relation between the LERW and simple random walks, combined with the fact that quantities related to simple random walks, such as hitting probabilities, converge to their continuous conformally invariant counterparts. 938
CONFORMAL INVARIANCE
947
The proof of each of our main theorems is naturally divided into two parts. The first part establishes the convergence to SLE with respect to a weaker topology than the topology induced by the metric p of paths; namely, we show that the Loewner driving process for the discrete random path converges to a Brownian motion. This part of the proof, which we consider to be the more important one, is essentially self-contained. The second part uses some regularity properties of the discrete processes from [38] to prove convergence with respect to the stronger topology. The method for the first part can be considered as a rather general method for identifying the scaling limit of a dependent system that is conjectured to be conformally invariant. It requires having some "observable" quantity that can be estimated well and a mild Markovian property, which we now describe. Suppose that to every simply connected domain D containing 0 there is associated a random path y from aD to 0 (e.g., the orientation reversal of the LERW). The required property is that if f3 is an arc with one endpoint in aD and we condition on f3 C Y (assuming this has positive probability, say), then the conditioned distribution of y \ f3 is the same as the random path in the domain D \ f3 conditioned to start at the other endpoint q of f3. [Thus, (D \ f3, q) is the state of a Markov chain whose transitions correspond to adding edges from y to f3 and modifying q appropriately.] Interestingly, among the discrete processes conjectured to converge to SLE, the LERW is the only one where the verification of this property is not completely triviaL [For the LERW it is not trivial, but not difficult; see part (iii) of Lemma 3.2.] The statement of this property for the UST Peano curve is given in Lemma 4.1. The fact that SLE satisfies this property follows from the Markovian property of its driving Brownian motion. The particular choice of observable is not so important. What is essential is that one can conveniently calculate the asymptotics of the observable for appropriate large-scale configurations. The particular observable that we have chosen for the LERW convergence is the expected number of visits to a vertex v by the simple random walk generating the LERW. Conformal invariance is not assumed but comes out of the calculation-hitting probabilities for random walks are discrete harmonic functions, which converge to continuous harmonic functions. One technical issue is to establish this convergence without any boundary smoothness assumption. Once the observable has been approximated, the conditional expectation and variance of increments of the Loewner driving function for the discrete process can be estimated, and standard techniques (the Skorohod embedding) can be used to show that this random function approaches the appropriate Brownian motion. Although Theorem 1.3 can probably be derived with some work from Corollary 1.2, instead, to illustrate our method, we prove it by applying again the same general strategy of the proof of Theorem 1.1, with the choice of a different observable. 939
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
Actually, it is easier to explain the main ideas behind the proof of Theorem 1.3. Fix some vertex v in D and a subarc al Ca. Let.A, be the event that the UST path (not the Peano path, but the path contained in the UST) from v to a hits al. By Wilson's algorithm the probability of.A, is the same as the probability that a simple random walk started at v reflected off fJ first hits a in al. The latter probability can be estimated directly. If y [0, n] denotes the restriction of the Peano path to its first n steps, then P[.A, Iy [0, n]], the probability of .A, conditioned on y [0, n], is clearly a martingale with respect to n. But, by the Markovian property discussed above, the value ofP[.A,ly[O, n]] may be estimated in precisely the same way that P[.A,] is estimated. The estimate turns out to be a function of the conformal geometry of the configuration (v, D \ y[O, n], yen), aI, fJ). Knowing that this is a martingale for two appropriately chosen vertices v is sufficient to characterize the large-scale behavior of y. As mentioned above, in the case of the LERW, the observable we chose to look at is the expected number of visits to a fixed vertex v by the simple random walk r generating the LERW y. The walk r can be considered as the union of y with a sequence ofloops r j based at vertices of y. We look at the conditioned expectation of the number of visits of r to v given an arc y of y adjacent to the boundary of the domain. This is clearly a martingale with respect to the filtration obtained by taking larger and larger arcs y C y. This quantity falls into two parts: the visits to v in the loops r j based at y, and those that are not. Each of these two parts can be well estimated by random walk calculations. Translating the fact that this is a martingale to information about the Loewner driving process for y inevitably leads to the identification of this driving process as appropriately scaled Brownian motion. Actually, we first had a longer proof of convergence of the LERW to the SLE2, based on the fact that it is possible to construct the hull of a Brownian motion by adding Brownian loops to SLE2. This can be viewed as a particular case of the restriction properties of SLEK with Brownian loops added, which we study in [31]. Let us also mention the following related open question. Consider a sequence of simple random walks Sk (n) on a lattice with lattice spacing Ok -+ 0, from Sk(O) = to au, and let yk denote the corresponding loop-erased paths. Theorem 1.1 shows that one can find a subsequence such that the law of the pair (yk, Sk) converges to a coupling of SLE2 with Brownian motion [i.e., a law for a pair (X, Y), where X has the same distribution as the SLE2 path and Y has the same distribution as Brownian motion]. The question is whether, in this coupling, SLE2 is a deterministic function of the Brownian motion. In other words, is it possible to show that this is not a deterministic procedure to erase loops from a Brownian motion?
°
2. Preliminaries. The reading of this paper requires some background knowledge in several different fields. Some background about Loewner's equation and SLE is reviewed in the next section. It is assumed that the reader is familiar 940
949
CONFORMAL INVARIANCE
with some of the basic properties of Brownian motion (definition, strong Markov property, etc.). Some of the basic properties of conformal maps (Riemann's mapping theorem, compactness, Koebe distortion) are also needed for the proof. This material may be learned from the first two chapters of [35], for example. In terms of the theory of conformal mappings, this suffices for understanding the argument showing that the driving process of the LERW converges to Brownian motion. For improving the topology of convergence, some familiarity with the notion of extremal length (a.k.a. extremal distance) is also required. A possible source for that is [2]. The reader also needs to know some of the very basic properties of harmonic measure. 2.1. Loewner's equation and SLE. We now review some facts concerning Loewner's equations and stochastic Loewner evolutions. For more details, see, for example, [27], [28], [36] and [38]. Suppose that D ~ C is a simply connected domain with ED. Then there is a unique conformal homeomorphism 1f; = 1f;D : D ---+ U which is onto the unit disk U = {z E C: Izl < I} such that 1f;D(O) = and 1f;b(O) is a positive real. If DC U, then 1f;b (0) ::: 1, and log 1f;b (0) is called the capacity of U \ D from 0. Now suppose that 1] : [0, 00] ---+ U is a continuous simple curve in the unit disk with 1](0) E au, 1](00) = and 1](0,00] cU. For each t ::: 0, set K t := 1][0, t], Ut := U \ K t and gt := 1f;ut • Since t f--+ g; (0) is increasing (by the Schwarz lemma, say), one can reparameterize the path in such a way that g;(O) = exp(t). If that is the case, we say that 1] is parameterized by capacity from 0. By standard properties of conformal maps (see [35], Proposition 2.5), for each t E [0, (0) the limit
°
°
°
W(t):= lim gt(z), Z-H/(t)
where
z tends to 1](t) from within U \ 1][0, t], exists. One can also verify that W: [0, (0) ---+
au
is continuous. Assuming the parameterization by capacity, Loewner's theorem states that gt satisfies the differential equation (2.1)
a
__
tgt(z) -
gt(z) + Wet) gt(z) gt(z) _ W(t)'
It is also clear that
(2.2)
Vz EU,
go(z)
= z.
We call (W(t), t::: 0) the driving function of the curve 1]. The driving function W is sufficient to recover the two-dimensional path 1], because the procedure may be reversed, as follows. Suppose that W: [0, (0) ---+ au is continuous. Then for every z E U there is a solution gt(z) of the ODE (2.1) with initial value go(z) = z up to some time T(Z) E (0,00], beyond which the solution 941
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
does not exist. In fact, if r(z) < 00 and z i= W(O), then we have limttr(z) gt(z) W (t) = 0, since this is the only possible reason the ODE cannot be solved beyond time r(z). Then one defines K t := {z E U: r(z) :s t} and D t := U \ K t is the domain of definition of gt. The set K t is called the hull at time t. If W arises from a simple path rJ as described in the previous section, then we can recover rJ from W by using rJ(t) = g;l(W(t)). However, if W: [0, 00) --+ U is an arbitrary continuous driving function, then, in general, K t need not be a path, and even if it is a path, it does not have to be a simple path. Radial SLEK is the process (Kt, t ~ 0), where the driving function wet) is set to be Wet) := exp(iBKt ), where B: [0, 00) --+ lR is Brownian motion. Often, one takes the starting point Bo to be random uniform in [0, 2n]. It has been shown in [36] that the hull K t is a.s. a simple curve for every t > if K :s 4 and that, a.s. for every t > 0, K t is not a simple curve if K > 4. For every K ~ 0, there is a.s. some random continuous path rJ: [0, 00) --+ U such that, for all t > 0, D t is the component of U \ rJ[O, t] containing 0. When K i= 8, this was proved in [36], while for K = 8 this will be proven in the current paper. This path is called the radial SLE path. Suppose that D is a simply connected domain containing 0. If Y is a continuous simple curve joining aD to with only an endpoint in aD, one can reparameterize the path rJ := 0/ 0 y according to capacity and find its driving function W, as before. The conformal map
°
°
gt =
o/D\y[O,tl: D \ y[O, t] --+ U
still satisfies (2.1), but this time, gO = 0/ D. (Here, the parameterization chosen for y is according to the capacity of 0/ 0 y[O, t].) Radial SLE in D is then simply the image under 0/D1 of radial SLE in the unit disk. Similarly, one can encode continuous simple curves rJ from to 00 in the closed upper half plane lHI via a variant of Loewner's equation. For each time t ~ 0, there is a unique conformal map gt from H t := lHI\ rJ[O, t] onto lHI satisfying the so-called
°
hydrodynamic normalization
(2.3)
lim gt(z) - z = 0, z---+oo
where z --+ 00 in lHI. If we write gt(z) = z + a(t)z-l + o(z-l) near 00, it turns out that aCt) is monotone. Consequently, one can reparameterize rJ in such a way that aCt) = 2t, that is, gt(z) = z + 2tz- 1 + o(z-l) when z --+ 00. This parameterization of rJ is called the parameterization by capacity from 00. (This notion of capacity is analogous to the notion of capacity in the radial setting; however, these are two distinct notions and should not be confused.) If g : lHI \ K --+ lHI is the conformal homeomorphism satisfying the hydrodynamic normalization, then limz---+ oo (gt (z) - z) z/2 is called the capacity of K from 00. Assuming that rJ is parameterized by capacity, the following analog of Loewner's 942
CONFORMAL INVARIANCE
951
equation holds: (2.4)
"It> 0, V Z E H t ,
2 atgt(z) = gt(z) - W(t)'
where the driving function W is again defined by Wet) := gt(l1(t». As above, 11 is determined by W. Conversely, suppose that W is a real-valued continuous function. For z E 1HI, one can solve the differential equation (2.4) starting with ga(z) = z, up to the first time T(Z) where gt(z) and W(z) collide [possibly, T(Z) = 00]. Let the hull be defined by K t := {z E 1HI: T(Z) ::s t}. Then gt : 1HI \ K t ~ 1HI is a conformal map onto 1HI, and ga(z) = z. In general, K t is not necessarily a simple curve. If Wet) = BKt , then (Kt, t :::: 0) is called chordal SLEK • It turns out (see [28], Section 4.1) that the local properties of chordal SLEK and of radial SLEK are essentially the same. [That is the reason the normalization aCt) = 2t was chosen over the seemingly more natural aCt) = t.] In particular, for every K, chordal SLEK is generated by a random continuous path, called the chordal SLEK path. At some points in our proofs, we will need the following simple observation. LEMMA 2.1 (Diameter bounds on Kt). There is a constant C > Osuch thal the following always holds. Let W: [0, (0) ~ lR. be continuous and let (Kt, t:::: 0) be the corresponding hull for Loewner's chordal equation (2.4) with driving function W. Set k(t) :=
v't + max{IW(s) -
W(O)I: s E [0,
tn.
Then
"It:::: 0,
C-1k(t)::s diamKt::S C k(t).
Similarly, when K t C U is the radial hull for a continuous driving function W: [0, (0) ~ au, then "It ::::0,
c- 1 min{k(t), 1}::s diamKt ::s C k(t).
PROOF. This lemma can be derived by various means. We will only give a detailed argument in the radial case. The chordal case is actually easier and can be derived using the same methods. It can also be seen as a consequence of the result in the radial setting (because chordal Loewner equations can be interpreted as scaling limits of radial Loewner equations). We start by proving the upper bound on diamKt . Let 8 :::: max{IW(s) W(O)I: s E [0, tn. Then, as long as Igt(z) - W(O)I :::: 38, we have latgt(z)1 ::s 1/8. Hence, if Iz - W (0) I :::: 48, then, for all t ::s 82 , Igt (z) - zl ::s 8 and therefore z ¢. K t . Hence, diam K t ::s 8k(t). 943
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O. F. LAWLER, O. SCHRAMM AND W. WERNER
In order to derive the lower bound, we will compare capacity with harmonic measure. It is sufficient to consider the case where diamKt < 1/10. Let fJ., denote the harmonic measure on K t U au from O. Because K t is contained in the disk of radius diam K t with center W (0) E K t n au, there is a universal constant c such that fJ.,(K t ) ~ cdiamKt • Hence, it suffices to give a lower bound for fJ.,(K t ). Since gt (z) / z is analytic and nonzero in a neighborhood of 0, the function h(z) = log Igt(z)1 - log Izl is harmonic in Ut := U \ K t . Note that h(O) = t. Because Igt(z)1 --+ 1 as z tends to the boundary of Ut , the mean value property of h 0 gt- l implies the following relation between harmonic measure and capacity: t = h(O) = J10g(I/lzl) dfJ.,(z). Since K t contains points in au and diamKt ~ 1/10, we have 10g(1/lzl) ~ c'diamK t for all z E K t . Therefore, t ~ c' fJ.,(K t ) diamKt ~ c"(diamKt)2. It now remains to compare fJ.,(K t ) and IW(t) - W(O)I. We still assume that diamKt < 1/10. Let At := au \ gt(U \ Kt). If z E au \ Ks and s ~ t, then (2.1) shows that au Igs+u (z) - W (s) I :::: 0 at u = O. This implies that (As, s ~ t) is nondecreasing. Hence, for all s ~ t, we have W(O) E Ao C At and W(s) E As C At so that IW(s) - W(O)I is bounded by the length of At, which is equal to 2n fJ.,(K t ). This completes the proof of the lemma. D 2.2. A discrete harmonic measure estimate. In this section, we introduce some notation and state an estimate relating discrete harmonic measure and continuous harmonic measure in domains in the plane. In order to get more quickly to the core of our method in Section 3.2, we postpone the proof of the harmonic measure estimate to Section 5. A grid domain D is a domain whose boundary consists of edges of the grid Z2. For an arbitrary domain Dee, and p E D define the inner radius of D with respect to p,
radp(D) := inf{lz -
pi : z ¢. D}.
Let ~ denote the set of all simply connected grid domains such that 0< rado(D) < 00 (i.e., D f. C and 0 ED). Points in ]R2 = C with integer coordinates will be called vertices, or lattice points. Let V(D) := D n Z2 denote the lattice points in D. Let D E ~ and let v be a vertex in aD. If a D contains more than one edge incident with v, then it may happen that the intersection of D with a small disk centered at v will not be connected. Hence, as viewed from D, v appears as more than one vertex. In particular, 1/1 = 1/ID does not extend continuously to v. This is a standard issue in conformal mapping theory, which is often resolved by introducing the notion of prime ends. But in the present case, there is a simpler solution which suffices for our purposes. Suppose v E Z2 n aD and e is an edge incident with v that intersects D. The set of such pairs w = (v, e) will be denoted Va(D). If 1/1: D --+ U is conformal, then 1/I(w) will be shorthand for the 944
CONFORMAL INVARIANCE
953
limit of 1/f(z) as z --+ v along e (which always exists, by [35], Proposition 2.14). Similarly, if a random walk first exits D at v, we say that it exited D at w if the edge e was used when first hitting v. A reader of this paper who chooses to be sloppy and not distinguish between v and w will not lose anything in the way of substance. We will not always be so careful to make this distinction. If a E V(D) and b E V(D) U Va(D), define H(a, b) = HD(a, b) as the probability that the simple random walk started from a and stopped at its first exit time of D visits b. For any wED and u E Va(D), we define (2.5)
A = A(W, u; D):=
1 -11/f(w)1 2 = Re(1/f(U) + 1/f(W)). 11/f(w) - 1/f(u) 12 1/f(u) - 1/f(w)
Note that A is also equal to the imaginary part of the image of w by the conformal map from D onto the upper half plane that maps 0 onto i and u to 00. It is also the limit when E --+ 0 of the ratio between the harmonic measure in D of the E neighborhood of u in aD, taken, respectively, at w and at 0 (i.e., it corresponds to the Poisson kernel). Therefore, A can be viewed as the continuous analog of H(w, u)/ H(O, u). Note that the function hew) = H(w, u)/ H(O, u) is discrete harmonic on V(D), which means that hew) is equal to the average of h on the neighbors of w when w E V(D). PROPOSITION 2.2 (Hitting probability). For every E > 0 there is some ro > 0 such that the following holds. Let DE 1) satisfy rado(D) > ro and let u E Va(D) and WE V(D). Suppose I1/fD(W)1 ::::; 1 - E and H(O, u) i= O. Then (2.6)
H(W,U)
I----A(w,u;D) H(O, u)
I <E.
The proof is given in Section 5. 3. Conformal invariance of the LERW. 3.1. Loop-erased random walk background. We now recall some well-known facts concerning loop-erased random walks.
LEMMA 3.1 (LERW reversal). Let DE 1) and let I' be a simple random walk from 0 stopped when it hits aD. Let f3 be the loop erasure of I' and let y be the loop erasure of the time reversal of I' . Then y has the same distribution as the time reversal of f3. See [25]. A simpler proof follows immediately from the symmetry of (12.2.3) in [26]. This result (and the proofs) also holds if we condition I' to exit aD at a prescribed u E Va(D), which corresponds to the event {y n aD = {uH = {f3 n aD = {u}} (assuming this has positive probability). 945
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
Throughout our proof, we will use the simple random walk r and the loop erasure Y = (Yo, Y1, ... , re) of its time reversal (so that Yo E aD and Ye = 0). We use Dj to denote the grid domains Dj := D \ U{~J[Yi' Yi+ll. Define, for j E {O, 1, ... ,e}, nj := min{n 2: 0: r(n) = Yj}
and note that n j+ 1 < n j for j = 0, 1, ... , e - 1 by the definition of y. Also set
r J·+1 :=r[nj+1,nj]. More precisely, consider r
j
as the grid path given by
rj(m):= rem +nj),
m =0, 1, ... ,nj-1-nj.
LEMMA 3.2 (Markovian property). Let j E N and let uo, ... , u j E 71.. 2 . Suppose that the probability of the event (Yo, ... , Yj) = (uo, ... , u j) is positive. Conditioned on this event, the following hold: (i) The paths r 1, ... , r j and r [0, n j] are conditionally independent. (ii) For k E {1, ... , j}, the conditional law of rk is that of a simple random walk in Dk-1 started from Uk and conditioned to leave Dk-1 through the edge [Uk, uk-ll. (iii) The conditional law of r [0, n j] is that of a simple random walk started from conditioned to leave D j at Uj, and Y [j, l] is the loop erasure of the time reversal ofr[O, n j].
°
PROOF. Since Y is the loop erasure of the reversal of r, the event (Yo, ... , Yj) = (uo, ... , Uj) is equivalent to the statement that, for each k = 0, 1, ... , j -1, the first hit ofr to {uo, ... , uk}UaD is through the edge [Uk+1, Uk]. Let Tk := min{n: r(n) E {Uo, ... , Uk} u aD}, k = 0, ... , j. The strong Markov property of r with the stopping times Tk now implies the lemma. D
The following simple lemma will also be needed. LEMMA 3.3 (Expected visits). Suppose that v E V(D) and that Uo and U1 are two vertices satisfying P[yO = Uo, Y1 = ull > 0. Conditioned on Yo = Uo and Y1 = U1, the expected number of visits to v by r1 is G(U1, v)H(v, U1). Here G (u, v) denotes the discrete Green's function, that is, the expected number of visits to v by a simple random walk started at u, which is stopped on exiting D. PROOF OF LEMMA 3.3. Let X be a simple random walk from U1 stopped on exiting D and let k be the last time such that X(k) = U1. Then r 1 conditioned on Yo = Uo and Y1 = U1 has the same distribution as X conditioned on X(k + 1) = Uo. But the path j t--+ X(k + j) is independent from X[O, k]. Consequently, the expected number of visits of X to v conditioned on X(k + 1) = Uo is equal to the expected number of visits to v of X[O, k]. The lemma follows. D 946
CONFORMALINVARlANCE
955
3.2. The core argument. We keep the previous notation and also use the conformal maps 1/tj : Dj ~ U satisfying 1/tj(O) = 0 and 1/tj(O) > o. Set Uj := 1/tj(Yj) and U := Uo. Note that Y can also be viewed as a continuously growing simple curve from aD to 0 and therefore can be represented by Loewner's equation. Let W: [0, (0) ~ au denote the (unique) continuous function such that solving the radial Loewner equation with driving function W (t) gives the path y. Note that Uj = W(tj), where tj is the continuous capacity of y[O, j] from 0 in D (i.e., the capacity of 1/t(y[O, j]) from 0 in U). We denote by (o-(t), t ~ 0) the continuous real-valued function with 0-(0) = 0 such that Wet) = W(O) exp(io-(t». We also define ~ j = o-(tj), so that Uj = U exp(i ~ j). PROPOSITION 3.4 (The key estimate). There exists a positive constant C such that,for all small positive 8, there exists ro = ro(8) such that the following holds. Let D E ~ satisfy rado(D) > roo For every uo E Va (D) with P[yO = uo] > 0, let y denote the random path from Uo to 0 obtained by loop erasure of the time reversal of a simple random walk from 0 to aD conditioned to hit aD in Uo. Let
m:= minU ~ 1 :tj ~ 82 or I~jl ~ 8}, where
~j
and tj are as described above. Then
(3.1) and (3.2)
Recall that Lemma 3.1 says that y has the same distribution as the chronological loop erasure of a random walk from 0 to aD conditioned to hit aD at uo. Here is a rough sketch of the proof. Let v E V(D) satisfy (3.3)
ht
rado(D)/200 < Ivl < rado(D)/5.
denote the number of visits to v by r. (This is the quantity which we Let referred to in Section 1 as the "observable.") The proof is based on estimating the two sides of the equality (3.4) The estimate for the right-hand side will involve the distribution of tm and ~m. We get the two relations (3.1) and (3.2) by considering two different choices for such a v. The estimates for the two sides of (3.4) are rather straightforward. Basically, each side is translated into expressions involving the Green's functions G j and the hitting probabilities Hj. These are then translated into analytic quantities using (2.6). Earlier versions of the proof required other estimates, somewhat more delicate, in addition to (2.6). Fortunately, it turned out that (2.6) is sufficient. Since 947
956
G. F. LAWLER, O. SCHRAMM AND W. WERNER
we came across several different variants for the proof, based on choosing different observables, it may be said that the proof is inevitable, rather than accidental (and this also applies to Theorem 1.3). Basically, the reason the proof works is that the expected number of visits to v in Uj=1 [' j given Y[0, m] can be estimated rather well given the rough geometry of y[O, m] in a scale much coarser than the scale of the grid. Similarly, it is important that E[ht] can be estimated given the rough geometry of D, but this fact is not surprising. In the following, we abbreviate the Green's functions and hitting probabilities in Dj by Gj := GDj and Hj := HDr The following lemma will be needed. LEMMA 3.5 (Green's function bounds). There is a constant C > 0 such that, for every D E 1) and v E V(D) satisfying (3.3), (3.5)
l/C:s GD(O, v):s C
holds. Also, given 8> 0, there is an r the notation of Proposition 3.4,
= r(8)
such that, ifrado(D) > r, then, with
(3.6)
PROOF OF PROPOSITION 3.4. Since tm - l < 82 , it follows from the Koebe 1/4 theorem that rado(Dm-l) > rado(D)/5 :::: ro/5 if 8 is small. (Apply [35], Corollary 1.4, with z = 0 to 1/10 1 and 1/I;;;~1') Moreover, the continuous harmonic measure in D at 0 of any edge e with a vertex on aD can be made arbitrarily small by requiring rado(D) to be large. [A Brownian motion started at 0 has probability going to 1 to surround the disk rado(D)U before hitting e, as rado(D) --+ 00.] By the conformal invariance of the harmonic measure, this implies that the diameter of 1/1 (e) can be made arbitrarily small. Applying this to the domains Dj and using Lemma 2.1, we see that we may take ro large enough so that, for all j < m, for all t E [tj, tj+l], Il?-(t) - l?- (tj) I :s 83 and Itj+l - tj I :s 83 . In particular, tm :s 82 + 83 and I~m I :s 8 + 83 . We also require ro/8 to be larger than the r(8) of Lemma 3.5. Suppose v E V(D) satisfies (3.3). Set Zj := 1/Ij(v) and Z := Zoo For each j E {I, ... , R.}, let h j denote the number of visits to v by [' j . Also let e
hj:=
L
hk,
k=j+l
which is the number of visits of v by [,[O,nj]. Let J....j := J....(v, Yj; Dj), where J....( v, v'; D j) is as in (2.5). Since, conditionally on Y [0, [,[0, n j] is a random walk in Dj conditioned to leave Dj at Yj,
n,
E[h -+: I [0 .]] 1 Y ,J
=
G ·(0 v)H ·(v y.) l' l' 1 H'(O') 1
948
'Yl
957
CONFORMAL INVARIANCE
and Proposition 2.2 [together with (3.5)] implies that if ro(8) is sufficiently large, then, for every j E {a, 1, ... , m},
j]] = Gj(O, V)Aj + 0(8 3 ).
E[hjIY[O,
[This 0 (.) notation is shorthand for the statement that there is an absolute constant C such that IE[hjIY[O, j]] - Gj(O, v)Ajl :s C8 3 . We freely use this shorthand below.] In particular, (3.7)
E
[~ hj ] ~ E[ht -
h!]
~ E[ColO, v)Ao -
C m (0, v)Am] + 0(83).
We will now get a different approximation for the left-hand side. Applying Lemma 3.3 to the domain Dj-1 gives E[hjIY[O,
j]] = Gj-1(Yj, V)Hj-1(V, Yj).
Proposition 2.2 implies that, for ro(8) large enough, (3.8)
E[hjIY[O,
j]] = (Aj-1 + 0(8»)G j -1(Yj, V)Hj-1(0, Yj).
Considering the same simple random walk starting at Dj or Dj-1 shows that (3.9)
Gj-1(0, v) - Gj(O, v)
But IUj -
and stopped when it exits
= Hj-1(0, Yj)Gj-1(Yj, v).
We now derive an a priori bound on max {IA j (3.10)
°
-
Am 1: j
:s m}. Recall that
Aj-Ao=Re(Uj+Zj _ U+Z). Uj - Zj U - Z
UI :s
0(8) for j
v j:S m,
:s m, and Loewner's equation shows that Zj
U+Z _ Z +tjO(8)
= Z +tjZ U
(3.11)
= Z + tjZ
U+Z U-Z
+ 0(8 3 ),
and, in particular, Zj = Z + 0(8 2 ). (The equation blows up when IU - ZI is small, and such estimates would not be valid in such a situation. However, this is not a problem here. First, 0/0(0) :s l/rado(D) by the Schwarz lemma applied to the restriction of % to rado(D)1IJ. Now the Koebe 1/4 theorem (the case z = in the left-hand inequality in [35], Corollary 1.4) gives %l((4/5)U) ~ (1/4)10/0(0)1- 1 (4/5)U ~ (rado(D)/5)U. In particular, IZI = 100o(v)1 :s 4/5 by (3.3). Since tm = 0(8 2 ), it is clear that if 8 is small and one starts flowing from Z according to Loewner's equation, it is impossible for Z to get close to au up to time tm .) Thus, we get our bound,
°
v j:s m, 949
958
G. F. LAWLER, O. SCHRAMM AND W. WERNER
Using (3.8), this implies E[hjIY[O, j]] =
(Jlom + 0(8))G j -I(Yj, V)Hj-I(O, Yj).
Now applying (3.9) yields
E[~h j] = E[(Am + o (S))(Go(O, v) -
Gm(O, v))],
and hence (3.6) implies
E[~ h j ]
= E[Am(GO(O, v) - Gm(O, v))]
Comparing with (3.7) gives Go(O, v)E[)'.. m - AO] E[A m - AO]
(3.12)
+ O(S3).
= 0(8 3 ), and hence (3.5) implies
= 0(8 3).
[The reader may wonder about the apparent miracle happening here, namely, that Aj turns out to be "almost" a martingale. In fact, this is not important for identifying the scaling limit. If the right-hand side in (3.12) turned out to be any other explicit quantity, up to 83 error terms, the proof would still work, but give a different limiting process. In Remark 3.6 below, we give a short proof of (3.12) and further comments.] Recall that this equation is valid uniformly over all choices of v. We now Taylorexpand Am - AO with respect to Um - U and Zm - Z, up to 0(8 3) error terms. As we have seen, Um - U = 0(8) and Zm - Z = 0(8 2), and hence only the firstorder derivative with respect to Zm - Z and the first two derivatives with respect to Um - U
= (e iLlm
-
I)U
= iU ~m
- U ~~/2 + 0(8 3)
come into play (the mixed derivatives can be ignored). Using (3.10) and (3.11), we get
Am-Ao=~mIm(
2ZU 2)+(2tm-~~)Re(ZU(U+;))+0(83), (U - Z) (U - Z)
and therefore (3.12) gives (3.13)
Im(
2ZU 2)E[~m] (U-Z)
+ Re(ZU(U + ;))E[2tm - ~~] = 0(8 3). (U-Z)
We claim that when ro(8) is large enough we may find VI, V2 E V(D) in the range (3.3) satisfying Il/I(VI) - U /301 < 83 and Il/I(V2) - i U /301 < 83 . Indeed, by Theorem 1.3 and Corollary 1.4 from [35], for every R E (0, 1), there is a C = C(R) < 00 such that Il/I'(z) 1 :::s C/radz(D) and radz(D) ~ C- l rado(D) hold for all z E l/I-I(RU). Let VI be a vertex closest to l/I-I(U/30). By integrating 950
CONFORMAL INVARIANCE
959
the above bound on 1/1,/ along the line segment from 1/I-l(U /30) to VI (whose length is less than 1), we get 11/r(Vl) - U /301 < 83 if rado(D) is large enough. Another application of Theorem l.3 and Corollary 1.4 from [35] now shows that VI satisfies (3.3). An entirely similar argument produces V2. Consequently, (3.13) holds with Z E {U /30, i U /30}. Plugging in these two values for Z produces two linearly independent equations in the variables E[2tm - ..6.~] and E[..6. m ] and thereby proves (3.1) and (3.2). D REMARK 3.6. Here is another proof of (3.12). Given a vertex V E V(D), let f3 = (f3o, f31, ... ) denote the loop erasure of the reversal of the simple random walk r v started from V and stopped on exiting D (i.e., the analog of y, but starting from v instead of 0). Abbreviate yn := (Yo, ... ,Yn) and similarly f3n := (f3o, ... , f3n). For a sequence of vertices u = (uo, Ul, ... , un), let an(u):= p[yn = u] and bn(u):= p[f3n = u]. Set Mn := bn(yn)/an(yn). (In other words, Mn is the Radon-Nikodym derivative of the law of f3n with respect to the law of yn.) It is easy to verify that Mn is a martingale: bn+l (ynw) an+l (ynw) _ Lw bn+l (ynw) - M E [Mn+l IYn] -_ " ~ n· n w
an+l(y W)
an(yn)
an(yn)
Lemma 3.2 implies that Mn = Hn(v, Yn)/ Hn(O, Yn), since, on the event that r v and r first hit {uo, ... , un} u aD at Un, we may couple them to agree after that first visit to Un. Now (2.6) implies (3.12). Although this proof is shorter than the first proof of (3.12), it is harder to motivate and less natural. For this reason, we chose to stress the first proof. Let us finally note that (as opposed to the martingale that shows up in the analysis of the UST Peano curve), the quantity corresponding to this martingale in the scaling limit is unbounded and converges almost surely to (it is not uniformly integrable), so that it cannot be interpreted as a conditional probability or a conditional expectation. Correspondingly, in the discrete setting, Mn is very large when the path hits v (if it does) and Mn is very small when the path hits 0.
°
3.3. Recognizing the driving process. The objective in this section is to show that W of the previous section is close to a time-scaled Brownian motion on the unit circle.
°
THEOREM 3.7 (Driving process convergence). For every T > and E > 0, there is an rl = rl(E, T) > Osuch that,forall D E ~ with rado(D) > rl, there is a coupling of y with Brownian motion B (t) starting at a random uniform point in [0, 2n] such that P[sup{W(t) - B(2t)l:t 951
E [0,
Tn > E] < E.
960
G. F. LAWLER, O. SCHRAMM AND W. WERNER
Recall that a coupling of two random variables (or random processes) A and B is a probability space with two random variables A' and B', where A' has the same distribution as A and B' has the same distribution as B. In the above statement (as is customary), we do not distinguish between A and A' and between B and B'. In order to deduce this theorem from Proposition 3.4, we will use the Skorohod embedding theorem, which is one of the standard tools for proving convergence to Brownian motion (one could work out a more direct proof but the following proof seems cleaner). LEMMA 3.8 (Skorohod embedding). If (Mn)n~N is an (:Fn)n~N martingale, with liMn - Mn-Illoo S 28 and Mo = 0 a.s., then there are stopping times o = TO S TI S ... S TN for standard Brownian motion (Bt , t :::: 0) such that (Mo, MI, ... , MN) and (Bro, Brl' ... , B rN ) have the same law. Moreover, one can impose, for n = 0,1, ... , N -1, (3.14) and
(3.15) The proof can be found in many probability textbooks, including [9] and [37]. Often, it is stated for just one random variable MI; for a statement in terms of martingales, see, for instance, [8] and [43]. The relation (3.15) is not stated explicitly in these references (since the assumption that the increments of Mn are bounded is weakened), but is a consequence of the proof. It can also be derived a posteriori from E[Tn+I - Tn] = E[(Mn+I - Mn)2] < 00, since the expected time for Brownian motion started outside an interval to hit the interval is infinite. PROOF OF THEOREM 3.7. Since the hitting measure of a simple random walk from 0 is close to the hitting measure for Brownian motion when rado(D) is large (see, e.g., Section 5), it is clear that W(O) is nearly uniform in au. It is therefore enough to show that iJ (t /2) is close to standard Brownian motion. Assume, without loss of generality, that T :::: 1. Pick 8 = 8 (E, T) > 0 smalL Let ro be as in Proposition 3.4 and take rt := 8 exp(20T)ro. Let yt denote the initial segment of y such that 1/ID(yt) has capacity t from O. By the Schwarz lemma, 1/Ib(O) S rado(D)-I. Therefore, the Koebe 1/4 theorem implies rado(D \ yt) :::: exp( -t) rado(D)/4. Hence, if rado(D) :::: rI, Proposition 3.4 is valid not only for the initial domain D, but also for the domain D slitted by subarcs of y, up to capacity 20T. As in Proposition 3.4, define m to be the first j = 1, 2, ... such that I~ j I :::: 8 or t j :::: 82 . Set mo := 0, m 1 := m and inductively let mn+ 1 be the first j :::: mn + 1 such that I~ j - ~mn I :::: 8 or tj - tmn :::: 82 , whichever happens first. Let :Fn denote the a-field generated by y[O, m n ]. Set N := rIOT 8- 21. 952
CONFORMAL INVARIANCE
961
Our choice of r1 ensures that tj+1 - tj ::: 28 2 for all j < N and that tN ::: 20T. Hence, Proposition 3.4 holds for all domains Dmn with n < N. Applying clause (iii) of Lemma 3.2 therefore gives (3.16) and (3.17) For n ::: N, set (3.18)
n-1 Mn := L(D.mj+! - D.mj -
E[ D.mj+!
j=O
- D.mj Irj D·
Clearly, Mo, ... , MN is a martingale for .1"0, ... , rN. The definition of mn and the choice of r1 imply that IIMn+1 - Mn 1100 ::: 28. By Lemma 3.8, we may couple (Mo, ... , MN) with a standard Brownian motion with stopping times TO ::: T1 ::: ... ::: TN such that Brn = Mn and (3.14) hold. Extend the coupling to include y (this clearly can be done). Note that the definition of tmn and (3.15) ensure that, for all n < N,
sup{IBt - Brn!:t
(3.19)
sup{I1J(t) - D. tmn I:t
E
E
[Tn, Tn+1l} ::: 28,
[tmn' tmn +!]} ::: 28
and (3.16) shows that
(3.20)
sup{lD. tmn -Mn l:n:::N}=0(8 3 N)=0(8T).
Hence, as Mn = Brn and B t is a.s. continuous, it remains to relate the capacities tmn with the stopping times Tn and verify that tmN > T with high probability. For this purpose, define
n-1 Yn = L(Mj+1 - Mj)2. j=o
We first show that Yn is close to 2tmn . Let Zn := Yn - 2tmn . By (3.18) and (3.16), we have, for n < N, IMn+1 - Mn - D. tm n+! + D. tm n I = 0(8 3 ). This implies IMn+1 - Mnl = 0(8) and hence also
Yn+1 - Yn = (Mn+1 - Mn) 2
= ( D.tmn+!
- D. tmn )2 + 0(8 4 ).
Consequently, (3.17) gives
E[Zn+1 - Znlrn]::: 0(8 3 ). From the fact that the increments of tmn and those of Yn are bounded by 0(8 2 ), 953
962
G. F. LAWLER, O. SCHRAMM AND W. WERNER
we also have E[(Zn+l - Zn)2IFn] :::: 0(8 4 ). Set Z~ := Zn Zj-lIFj-IJ. Since this is an Fn-martingale, we have E[Z~,2] =
LJ=1 E[Zj L7=1 E[(Zj
-
Zj_l)2] and the above estimates give E[Z~ 2] = O(N 84 ). Assuming N8 3 <
8 1/ 2/2, without loss of generality, and applying Doob's maximal inequality (see [37],11.1.7) for L2-martingales to Z~, we get (3.21) By the definition of the tmn , we have Yn +l - Yn + tmn+l - tmn ~ 82. Summing gives Y N + tmN ~ N 82 ~ 1OT. Therefore, (3.21) implies (3.22)
P[tmN < 2T] = OCT 8).
We now show that with high probability Tn is also close to Yn for every n :::: N. By (3.15), it is clear that E[(Tn+l - Tn)2IB[0, Tn]] = 0(8 4 ) and therefore E[(Tn+1 - Yn+l) - (Tn - Yn))2IB[0, Tn]] = 0(8 4 ).
Also, (3.14) gives E[(Tn+1 - Yn+l) - (Tn - Yn)IB[O, Tn]] = 0.
Doob's inequality therefore implies p[maxlTn - Ynl > 8 1/ 2 ] = 0(T8). n~N
Combining this with (3.21) leads to p[maxlTn - 2tmn I > 8 1/ 2] n~N
= OCT 8).
Since Bt is a.s. continuous, together with (3.22), (3.19) and (3.20), this completes the proof. D
3.4. Convergence with respect to a stronger topology. Theorem 3.7 provides a kind of convergence of the loop-erased random walk to SLE2. As we will see in this section, this kind of convergence suffices, for example, to show that the scaling and the LERW in U limit with respect to the Hausdorff metric of the union of and the SLE2 path. is the union of Let a: [0, 1] --+ C and f3: [0,1] --+ C be two continuous paths. Define
au
au
pea, (3) := inf sup la(t) - f3 CPE tE[O, 1]
0
¢(t)l,
where
CONFORMAL INVARIANCE 1>1,1>2
E
963
<1>. Since pea, fJ) does not depend on the particular parameterization of
a or fJ, the metric p is also defined for paths on intervals other than [0, 1].
To explain our present goal, let us point out that there is a sequence of paths an from 1 to 0 in U such that their Loewner driving functions Wn (t) converge uniformly to the constant 1 but an does not converge to the path aCt) = 1 - t, t E [0,1], in the metric p, although the driving function for a (reparameterized by capacity) is the constant 1. For example, we may take an as the polygonal path through the points aI, bl +in- 2, a2, b2 -in- 2, a3, b3 +in- 2, ... , a Ln/2J, 0, where aj := 1 - n- 1 + (jn)-1 and bj := 1 - j In. THEOREM 3.9 (LERW image in U converges). For any sequence Dn E ;.v with rado(Dn) ---+ 00, if ILn denotes the law of yn := VtDn 0 yn, where yn is the time reversal of the LERW from 0 to aDn , then ILn converges weakly (with respect to the metric p) to the law of the radial SLE2 path started uniformly on the unit circle. The outline of the proof is as follows. We define a suitable family of compact subsets of the space of simple paths from au to 0 in U, which we can use to show that the sequence ILn is tight. (See, e.g., [9] for background on weak convergence and the notion of tightness.) This implies that a subsequence of ILn converges weakly to some probability measure. Theorem 3.9 then shows that the law of SLE2 is the unique possible subsequential limit. In order to prove tightness, we will use properties of the loop-erased random walk proved in [38]. The actual details will require some background in the geometric theory of conformal maps. In particular, some properties of extremal distance (a.k.a. extremal length) will be used. See, for example, [2] for background. The basic ideas that are used in the proof are taken from [3] and [38]. For a simply connected D ~ C containing 0, let Xo(D) denote the space of all simple paths y : [0, 00] ---+ D from aD to 0 in D, which intersect aD only at the starting point. Given a monotone nondecreasing function Y: (0, (0) ---+ (0, 1], let Xy(D) C Xo(D) denote the space of all simple paths y E Xo(D) such that, for every O:s SI < S2, dist(y[O, sIl
u aD, Y[S2, ooD/rado(D) ~ Y(diam(y[sl, s2D/rado(D»).
Note that whether y is scaling invariant.
E
Xy(D) does not depend on the parameterization of y and
LEMMA 3.10 (Compactness). Let Y: (0, (0) ---+ (0, 1] be monotone nondecreasing. Then Xy(U) is compact in the topology of convergence with respect to p. PROOF. We use an idea from [3]. For all n EN, let Zn be a finite collection of points such that the open balls 93(z, 2- n), z E Zn, cover U. Given a setK C U 955
964
G. F.
LAWLER, O. SCHRAMM AND W. WERNER
and a point Z E Zn, let s(K, z, n) denote the diameter of K n 93(z, 2 1- n ) in the metric obtained from the Euclidean metric on the disk 93(z, 21-n) by collapsing the boundary a93(z, 2 1- n) to a single point. [In other words, this metric d(x, y) is defined as d(x, y) = min{lx - yl, dist(x, a93) + dist(y, a93)}, where 93 = 93(z, 2 1- n).] Fix y E XT(U). Given t 2: 0, let s(t)=Sy(t):=
L L nEN ZEZn
s(y[O,t],z,n). 1Zn 1
Clearly, Sy(t) :::: Ln:::O 2 2- n = 8, and s: [0, 00] --+ [0, (0) is continuous and strictly monotone increasing. (To verify that s is strictly monotone increasing, note that, if t2 > t1 2: 0, then there is some n E N such that dist(y (t2), Y[0, tIJ) 2: 2 2- n , and so s(y[O, t2], z, n) 2: s(y[O, tIJ, z, n) + 2-n if z E Zn satisfies y(t2) E 93(z, 2-n ).) Let yes) be y parameterized by s; that is, y = y 0 s-l. Let Sl < S2 and set E:= diam9[sl, S2] > 0. Then dist(y(s2), y[O, sIJ u aU) 2: I(E). By the argument for strict monotonicity given above, this shows that S2 - Sl 2: 2- n/IZn I, where n := min{ kEN: 22- k :::: I (E)}. Therefore, y satisfies an equicontinuity estimate. By the Arzela-Ascoli theorem, it follows that the closure of XT(U) is compact in the p metric. It is also clear that X T (U) is closed. D Our next goal is to use these compact sets to prove tightness, and we start by observing that the diameter is tight.
° °
LEMMA 3.11 (Diameter is tight). There are constants c, C > such that for every D E :D and every r 2: 1 the simple random walk r starting from and stopped on hitting aD satisfies P[diam(r) 2: rrado(D)] :::: Cr- c . Consequently, the same estimate holds for the loop erasure y.
The first statement is an easy well-known fact. Since the complement of D is connected and unbounded, if the random walk makes a loop separating the circle rado(D)aU from the circle (r /2) rado(D) au before hitting the latter circle, then it must hit aD before (r/2)rado(D)aU. Thus, the lemma is easily proved directly and also follows from the convergence of a simple random walk to Brownian motion. A rather precise form of this estimate for the random walk, where c = 1/2, is known as the discrete Beuding theorem (see [25], Theorem 2.5.2).
°
LEMMA 3.12 (Tameness). For every E > 0, there is some monotone nondecreasing I: (0, (0) --+ (0,1] and some ro > such that for every D E:D with rado(D) 2: ro its time-reversed loop-erased walk y = YD satisfies P[y E XT(D)] 2: 1 - E. 956
965
CONFORMAL INVARIANCE
PROOF. The proof is essentially contained in the proof of [38], Theorem 1.1, where it is established that every subsequential scaling limit of LERW is a.s. a simple path. We will not repeat the complete proof from [38] here, but indicate how it may be adapted to yield the statement of the lemma. Let E > O. Clearly, y E Xo(D).1f y fj. Xy(D), then there are O:s Sl < S2 < 00 such that the distance between y[O, sIl u aD and Y[S2,00] is smaller than rado(D)Y(diarny[sl,s2]/rado(D». Let us first deal with the case where the distance between Y[S2, 00] and aD is small. Let 1 be the walk generating the time reversal of y, and let tn be the first time t where the distance from 1 (t) to aD is smaller than 2- n rado(D) and let r = inf{t: let) E aD}. By the Markov property of 1 at time tn and Lemma 3.11,
P[diam [,[tn, r] > 2-n/2rado(D)] :s C2- cn / 2. Consequently, there is an N = N (E) such that with probability 1 - E/2 for every integer n 2: N we have diam['[tn , r] :s 2-n/2rado(D). In this case, if diam y [0, S2] > 2 -n /2 rado (D), where n > N, then y [0, S2] is not contained in [,[tn, r], which implies that Y[S2, 00] C [,[0, tn ] and gives dist(Y[S2, 00], aD) 2: 2-n rado(D). In other words, if Y satisfies (3.23)
yet) < min{t 2, 2- 2N }j4,
then with probability at least 1 - E/2, for every Sl , S2 (3.24)
E
[0, 00],
dist(a D, y [S2, 00]) 2: rado(D) Y (diam y [Sl, S2]/ rado(D»).
We now focus on the case where the distance between y[O, sIl and Y[S2, (0) is small. We shall say that y has a ({3, a)-quasi-Ioop if there are 0 < Sl < S2 < 00 such that Iy(sl) - y(s2)1:s arado(D) but diamy[sl,s2] 2: {3rado(D). Note that if there are 0 < Sl < S2 < 00 such that dist(y[O,sIl, Y[S2, 00]) < arado(D) and diamy[sl,s2] 2: {3rado(D), then y has a ({3,a)-quasi-Ioop. Let .A,({3, a) denote the event that y has a ({3, a)-quasi-Ioop. Assume, for the moment, that, for all n 2: 0, (3.25)
lim P[.A,(2- n , a)] = 0,
0:\,0
uniformly in D. Then we may take a decreasing sequence an ~ 0 such that L~l P[.A,(2- n , an)] < E/2 holds for every D E ~. Then with probability at least 1 - E/2, y has no (2-n, an)-quasi-Ioop for any n = 1,2, .... Assuming that yet) < an holds whenever t:s 2 l - n, n EN, and yet) < al for all t, on this event we also have dist(y[O, sIl, Y[S2, 00]) :s rado(D)Y(diam Y[Sl, S2]/ rado(D») for all 0 < Sl < S2 < 00. If we also assume (3.23), then together with (3.24) we get P[y E Xy(D)] 2: 1 - E, completing the proof of the lemma. Thus, it remains to verify (3.25). 957
966
G. F. LAWLER, O. SCHRAMM AND W. WERNER
Let .A, (ZO, {3, a) denote the event that there are 0 < sl < s2 < 00 such that Iy(sl) - y(s2)1 :s arado(D), y(Sl), y(S2) E $(zo, {3rado(D)/4) and diam(y[sl, S2]) 2: {3rado(D). In particular, this implies that y[Sl, S2] is not contained in the interior of $(zo, {3 rado(D)/2). Assume that 8a < {3. By Lemma 3.11, there is an R = R(E) > 0 such that with probability at least 1 - E/2 we have y[O,oo] C $(0, Rrado(D)). There is a collection {Zl, Z2, ... , Zk} of points such that every disk of radius 2a rado (D) with center in $ (0, R rado (D)) is contained in one of the k balls $(Zj, {3 rado(D)/2), j = 1,2, ... , k, and we may take k < c((R/{3)2 + 1), where c is an absolute constant. On the event y[O, 00] C $(0, Rrado(D)), we have .A,({3, a) C U~=l .A,(Zj, {3, a). Since E > 0 was arbitrary and P[y[O, 00] C $(0, Rrado(D))] 2: 1 - E/2, it is therefore sufficient to show that P[.A,(Zj, {3, a)] ---+ 0 as a ---+ 0 uniformly in D. The proof of this statement is given (with minor changes in the setup) in [38], Theorem 1.1. D Let Xy (D) denote the set of paths y E X y (D) that are contained in the ball of radius r rado(D) about O. Given y E Xo(D), let y*: [0, 00) ---+ U denote the path 1/1D 0 y, parameterized by capacity. LEMMA 3.13 (Tameness invariance). For every monotone nondecreasing Y: (0, 00) ---+ (0, 1] and every r > 1, there is a monotone nondecreasing Y*: (0, 00) ---+ (0, 1] such that,forall D E ~ and y E Xy(D), y* E XY*(U). PROOF. Let D E~, Y E Xy(D) and O:s si < s~ :s 00. Note that there exist Sl and S2 satisfying si :s Sl :s S2 :s s~ such that
diam(y*[sl, S2]) 2: diam(y*[si, s~])/4 and (3.26)
dist(O, y*[Sl' S2]) 2: diam(y*[sl, S2]).
Since (3.27)
dist(y*[O, siJ u au, y*[s~, 00)) 2: dist(y*[O, sll
u au, y*[S2, 00)),
it is sufficient to give a lower bound of the right-hand side of (3.27) in terms of E := diam(y*[sl, S2]). The Schwarz lemma gives 1/I'v(0) :s l/rado(D). Therefore, by the Koebe 1/4 theorem (applied to the restriction of 1/1£/ to EU) and (3.26), dist(O, y[Sl, S2]) > q rado(D), where q = E/4. On the other hand, the harmonic measure in U from 0 of y*[Sl, S2] is at least C2, where C2 = C2(E) > 0, so that the harmonic measure in D from 0 of y[Sl, S2] is at least C2. Hence, (3.28)
diam y [Sl, S2] 2: C3 rado(D),
958
967
CONFORMAL INVARIANCE
Also set 8 := dist(y*[O, siJ
u au, y*[S2, 00]). Since
diam y*[S2, 00] 2: dist(O, y*[Sl' S2]) 2: E, the extremal distance between y*[O, Sl] uau and y*[S2, 00] is at most ¢l (8, E) > 0, where ¢l is some function satisfying ¢l (8, E) ---+ as 8 0. By the conformal invariance of the extremal distance, this implies that the extremal distance between y[O, siJ u aD and Y[S2, 00] is at most ¢l (8, E). Because y is contained in the disk of radius r rado(D) about 0, this implies that
°
dist(y[O, sll
where ¢2 ---+
°
as 8
+
u aD, Y[S2, 00]) :s ¢2(8, E)rrado(D),
+0. Because y E Xy (D), (3.28) and this together imply ¢2(8, E)r 2: I(C3(E»,
which gives a positive lower bound for 8 = dist(y*[O, siJ u au, y*[S2, 00]) in terms of I, r and E = diam(y*[sl, S2]). This completes the proof. D LEMMA 3.14 (Convergence relations). Suppose W n , Ware continuous functions from [0, 00) to au such that W n ---+ W locally uniformly. Let g~, gt be the corresponding solutions to Loewner's radial equation and set f tn = (g~)-l, ft = gtl. Then ft ---+ ft locally uniformly on [0,00) x 1U. If there are continuous curves yn: [0, 00) ---+ U such that, for all t 2: 0, the image of f tn is the component ofO in U \ yn[o, t] and there is a y : [0, 00) ---+ U such that yn ---+ y locally uniformly on [0, 00), then for all t 2: the image of ft is the component of in U \ y[O, t].
°
°
PROOF. Since gt is obtained by flowing along a vector field depending on W, the inverse ft is obtained by flowing along the opposite field, with the time reversed. Hence, the first statement is an immediate consequence of the principle that solutions of ODE depend continuously on the parameters of the ODE. The second statement is an immediate consequence of the Caratheodory kernel theorem (see [35], Theorem 1.8). D PROOF OF THEOREM 3.9. Let Wn denote the Loewner parameter of 9n and let fln denote the law of the pair (9 n , wn). By Theorem 3.7, we know that the law of Wn tends weakly to the law of Brownian motion. Lemmas 3.10-3.13 show that the set of measures {JLn} is tight with respect to the metric p. Consequently, the sequence fln is also tight. Prokhorov's theorem (e.g., [9] and [37]) implies that there is a subsequence such that fln converges weakly along the subsequence. Let fl be any subsequential weak limit and let (9, W) be a sample from fl. The lemmas show that 9 is a.s. a simple path, and Theorem 3.7 shows that W is Brownian motion (with time scaled). By the properties of weak convergence, we may couple the subsequence of pairs (9 n , wn) and (9, W) so that a.s. p(9 n , 9) ---+ and W n ---+ W locally uniformly.
°
959
968
G. F. LAWLER, O. SCHRAMM AND W. WERNER
Recall that the capacity is continuous with respect to the metric p; that is, if f3, f3n : [0, 1] ~ 1U \ {OJ and p(f3n, (3) ~ 0, then the capacity of f3n[O, 1] tends to the capacity of f3[0, 1]. (In fact, it is enough that f3n[O, 1] tends to f3[0, 1] in the Hausdorff metric.) Indeed, this follows immediately from CaratModory's kernel theorem (see [35], Theorem 1.8) and the fact that the local uniform convergence of conformal maps implies the convergence of the derivatives (by Cauchy's formula for the derivative). Since ji is almost surely a simple path, the capacity of ji increases strictly, and one can parameterize the path continuously by its capacity. We also parameterize the paths jin by capacity. The next goal is to show that jin ~ ji locally uniformly on [0, (0). Since p(ji, jin) ~ 0, there are strictly monotone continuous onto maps Sn : [0, (0) ~ [0, (0) so that jin 0 Sn ~ ji locally uniformly. If tn E [0, (0) and tn ~ t E [0, (0), then it follows from the continuity of capacity with respect to p that snCtn) ~ t [because if s is a subsequential limit of snCtn), then the capacity of ji(s) must be t; i.e., s = t]. This implies that Sn converges to the identity map t ~ t, locally uniformly. By the continuity of ji, it follows that ji 0 S;; 1 ~ ji locally uniformly. This gives jin ~ ji locally uniformly. We can now finally apply Lemma 3.14 to show that ji is the SLE2 path. As the law of the limit ji does not depend on the subsequence, the theorem follows. D In the following proof of Theorem 1.1, the main technical point is that we do not make any smoothness assumptions on aD. If aD is a simple closed path, the theorem follows easily from Theorem 3.9, because the suitably normalized conformal maps from 1U to the discrete approximations of D converge uniformly to the conformal map onto D. PROOF OF THEOREM 1.1. Let D8 be the component of 0 in the complement of all the closed square faces of the grid 0Z2 intersecting aD. Let Y8 be the time reversal of the loop-erased random walk from 0 to aD8 and let f3 be the radial SLE2 path in 1U. Let CP8 : 1U ~ D8 be the conformal map satisfying CP8 (0) = 0 and cP~ (0) > 0 and let cP : 1U ~ D be the conformal map satisfying cP (0) = 0, cp' (0) > O. Theorem 3.9 tells us that we may couple f3 with each of the paths Y8 such that p(cpi 1 0 Y8, (3) ~ 0 in probability as 0 O. Moreover, the proof shows that if we use the capacity parameterization for both, then, in probability,
+
sup{lcpi l
0
Y8(t) - f3(t) I : t 2:: O} ~ O.
(There is no problem with convergence in a neighborhood of t = 00, because we know that the weak limit of cpi 1 0 Y8 with respect to p is a simple path tending to 0 as t ~ 00.) The CaratModory kernel theorem (see [35], Theorem 1.8) implies that CP8 ~ cP uniformly on compact subsets of 1U as 0 ~ O. Consequently, the above gives (3.29)
"Ito> 0,
SUP{IY8(t) - cP 960
0
f3(t)1 : t 2:: to}
~
0,
969
CONFORMALINVARlANCE
in probability. Let E > 0 be small. Then, by Lemma 3.11, there is an E' > 0 such that for every D' E :D the probability that a simple random walk from 0 gets to distance rado (D') / (2E') before hitting a D' is less than E/2. Let A be the connected component of 0 in the set of points in D n (rado(D)/E')1U having distance at least EE' rado(D) from aD. By considering the first point where the random walk generating Y8 exits A, it follows that, with probability at least 1 - E, the diameter of Y8[0, 00] \ A is at most E rado(D) + o. Now note that there is a compact A' C 1U such that ¢i 1 (A) c A' for all sufficiently small 0, since ¢8 --+ ¢ uniformly on compacts. Therefore, there is some t1 > 0 such that Y8 [0, tIl n A = 0 a.s. for all sufficiently small 0 > O. In particular, (3.30)
P[diam Y8[0, tIl
> E rado(D) +
0]
< E.
If we take t2 E (0, t1), then taking 0 '\. 0 in (3.29) implies P[diam¢ 0 ,8 [t2, tIl> 2Efado(D)] < E. Since this holds for every t2, it follows that P[diam¢ 0 ,8(0, tIl> 2Efado(D)] < E. Using this with (3.30) and choosing to = t1 in (3.29) gives P[SUp{IY8(t) - ¢
0
,8(t) I :t > O} < 3Efado(D)] --+ 1.
Since this holds for every E > 0, the theorem follows.
D
4. The UST Peano curve. 4.1. Setup. The UST Peano curve is obtained as the interface between the UST and the dual UST. The setup which corresponds to chordal SLEs is where there is symmetry between the UST and the dual UST. Loosely speaking, the UST is the uniform spanning tree on the grid inside a domain D but with an entire arc a C aD on the boundary identified (wired) as a single vertex, and the dual UST also has an arc ,8 CaD on the boundary which is identified. The arcs a and ,8 are essentially complementary arcs. See Figure 5, where D is approximately a rectangle. As mentioned in Section 1, it was conjectured in [36] that, for an analogous setup, the interface defined for the critical random cluster models with q E (0,4] converges to SLEK , where K = K(q) E [4,8). A combinatorial framework is necessary in order to be more precise. There are several different possible setups that would work, and the following is somewhat arbitrary. If a tree T lies in the grid 7i}, then its dual tree Tt will lie in the dual grid (Z + 1/2)2, and the Peano path Y will lie in the graph G whose vertices are (1/4 + Z/2)2 and where v, u neighbor iff Iv - ul = 1/2. We have three kinds of vertices: elements of Z2 are the primal vertices, elements of (1/2+ Z)2 are the dual 961
970
G. F. LAWLER.O. SC HRAMM AN D W. WER NER
dual
tl'CC
tree Pcallo
a ~============~~~ FI G.5. The tree. dual free and 'Jeano UST path y. vertices and e lements of ( 1/ 4 + '£./2)2 are the Peano vertices. If wi-v are vertices of any kind , 110t necessarily the same, we say that they are adjacellt if the distance between them is as small as it can be for distinct vert ices of these particular kinds. In other words, if they are of the same kind, this means thal lhey are neighbors, if v E (1 / 4 + Z/2)2 and WE Z 2 U( I/ 2+ Z)2, this means Ilv - wlloo = 1/ 4 , while if v E Z2 and W E (1/2 + Z )2, thi s means IIv - wll oo = 1/ 2. Since there is no added comp licat ion, we consider a more general case where a and f3 are trees , rather than arcs . Let a be some finite tree in the primal grid Z2 and let fJ be a finit e tree in the dual grid (1 / 2 + Z)2. Suppose that no edge of a inte rsects an edge of {J . Further suppose that there are two Peano vert ices a , b E (1 / 4 + Z/2)2 such that a is adjacent to both a primal ve rtex et(l E a and a dual ve rtex {J1l E {J, and b is adjace nt 10 both a primal vertex etb E a and a dual vertex {Jh E {J. See Figure 6. Note that the line segme nt [et(l, {Jill has a as its midpoint, and the line segment lab ,{JhJ has b as its midpoint. Let D = D(a ,{J, a , b) be the (unique) bounded
+
1 FI G. 6.
The bOlll/dary daw Glulthe Peallo grid.
962
971
CONFORMAL INVARIANCE
connected component of C \ (a U [ab, t3b] U t3 U [t3a, aaD. Let Vp = Vp(D) denote the collection of all Peano vertices in D, and, as before, V(D) denotes the collection of all primal vertices in D. Let l = leD) denote the cardinality of Vp \ {a, b}. By switching the role of a and b, if necessary, assume that D lies to the immediate right of the oriented segment [aa, t3a]. Let;D* denote the collection of all domains obtained in this way. Let H = H(D) denote the subgraph of Z2 whose vertices are the vertices of a and V(D) and whose edges are those edges on this set of vertices which do not intersect t3. Since t3 is a tree, H is connected. Since H is connected, there is at least one spanning tree T of H which contains a. If we replace a by T and apply the dual argument, it follows that there is also a tree Tt in the dual grid (1/2 + Z)2, which is disjoint from T, contains t3 and whose vertices are the dual vertices in t3 and the dual vertices in D. In fact, Tt contains every dual edge lying in D \ T. We now need to give an orientation to the Peano grid G. Every edge in G is either on the boundary of a square face of G centered on a primal vertex or on the boundary of a square face of G centered on a dual vertex, and these two possibilities are exclusive. We orient the edges of G by specifying that the square faces of G containing a primal vertex are oriented clockwise, while those containing a dual vertex are oriented counterclockwise. When we want to emphasize the orientation of the edges, we write G ---+ instead of G. Note that the edges of G contained in a horizontal or vertical line all get the same direction in G ---+, and consecutive parallel lines get opposite orientations. For this reason, G ---+ is often called the Manhattan lattice. Let Y = Y (T) denote the set of all edges of G ---+ which do not intersect T U Tt and which have at least one endpoint in D. Let v E Vp \ {a, b} be some Peano vertex in D. Note that there are precisely two oriented edges of G---+ with initial point v, say el and e2, where one of these, say el, intersects an edge h of the primal grid Z2, and the other intersects an edge 12 of the dual grid (1/2 + Z)2. Note also that h n 12 i- 0. It therefore follows that exactly one of the edges h, 12 is in T U Tt . Consequently, exactly one of the edges el, e2 is in y. This shows that y has out-degree I at every v E Vp \ {a, b}. An entirely similar argument shows that y has in-degree I at every such v. In particular, this shows that y does not contain the entire boundary of a square face of G that does not contain a primal or dual vertex. If y had a cycle, the cycle therefore would have to surround some primal or dual vertex. But as T and Tt are connected and disjoint from y, this is impossible. It therefore follows that y is an oriented simple path (i.e., self-avoiding path), and the endpoints of y are a and b. Since we are assuming that D lies to the right of [aa , t3a], the initial point of y is a and the terminal point is b. Conversely, suppose that y* = (YO', ... , Ye*+l) is any oriented simple path in G---+, respecting the orientation of G---+, from a to b, whose vertices are Vp. For n E {O, ... , l + I}, let Vn be the (unique) primal vertex adjacent to and
y:
963
972
G. F. LAWLER, O.
SCHRAMM AND W. WERNER
let vJ be the dual vertex adjacent to y;. Note that Vn and Vn+1 are either the same vertex or adjacent vertices when n = {O, ... ,£}. Let an = an (y*) denote the union of a with the collection of all edges [Vk, vk+rJ for k < n such that Vk i= Vk+I and, similarly, let 13 = f3n(Y*) denote the union of 13 with the collection of all dual edges [vk, Vk+I] for k < n such that vk i= vk+1. Then T(y*) := ae+1(Y*) and Tt (y*) := f3e+1 (y*) are obviously connected, and there are no edges in T (y*) intersecting edges in Tt (y*). Now, T(y*) cannot contain a cycle, for such a cycle would have to separate Tt(y*). Hence, T(y*) is a spanning tree of H containing a.1t is also clear that y* = y(T(y*». That is, T ~ y(T) is a bijection between the set of spanning trees of H containing a and the set of oriented paths in G --+ n D from a to b containing V p. Hence, when T is the UST on H conditioned to contain a, y is uniformly distributed among such Peano paths; it is the UST Peano path associated with (a, 13, a, b). Let (a = Wo, WI, ... , WHI = b) be the order of the vertices in the UST Peano path y. For n E {O, 1, ... , £}, let y[O, n] denote the initial arc of y from Wo to Wn . Since y is uniformly distributed among simple oriented paths in G --+ from a to b which contain V p, we immediately get the following Markov property. LEMMA 4.1 (Markovian property). Fix any n E {I, 2, ... , £}. Conditioned on y[O, n], the distribution of (y \ y[O, n]) U {w n } is the same as that of the UST Peano curve associated with (an(y), f3n(y), W n , b). This lemma will play the same role in the proof as Lemma 3.2 in the case of the LERW. We will also use the convergence of certain discrete harmonic functions toward their continuous counterparts. To facilitate this, we have to set the combinatorial notation for the discrete Dirichlet-Neumann problem. Let H be a finite nonempty connected sub graph of Z2 with vertices VHand let E a denote the set of oriented edges in Z2 whose initial endpoint is in VH , but whose unoriented version is not in H. Suppose Ea = Eo U EI U E2 is a disjoint union, where Eo U EI i= 0. Suppose also that h: VH ---+ [0,1] is some function. For v E VH, set llH,Eo,E}.E2h(V) := Ldh[v, u], where the sum is over all neighbors u of v in Z2, and dh[v, u] := h(u) - h(v) when [v, u] ¢. Ea, dh[v, u] := h(v) when [v, u] E Eo, dh[v, u] := 1 - h(v) when [v, u] E EI and dh[v, u] := when [v, u] E E2. Note that there is a unique h: VH ---+ [0, 1] such that llH,EO,E],E2h(V) = in VH: h(v) is the probability that a simple random walk on H U Eo U E 1 started from v will use an edge in E 1 before using an edge of Eo. This h will be called the llH,EO,E],E2-harmonic function.
°- °
°
PROPOSITION 4.2 (Dirichlet-Neumann approximation). For every E > 0, there is an ro = rO(E) such that the following holds. Let Dee be a simply connected domain satisfying rado(D) :::: roo Let Ao, Al c au be two disjoint arcs, each of length at least E, and set A2 := au \ (Ao U Ad. Let 1} C D be a simple 964
973
CONFORMAL INVARI ANCE
closed path which surrounds 0, sllch that each point of 1) is within distance 5 from aD. Suppose that A~ , AJ C 1) are two disjoint arcs, A z ;= 'I \ (Ao U A J) and the triple (Ab, A'l ' A z) corresponds to (AO , AI , A2) under VrD, in the sense that for each j = 0, 1, 2 and each p E Aj there is a contilluOIl S path a : [0 , 1) -+ D satisfying diam a [0, I) :::: 5, a(O) = p, alld limst 1 t/I D 0 a (s) exists and is in A j. Let H be the component of 0 in the set of edges of ,!} that do not intersect 1). For j = 0, I, 2, let E j dellote the set of oriented edges [v , Il ] intersecting 1). where v is in H , and the first point of intersection fivm the direction of v is in A j . Let I~ denote the t:1H.Eo. El.E2-harmonic filllction. Let h : 1U -+ [0, I] be the continuous harmonic fllnction which has boundary vallll! 0 Oil Ao, 1 all A I, and satisfies the Neu mann boundary condition on A2. Then Ih(O) - h(O)1 < E.
The proof will be given in Section 5.4. 4.2. Driving process convergence. Let a , {J,a, band D = D(a , {J , a , b) be as above and suppose now that 0 E D. As before, let e denote the number of Peano vertices in D and let y = (y(O), ... , y(e + I» be the UST Peano path from a to b in G -l> n D. For eac h /I :::: e, there are two domai l2s that are nalural ly associated to yI.O, nj . The fi rst one (as in Lemma 4. 1) is Dn := D(a n , /3n ,y(n) , b) (see Figure 7). But Dn is not so useful if we want to make estimates using Loewner's equati on. We therefore also defin e Dn ;= D \ y [O, /I ]. Let 4>0 = ¢; D -+ 1HI be the conformal map which takes D to lHI, takes b to 00, takes a to 0 and satisfi es I¢(O) 1= I . Let ¢n : Dn -+ 1HI be the conformal maps satisfying ¢n (z ) - 4>o(z ) -+ 0 as z -+ b within Dn. De fin e Wn := ¢n(y(n» E lR. Also let tn denote the capac ity fro m 00 in 1HI of 4>0 0 y [O ,II ] , so that ¢no¢;;I(Z)=z+2tn/z+o( l /z) when Z -+ 00 in 1HI.
b
, I I I I I
•
1
i
1'(n)-;-
I I
I I
i
i: I I
: ..... : .----.
J
iI (In!I [=~~a ~--------------~----F IG. 7.
Tlte domaill
965
btl
is shaded.
974
G. F. LAWLER, O. SCHRAMM AND W. WERNER
We now prove the analog of Proposition 3.4 for the UST Peano curve. Let SJD(Z, A) denote the continuous harmonic measure of A from Z in the domain D\A. PROPOSITION 4.3 (The key estimate). For every sufficiently small 8, E > 0, there is some ro = ro(8, E) such that the following holds. Let y, Dn, ¢n, Wn and tn be as above, let kEN and let m be the first n :::: k such that IWn - Wk I :::: 8 or tn - tk :::: 82 . Then (4.1)
and (4.2)
PROOF. Assume first k = O. Let Vo E V(D) be some vertex such that I¢ (vo) I ::: 2 and 1m ¢ (vo) :::: 1/2, say. [As we have seen in Section 3.2, there is such a Vo when rado(D) is large.] If Q = [q, q'] is a line segment where qED is a dual vertex and q' E a is the midpoint of a dual edge containing q, then let ¢* (Q) E lR+ denote the limit of ¢(z) as z tends to aD along Q (which always exists by [35], Proposition 2.14). Fix such a Qo satisfying U := ¢*(Qo) E [1/2,2]; there clearly is such Qo when ro is large, because the harmonic measure from 0 of any square of the dual grid adjacent to the boundary of D is small. Let T/ C D be the set of points within distance 1/10 from aD. Then T/ is a simple closed path. Consider it as oriented counterclockwise around the bounded domain of C \ T/. Let Po be the point of T/ closest to a, PI the point in T/ n Qo and P2 the point of T/ closest to b. Let be the positively oriented subarc of T/ from Po and PI, the positively oriented arc from PI to P2 and A~ the positively oriented arc from P2 to po. Let.A, be the event that the path in the tree T (y) = aH I (y) from Vo to a hits AI. We will now estimate both sides of the identity
Ao
Al
P[.A,] = E[P[.A,IDm ]]
(4.3)
using Proposition 4.2. By Wilson's algorithm, P[.A,] is the probability that a simple random walk on the graph H (D) started at Vo stopped on hitting a will cross AI. This is exactly h(vo), where the function h is as defined in Proposition 4.2. Set h(z):=
~coCI( l-lzl ) n
21m.Jz
=
~coCI( n
l-r
2,Jrsin(8/2)
),
where z = re i8 and we take the value of coe l between 0 and n. Note that h is harmonic in 1HI, is equal to 0 on (0,1), is equal to 1 on (1, (0), and ayh = 0 on (-00,0). (Of course, we found the map h satisfying these boundary conditions by reflecting the domain along the negative real axis, mapping this larger domain to 1HI 966
975
CONFORMAL INVARIANCE
with z t-+ "fi and then using a conformal map from 1HI to 1[J to calculate the hitting probabilities.) Consequently, Proposition 4.2 shows that if ro is sufficiently large, then (4.4)
P[rA] = h(c/>(vo)/ U)
+ 0(8 3 ).
Set Vj := c/>j(vo) and Uj := c/>j 0 c/>Ol(U). By the chordal version of Loewner's equation and the definition of m, we have (4.5)
Vm = Vo
+ -2tm + 0(8 3),
Um = Uo
Vo
2tm Uo
+-
+ 0(8
3
).
Note also that rado(Dm) > rado(D)/2, Vo E Dm and Um E [1/4,4] provided that 8 is small enough. We now employ a similar argument to estimat~ P[rAIDm]. Recall that On = D(an , fin, yen), b). Assume that Qo intersects Dn , which will be the case if Un > 1/4, say. Let 1Jn be the set of points in On at distance 1/10 from oOn. Again, 1Jn is a simple closed path, and we write 1Jn = A~(n) U A~ (n) U A~(n), where A~(n) is the arc of 1Jn from the closest point to y (n) to the point of intersection of Qo with 1Jn, A~ (n) is the arc of 1Jn from the point in Qo n 1Jn to the point of 1Jn closest to b and A~(n) is the remaining part of 1Jn. By Lemma 4.1, P[rAIDn] is the same as the quantity hn(vo), where h n is the function h defined in Proposition 4.2, but with A~(n), A~ (n), A~(n) an_d 1Jn replacing A~, A~, A~ and 1J and Dn replacing D. (It is Dn replacing D, not Dn. The conditions of Proposition 4.2 hold for either of these, but the conformal map we consider is defined on Dn.) Proposition 4.2 therefore gives (4.6) Write feU, V, W) := h((V - W)/(U - W». We Taylor-expand the right-hand side in (4.6) to second order in Wm and to first order in Vm - Vo and Um - Uo. Together with (4.3)-(4.5) this gives 0= E[P[rAIDmJ] - P[rA] =
~o?v fE[W~] + Ow fE[Wm] + ov f2E~~m] + ou f2E~~m] + 0(8 3 ).
Here, the derivatives of f are evaluated at (Vo, Wo, Uo). (Note that V is complex valued, and we interpret ov f as an lR-linear map from CC to JR.) If we plug in Vo = i + 0(8 3 ) and Uo = 1 + 0(8 3 ) [as we have seen in Section 3.2, one can certainly find Vo and Uo satisfying c/>(vo) = i + 0(8 3 ) and c/>(uo) = 1 + 0(8 3 ) if ro is large], then after some tedious but straightforward computations the above equality simplifies to
967
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while Vo = 2i
LAWLER, O. SCHRAMM AND W. WERNER
+ 0(8 3 ) and Uo = 1 + 0(8 3 ) give 3E[W~] + 8E[Wm ] - 24E[tm] =
08 3 ).
Combining these two relations implies (4.1) and (4.2) in the case k = O. For k > 0, the proof is basically the same; the only essential difference is that one must use T/k in place of T/. D THEOREM 4.4 (Driving process convergence). For every positive E1,E2, E3 and i, there is some positive r1 = r1 (E1, E2, E3, i) such that the following holds. Let D = D(a, {3, a, b) E i)* satisfy rado(D) > r1 and SJD(O, a) E [E1, 1 - Ed. Let y be the corresponding UST Peano path, let E2] < E3. PROOF. The proof is almost identical to the proof of Theorem 3.7, where we used Skorohod's embedding, but one has to be a little careful because it may happen that 0 is "swallowed" before time i. Let us first assume that
i:::: 1/100 and P[ B[O, i] c [-1/10,1/10]] > 1 -
E3/3,
where B is standard Brownian motion. Take 8 = 8(E1, E2, E3) > 0 small and E = 1/10. Define rO(E, 8) as in Proposition 4.3. Let kEN be the first integer where rado(Dk) :::: ro or SJDk(O, ak) fj. [E, 1- E] and define io:= min{i, td, where tn is as in the proposition. Exactly as in the proof of Theorem 3.7, Proposition 4.3 implies that we may couple W with a Brownian motion B in such a way that P[sup{IW(t) - B(8t)l:t E [O,to]} > E2/3] < E3/3,
if rado(D) ~ r1 and r1 is large enough. By our assumptions regarding i, we have with high probability that, for all t E [0, io], Wet) E [-1/5, 1/5]. If we choose TI large enough, this guarantees that P[io i= i] < E3/3 and proves the theorem when (4.7) is satisfied and 0 be some constant satisfying (4.7) and let zo :=
CONFORMALINVAruANCE
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4.3. Uniform continuity. In order to prove convergence with respect to a stronger topology, tightness will be needed, and we therefore derive in this section some regularity estimates for UST Peano curves with respect to the capacity parameterization. Some results from [38] will be used. Let Dee be a simply connected domain containing 0, whose boundary is a C I-simple closed path. Let a and b be two distinct points on aD. In this section, we consider for large R the UST Peano curve from a point near Ra to a point near Rb on a grid approximation of RD. One reason not to consider arbitrary domains is that we need to partially adapt to the framework of [38] in order to quote results from there. Also, it is natural (since the UST Peano curve is asymptotically space filling) to impose regularity conditions on aD in order to get uniform regularity estimates for the UST Peano curve. Let aD and fJ D be, respectively, the clockwise and anticlockwise arcs of aD from b to a. Given R large, let DR = D(a R , fJR, a R , b R ) E 1'* be an approximation of (RD, RaD, RfJD) in the following sense. Fix some sufficiently large constant C > 0; for example, C = 10 would do. We require a R to be a simple path in 71.,2 satisfying p(a R , RaD) :::: C and require fJR to be a simple path in the dual grid (1/2 + 71.,)2 satisfying p (fJ R , RfJ D) :::: C. We also require that fJR n a R = 0, of course, and that each of a R , b R is a Peano vertex adjacent to an endpoint of a R and an endpoint of fJR. Let y = yR be the UST Peano path in DR. Let ¢: D -+ H be the conformal homeomorphism satisfying ¢(a) = 0, ¢(b) = 00 and I¢(O)I = 1. Let ¢R: DR -+ H be the conformal homeomorphism satisfying I¢R (0) I = 1, taking a R to 0 and b R to 00. Then limR---+oo R-l¢"Rl(Z) = ¢-l(Z) uniformly in H. (This follows, e.g., from Corollary 2.4 in [35].) Let y := ¢R 0 y, parameterized according to capacity from 00. Let gt: H \ 9[0, t] -+ H be the conformal map with the usual normalization gt(z) - z -+ 0 when Izl-+ 00.
PROPOSITION 4.5 (Uniform continuity estimate). For every E > 0 and t > 0, there are some positive Ro = Ro(D, t, E) and 8 = 8(D, t, E) such that, for all R >Ro,
We first prove a slightly modified version of this proposition. LEMMA 4.6. For 0 < tl < t2 < 00, let Y(tl, t2) := diam(gtl 0 y[tl, t2]). For every E > 0, there is a 8 = 8(D, E) > 0 and an Ro = Ro(D, E) > 0 such that,for all R 2: Ro, (4.8)
P[sUp{ly(t2) - y(tdl: 0:::: tl :::: t2:::: r, Y(tl, t2):::: 8} 2: E] < E,
where r := inf{t 2: 0: Iy(t) I =
E- l }.
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The proof will use Theorems 10.7 and 11.1(ii) of [38]. As explained there, the proofs of these theorems are now easier, because we have established the conformal invariance of the UST; see Corollary 1.2. PROOF OF LEMMA 4.6. Let OR be a positive function of R such that limR--+oo OR = 0. It suffices to show that (4.8) holds for all sufficiently large R with OR in place of O. Let Z denote the semicircle 2E- 1aU n lHI, say. For R large, let tl and t2 be such that 19(t2) - 9(tl)1 is maximal subject to the constraints 0:::: tl :::: t2 :::: r and Y(tl, t2) :::: OR. Note that mint
°
°
(4.9) (4.10) (4.11)
"lsI> 0, 3Ro > 0, V R > Ro,
P[.A \ XI(SI)] < E,
V SO > 0, 3 SI > 0, 3 Ro > 0, VR > Ro, P[.A n Xl (SI) \ Xo(so)] <
E,
P[.A n Xo(so)] <
E.
3so > 0, 3 Ro > 0, V R > Ro,
U sing these statements, the proof of the lemma is completed by choosing so according to (4.11), then choosing SI according to (4.10) and, finally, choosing Ro according to (4.9), (4.10) and (4.11). We start with (4.9). Fix some SI > and assume that .A \ Xl (SI) holds. We also assume that E < SI. There is no loss of generality in that assumption, since .A is monotone decreasing in E. Since limR--+oo length(1]) = 0, for large R the two endpoints of 1] must be in 9[0, td. Because 9 tends to 00 with t, it is clear that 9[t2, 00) n 1] =f. 0. In fact, the crossing number of 9[t2, 00) and 1] must be ±1, since 9 and 1] are simple curves. Consider the concentric annulus A whose inner circle is the smallest circle surrounding 1] and whose outer circle has radius E/4. Let 33 denote the open disk bounded by the outer circle of A and note that 33 C lHI, by our assumption E < SI. On the event .A, there is a t* E [tI, t2] such
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that the distance from 9 (t*) to 1] is at least E /2. In particular, 9 (t*) ¢. $. Now, 1] separates 9(t*) from 00 in lHI \ 9[0, til. Therefore, if .A, \ Xl (s) holds, then 9[0, til U 1] separates 9(t*) from 00. Since 9 is a simple path, this implies that the arc of 9[0, til between the two points 11 n 9[0, til does not stay in $. Hence, 9[0, (0) n $ has three distinct connected components, say 91, 92, 93, each of which intersects the inner circle of A such that 91, 93 C 9 [0, t*] and 92 C 9 [t*, (0) and 92 separates 91 from 93 within $. See Figure 8. Note that adjacent to one side of
°
inner boundary of R-l
°
FIG.
°
8. The paths 91,92 and 93 and the annulus A. 971
°
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
To prove (4.11), let s > 0 and let v E 71} be a vertex closest to ¢"R 1 (is). Let v t E (Z + 1/2)2 be a dual vertex adjacent to v. Let X be the simple path from v to OlR in T and let X t be the simple path from v t to f3R in Tt. We may sample X by running a simple random walk from v on H(DR) stopped on hitting OlR and loop-erasing it. It therefore follows by Proposition 4.2 that if s is sufficiently small, then the diameter of ¢ R (X) is smaller than E /10 with probability at least 1 - E/lO. Moreover, there is some So > 0 such that for all sufficiently large R with probability at least 1 - E/lO the distance from ¢R(X) to 0 is at least so, and the same two estimates will hold for X t. Let D' be the domain bounded by [v, v t] U X U X t U a DR which has the initial point a R of y on its boundary. Note that y crosses the boundary of D' exactly once, through the segment [v, v t ]. In particular, if diam(¢R(D')) < E and .A, holds, then T/ is not contained in ¢R(D'). This proves (4.11) and completes the proof of the lemma. D PROOF OF PROPOSITION 4.5. Theorem 4.4 implies that we may find some r > 0 such that P[ sup{ IW (t) I : t E [O,t]} 2: r] < E /4 for all sufficiently large R. Let E' := min{E, r- 1 } and let 8' denote the 8 obtained by using Lemma 4.6 with E' in place of E. Since Brownian motion is a.s. continuous, Theorem 4.4 implies that there is some 8 > 0 such that if R is large enough we have P[sup{IW(t1) - W(t2) I :t1, t2 E [0, t], It1 - t21 < 8} 2: 8'] < E/4. Lemma 2.1 applied to the path t
1--+
gtl
0
yet - t1) - W(t1) now implies
P[SUp{Y(t1, t2):0::::: t1::::: t2::::: t, It1 -t21 < 8} 2: C(8 1/ 2 +8')] < E/4. Now the proof is completed by using Lemma 4.6.
D
4.4. Consequences. In this section we gather some consequences, starting with the following two theorems.
THEOREM 4.7 (Chordal SLE8 traces a path). Let gt denote the chordal SLE8 process driven by B(St), where B(t) is standard Brownian motion. Then, a.s.for every t > 0, the map g;l extends continuously to IHI and yet) := g;l (B(St)) is a.s. continuous. Moreover, a.s. g;l(IHI) is the unbounded componentofIHI \ y[O, t]for every t 2: o. THEOREM 4.S (Peano path convergence). Let D C
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Here, we think of 9 and 9 as elements of the space of continuous maps from [0, (0) to lHI, with the topology of locally uniform convergence. A consequence of the theorem is that R- 1 y is close to R- 1¢ii1 09. That is, we may approximate the UST Peano path y by the image of chordal SLEs in DR. The analog of Theorem 4.7 was proven in [36] for all K i- S, but the particular case K = S could not be handled there. It is fortunate that the convergence of the UST Peano path to SLEs settles this problem. By Remark 7.5 in [36], it follows that with the notation of Theorem 4.7 for K :::: S we have gt 1 (lHI) = lHI \ 9[0, t] for every t :::: a.s.
°
PROOF OF THEOREMS 4.7 AND 4.S. Let Wet) = WR(t) denote the chordal Loewner driving process for 9. Fix a sequence Rn ---+ 00. First, note that the family of laws of 9 is tight, because of Proposition 4.5 and the Arzela-Ascoli theorem (see, for instance, [17], Theorem 2.4.10). Also, Theorem 4.4 implies that the law of W converges weakly to the law of B(St). Hence, there is a subsequence of Rn such that the law of the pair (9, W) converges weakly to some probability measure J-l. Let (y*, W*) be random with law J-l. Then we may identify W*(t) with B(St). By the chordal analog of Lemma 3.14, which is valid with the same proof, it follows that, for all t > 0, gt 1 (lHI) is the unbounded component of lHI \ y*[O, t]. Since y* is continuous, elementary properties of conformal maps imply that gt 1 extends continuously to lHI (e.g., Theorem 2.1 in [35]). It is easy to verify that, a.s. for every t > 0, gt 1 (B(St» = y*(t), using the fact that y*[t, t'] is contained in a small neighborhood of y*(t) when t' - t > is smalL This proves Theorem 4.7. Because the law of the limit path y * does not depend on the subsequence, the original sequence converges, and so Theorem 4.S is proved as well. D
°
We now list some easy consequences of Theorems 4.7 and 4.S. COROLLARY 4.9 (Radial SLEs traces a path). Let gt denote a radial SLEs process driven by Wet) = exp(iB(St», where B is standard Brownian motion. Then, almost surely, for every t > 0, the map gt 1 extends continuously to 1U. Moreover, gt 1 (W(t» is almost surely continuous. PROOF. This follows readily from Theorem 4.7 and the absolute continuity relation between radial and chordal SLEs derived in [2S], Proposition 4.2. D PROOF OF THEOREM 1.3. Define G8 = 8D 1/ 8, where D1/8 is defined as in Theorem 4.S. Consider the situation of Theorem 4.S. As previously remarked, it follows from [35], Corollary 2.4, that limR-Hlo R- 1¢i/(z) = ¢-1(Z) uniformly in lHI. Consequently, Theorem 4.S shows that, for all t > 0, the UST Peano curve scaling limit up to capacity t from b is equal to ¢ -1 0 9 up to time t. It therefore suffices to prove that for all E > there is an E' > such that for all sufficiently
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large R with probability at least 1 - E the part of y after the first time it hits the Ef -neighborhood of b stays within the E-neighborhood of b. This is easily proved by the same argument used to prove (4.11) applied to the reversal of the UST Peano path, which is also a UST Peano path. D COROLLARY 4.10 (Path reversal). The law of the chordal SLE8 curve is invariant under simultaneously reversing time and inverting in the unit circle, up to a monotone increasing time change. More precisely, if y is the chordal SLE8 curve from 0 to 00 defined in Theorem 4.7, then a time change of( -l/y(1/t), t ~ 0) has the same law as y. PROOF. This follows immediately from the fact that the reversed UST Peano curve is also a UST Peano curve. D
5. Random walk estimates. The goal of this section is to prove the remaining random walk estimates and thereby complete the proofs of the theorems. Basically, we show that, under certain boundary conditions, discrete harmonic functions converge to continuous harmonic functions satisfying corresponding boundary conditions, as the mesh of the grid goes to O. This general principle is not new, of course (see, e.g., [6]), but it seems that the precise statements which are needed here do not appear in the literature. In particular, our results make no smoothness assumptions on the boundary. It should perhaps be noted that some of the following proofs (and most likely the results, too) are special to two dimensions. 5.1. Preliminary lemmas. We now state some lemmas on discrete harmonic functions, which will be helpful in the proofs of Proposition 2.2, Lemma 3.5 and Proposition 4.2. For 8 > 0, define the discrete derivatives a~f(v) := 8- 1 (j(v
+ 8) -
a~f(v) := 8- 1 (j(v
+ i8) -
f(v)), f(v)).
Let :D8 := {8D: D E :D}, that is, domains adapted to the grid 871 2 . Similarly, for D8 = 8D E :D8, define V 8 (D8) := D8 n 871 2 = 8V(D) and Vt(D8) := 8Va(D). LEMMA 5.1 (Discrete derivative estimate). There is a constant C > 0 such that,for every D E:D and every boundedfunction h: V(D) U Va(D) --+ JR that is harmonic in V(D), (5.1)
a;h(O):::: Crado(D)-lllhlloo,
a;h(O):::: Crado(D)-ll1hlloo.
This lemma is proved using the Green's functions in [25], Theorem 1.7.1; see also [16], Lemma 7.1, for a proof of the analogous statement in the triangular lattice using the maximum principle. In Section 6, we rewrite and adapt the proof from [25] to more general walks on planar lattices. One can also rather easily prove the lemma using coupling. 974
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LEMMA 5.2. For all E > 0 and kEN, there exists a c = q(E) > 0 such that the following always holds. Let 8 E (0, c- I ) and let DE:D8 satisfy rado(D) 2: 1/2. Let a~l' ... , a~k E {a~, a~}. Let h: V 8 (D) U Vg(D) ---+ [0, (0) be nonnegative and harmonic in V 8 (D). If V E V 8 (D) satisfies 11/ID(V) I :::: 1- E, then
(5.2)
la~l a~2 ... a~kh(v)1 :::: ch(O).
Note that the case k = 0, which is included, is a kind of Harnack inequality. It is easy to give quantitative estimates for q(E), but they will not be needed here. Only k :::: 3 will be used in the rest of the paper. In the proof of the lemma, the following simple conformal geometry consequences of the Koebe distortion theorem (see [35], Theorem 1.3) will be needed. Let D, E and v be as in the statement of the lemma. First, note that 1/4 :::: rado(D)1/Ib(0) :::: 1 follows from the Koebe 1/4 theorem and the Schwarz lemma, respectively. Let l = leE) be large and set Zj := j 1/ID(v)/l and Wj := 1/Ii/(zj), j = 0,1, ... , l. The Koebe distortion theorem gives upper and lower bounds for rado (D) 11/Ib I on the preimage of the line segment [0, Zi]. This implies that there is a constant Cl = Cl (E) > 0 such that radwj(D) 2: Cl rado(D) and that if l = leE) is large, then, IWj - wj-Il:::: Clrado(D)/20, j = 1, ... ,£. In particular, if Vj is the vertex in V 8 (D) closest to W j, then, provided that 8 is sufficiently small, IVj - vj-Il :::: rad vj _1 (D)/lO. PROOF OF LEMMA 5.2. We start with k = O. Suppose first that Ivi :::: rado(D)/lO. Let W C V 8 (D) be the set of vertices W satisfying hew) 2: h(v). Then W contains a path from v to aD. But the probability p that the path traced by a simple random walk from 0 before exiting D separates v from aD is bounded away from o. On that event, the simple random walk hits W before exiting D. Consequently, h(O) 2: ph(v), as needed. For arbitrary v E V 8 (D) satisfying 11/1D (v) I :::: 1 - E, as we have noted, the Koebe distortion theorem implies thatthere is an l = l (E) depending only on E and a sequence 0 = VO, VI, ... , Vi = v in V 8 (D) with l:::: leE) such that IVj - vj-Il :::: radv/D)/l0 for each j = 1, ... , £. Consequently, iterating the above result gives h(O) 2: pih(v) and proves the case
k=O. Using the above, we know that hew) :::: c'h(O) on the set of vertices W E V 8 (D) such that Iw - vi:::: radv(D)/lO, where c' = C'(E) is some constant depending only on E. Consequently, the case k = 1 now follows from Lemma 5.1 applied with v translated to O. For k > 1, the proof is by induction. By the above, we may assume v = O. Let M be the maximum of la~kh(w)l/ h(O) on the set V of vertices W E V 8 (D) satisfying Iwi :::: rado(D)/lO. The above shows that M is bounded by a universal constant. Since a~k h is discrete harmonic on V, the proof is completed by applying the inductive hypotheses to the function agkh(w) + Mh(O). D 975
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LEMMA 5.3 (Continuous hannonic approximation). For every E > 0, there is some ro = rO(E) > 0 such that the following holds. If D E ~ satisfies rado(D) ::: ro and h: V(D) U Va(D) -+ [0, (0) is discrete harmonic in V(D), then there exists a harmonic function h*: D -+ [0, (0) such that (5.3)
Ih*(v) - h(v)1 ~ Eh(O)
holds for every vertex v E V(D) satisfying IVrD(V)1 < 1 - E.
PROOF. Suppose that the lemma is not true. Then there exists E > 0 and a sequence of pairs (Dn, h n ), where Dn E ~ satisfies rado(Dn) ::: n and h n > 0 is discrete harmonic in V(D n ), satisfies hn(O) = 1, but (5.3) fails for every hannonic function h. Set 0 = On := 1jrado(Dn). Our objective is to apply compactness to show that the maps h n 0 Vr D1 converge locally uniformly in U as n -+ 00 along some subsequence to some harmonic h, so that (5.3) does hold for some n. We put hn(v):= h n (vj8n). First, standard compactness properties of conformal maps say that one can take a subsequence such that the maps On Vr D~ converge locally uniformly in U to some conformal map, say ¢. (This follows, e.g., from the Arzela-Ascoli theorem, together with [35], Corollary 1.4, with z = 0 and part two of [35], Theorem 1.3.) If K c U is compact, then Lemma 5.2 shows that there is a constant C > 0 such that, for all sufficiently large n in the subsequence, the discrete derivatives Io~ h n I and lo~hnl are bounded by C in ¢(K) n V(oDn). By a variant of the Arzela-Ascoli theorem, it then follows that there is some continuous h*: ¢(U) -+ [0, (0) and a further subsequence such that, for every compact K C ¢ (U), sup{lhn(v) - h*(v)l: v E K
n OlE?} -+ 0
along the subsequence. The same argument may also be applied to prove the convergence of the discrete derivatives of h n to arbitrary order, possibly in a further subsequence. Obviously, the discrete derivatives of h n will converge to the corresponding continuous derivatives of h*; that is, (5.4)
sUp{iO~l ... o~khn(v) - oal ... oakh*(v)i: v
E
K
n oZ2} -+ 0,
where ogj E {o~, o~} and Oaj E {ox, Oy} is the corresponding continuous derivative, j = 1,2, ... ,k. Thefactthath n is discretehannonic translates to (o~)2hn(v-0)+ (o~)2hn(v - io) = O. Therefore, (5.4) shows that h* is hannonic. This completes the proof. D LEMMA 5.4 (Boundary hitting). For every El, E2 > 0, there is a 0 = 0 such that if D E ~ and W E V(D) is a vertex satisfying IVrD(W)1 ::: 1 - 0, then the probability that the simple random walk started at w will hit
O(El, E2) >
{v E V(D): IVrD(V) -VrD(w)1 > before hitting oD is at most E2. 976
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PROOF. We first prove the lemma in the case where E2 is very close to 1. Let 8 > 0 be much smaller than EI. Fix some vertex w E V(D) and suppose that IVrD(W)1 2: 1 - 8. Let
a := {z ED: IVrD(Z) -VrD(w)1 =
Ed.
be a point in aD closest to w and set r := dist(w, aD) = IZI - wi. Let Al be the line segment [w, ztJ. Let Q be the connected component of C(ZI, r) n D which contains w, where C (z, r) denotes the circle of radius r and center z. Then Q is an arc of a circle. Let A2 and A3 denote the two connected components of Q \ {w}. See Figure 9. For j = 1,2,3, let Kj be the connected component of D \ (A I U A2 U A3) which does not have A j as a subset of its boundary. Because 8 is small compared to EI, the Koebe distortion theorem (e.g., Corollaries 1.4 and 1.5 in [35]) shows that a n C (w, r /8) = 0. For j = 1, 2, 3, let -8 j be the collection of all paths which stay in K j from the first time they hit C(w, r/8) until they first exit from D. Let B(t) denote Brownian motion started from w. It is easy to see that there is a universal constant ct > 0 such that P[B E -8j] > ct for j = 1,2,3. For example, to prove this for j = 3, observe that the collection of Brownian paths which first hit C (w, r /8) in K 3 and later hit A3 before Al U A2 has probability bounded away from O. Suppose for the moment that a intersects Al and A2. Consider a subarc a' C a whose endpoints are in Al and A2, which is minimal with respect to inclusion. Then a' C K3 or a' C KI U K2. If a' C K3, then a' separates C(w, r/8) from aD in K 3. Consequently, on the event B E -8 j, B hits a before hitting aD. However, by choosing 8 to be sufficiently small and invoking the conformal invariance of the harmonic measure, we may ensure that the latter event has probability smaller than ct. An entirely similar argument rules out the possibility that a' C K I U K2. Similarly, it is not possible that a intersects both Al and A3 or that a intersects Let
Zl
w FI G.
9.
The arcs A j and the components K j
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G. F. LAWLER, O. SCHRAMM AND W. WERNER
both A2 and A3. Hence, there is some j E {1, 2, 3} such that a n Kj = 0. Let j' be such a j. By the convergence of a simple random walk to Brownian motion, it is clear that there is some universal constant ro > 0 such that if r > ro, then the probability that the simple random walk started from w is in -8 j' is at least q 12. This establishes the lemma in the case where E2 E (1 - q/2, 1] and r > roo Suppose r ~ roo Then there are two grid paths of bounded length starting from w to aD that are disjoint except at w. If a intersects both these paths, then this gives a lower bound for the continuous harmonic measure of a from w. Consequently, by making 8 small enough, we can make sure that this does not happen. Thus, again, with probability bounded away from 0, the random walk from w hits aD before a, since it may follow anyone of these two paths. This proves the lemma in the case where E2 E (c2, 1], where e2 is some universal constant. The Koebe distortion theorem implies that there is a constant e > 0 such that if VI, V2 E V(D) are neighbors, then 1 - I1/ID(V2) I ~ e(l - 11/ID(VI)l). (See, e.g., Corollaries 1.4 and 1.5 in [35].) Consequently, we may iterate the above restricted case of the lemma and use the Markov property, thereby proving the lemma for arbitrary E2 > O. D 5.2. The hitting probability estimate. PROOF OF PROPOSITION 2.2. Let EI > 0 be much smaller than E. We consider the discrete harmonic function hew) := H(w, u)1 H(O, u). For 8 > 0, let
V(8, EI) := {z E V(D): I1/ID(z) I 2: 1 - 8, 11/ID(Z) -1/ID(u)1 > Ed·
Our first goal is to show that, for every EI and some ro = ro(EI) > 0 such that (5.5)
E
(0, 114), there is some 8 = 8 (E I) > 0
max{h(z):z E V(8, EI)} < EI,
provided that rado(D) > roo This will be achieved by first showing that h is not too large on the set
W:= {z
E V(D): El/2
2: 11/ID(Z) -1/ID(u)1 2: EI/3}
and then letting 8 go to 0 and appealing to Lemma 5.4. Assume that rado(D) is sufficiently large so that any nearest neighbor path from o to u in D has a vertex in W. Let M denote the maximum of h on W. We claim that M is bounded by a constant e = e(EI) depending only on EI. Indeed, let K be the set of all V E V(D) satisfying h(v) 2: M 12 and let K' be the union of all edges where both endpoints are in K U {u}. Then the maximum principle shows that K' is connected and contains a simple nearest neighbor path J joining W to u whose vertices are in {z E V(D): 11/ID(Z) -1/ID(u)1 ~ E1/2} U {u}. Note that, in particular, diam(1/ID(J)) 2: E1/3. Consequently, the continuous harmonic measure from 0 of J in D is bounded from below by some constant q (EI) > O. 978
CONFORMAL INVARIANCE
987
We claim that the discrete harmonic measure HD(O, J) of J at the origin is also bounded away from 0 if ro is large enough. Indeed, let D' = D \ J and let A be the arc on au corresponding to J under the map l/fD" The length of A is bounded from below, since it is equal to 2n times the harmonic measure of J. Let A' denote the middle subarc of A having half the length of A. By Lemma 5.4 applied to the domain D', it follows that there is a C2 = C2(E) E (0, 1/10), such that HD'(V, J) ~ 1/2 on vertices v such that l/fD'(V) is within distance C2 of A'. Using Lemma 5.3 with E replaced by C2/4, we find that if ro is large, there is a nonnegative harmonic function h~ : D' ---+ [0, (0) such that Ih~(v) - HD'(V, J)I :::: C2/4 for all v E V(D') satisfying Il/fD' (v) I < l-c2/4. Take a E A' and z := (1 - c2/2)a. Then it follows from the Koebe distortion theorem (as in the argument toward the end of the proof of Proposition 3.4) that we may find a vertex v E V(D') such that Il/fD'(V) - zl < C2/4, assuming that ro is large enough. Thus, h~(v) > HD'(V, J) - C2/4 ~ 1/2 - C2/4 > 1/4. By the Harnack principle applied to h~, there is a universal constant C3 > 0 such that h~(z) ~ C3. Since this applies to every z E (1 - c2/2)A', the mean value property for h~ gives h~(O) ~ c3Iength(A')/(2n). Since Ih~(O) - HD'(O, J)I < C2/4, our claim that H'v (0, J) is bounded away from 0 is established. Since h is positive, harmonic h(O) = 1 and h ~ M/2 on J, this also gives the bound M:::: C(E). Since h is harmonic, Lemma 5.4 with EI = EI/2 and E2 := EI/C(E) instead of EI, E2 implies that if 8 = 8(EI) is sufficiently small, and rado(D) is large enough to guarantee that W separates V (8, E I) from u (in the graph-connectivity sense), then h(z) :::: ME2 < EI for all Z E V(8, EI): (5.5) holds. Now apply Lemma 5.3 again to conclude that there is a harmonic function h :U ---+ [0, (0) such that
Ih 0
l/fD(Z) - h(z)1 < EI _
A
for all z E V(D) such that Il/fD(z) I < 1 - 8/4. Set h(z) := h((1 - 8/2)z). We know that h ~ 0 in au, h(z') :::: 2EI on the set S:= {z' E au: Iz' -l/fD(u)1 ~ 2EI}. Consequently, the Poisson representation of h gives
-
h(z) = O(EI)
+ fa
au\s
-
h(z')
1 -lzl 2 2 Idz'l. Iz - z'l
Since h(O) = 1 + O(Ed and EI is arbitrary, the proposition follows.
0
5.3. Some Green's function estimates. As opposed to Proposition 2.2, Lemma 3.5 requires only crude bounds. It is actually possible to prove that Go(O, O) - G m (0,0) is close to tm , but we do not need this result here. PROOF OF LEMMA 3.5.
We start with (3.5). Let S be the set of vertices in
V(D) satisfying (3.3) and assume S"# 0. For a random walk starting from a vertex 979
988
G. F. LAWLER, O. SCHRAMM AND W. WERNER
in S, there is probability bounded away from exit D. This gives (5.6)
L
°that within rado(D)2 steps it will
GD(O, w)::::: 0(1) rado(D)2.
WES
°
On the other hand, with probability bounded away from 0, the number of steps into vertices in S for the random walk started at that is stopped on exiting D is greater than rado(D)2. Therefore, (5.7)
0(1)
L
GD(O, w) ::: rado(D)2.
WES
By reversing the walk, we know that GD(O, w) = GD(W, 0). Since GD(W, 0) is harmonic on V(D) \ {OJ, the Harnack principle [i.e., k = in (5.2)] can be used to show that GD(W, O)jGD(W', 0) = 0(1) when w, w' E S. Combining this with GD(O, w) = GD(W, 0) and the estimates (5.6), (5.7) gives (3.5). By Lemma 2.1, we have
°
(5.8)
diam(VrD
0
y[O, m]) = 0(8).
°
In the following, we fix y[O, m] (i.e., it will be considered deterministic). Let z be the vertex where a simple random walk from first exits Dm. By considering what happens to the random walk after first hitting z, we get the identity Go(O, v) - Gm(O, v) = E[Go(z, v)] [where Go(z, v) = for z fj. V(D), by definition]. Consequently,
°
Go(O, v) - Gm(O, v) ::::: P[z E y[O, mJ] max{E[Go(Yj, v)]: j = 1, ... , m}.
°°
By (5.8), the continuous harmonic measure from of VrD 0 y[O, m] in U is 0(8). Therefore, the continuous harmonic measure from of Y[O, m] in D is also 0(8). As in the argument given in Section 5.2, this implies that if rado(D) is large enough, P[z E y[O, m]] = 0(8). Let K denote the disk {w ED: Iw - vi < rado(D)j10} and fix some j E {I, 2, ... , m}. Since VrD 0 Y[O, m] is contained in U \ (1 - 0(8»U. It follows that the continuous harmonic measure of K from Yj in D is 0(8). If VrD(Yj) is sufficiently close to au (how close may depend on 8), then we can make sure that the corresponding discrete harmonic measure HD(Yj, K) is less than 8 by Lemma 5.4. If VrD(Yj) is not close to au, then when rado(D) is large the bound HD(Yj, K) ::::: 0(8) follows by the convergence of the discrete harmonic measure to the continuous harmonic measure, as we have seen before. If W E V(D) n K neighbors with a vertex outside of K, then Go(w, v) = 0(1) follows from (5.7) by translating W to 0. Hence, Go(Yj, v) = 0(1) HD(Yj, K) = 0(8). Putting these estimates together completes the proof. D 980
CONFORMAL INVARIANCE
989
5.4. Mixed boundary conditions. Recall that Proposition 4.2, which we will now prove, is not used in the proof of Theorem 1.1.
PROOF OF PROPOSITION 4.2. Suppose first that the distance between Ao and A 1 is at least E. Let Ao and A i denote the two connected components of 'fI \ (A~ U A~) such that the sequence (A~, Ao, A~, Ai) conforms to the counterclockwise order along 'fl. This induces a corresponding partition E2 = Eo U Ei of E2, according to whether or not the first point on the edge is in Ao . A*1. orm We need to use the discrete harmonic conjugate function k of h. To be perfectly precise, it is necessary to set some combinatorial infrastructure: we first define a (multi-) graph fi, and k will be defined on the planar dual fit of fi. The vertices of fi are VH U {vo, vd (where Vo and VI are new symbols not appearing in VH). As edges of fi, we take all the edges of H, and, additionally, for every j = 0, 1 and every directed edge [v, u] in E j, there is a corresponding edge [v, v j] in Finally, there is also the edge [vo, vIJ in fi. Consider a planar embedding of H which extends the planar embedding of H such that Vo and VI are in the unbounded component of C \ H. Let fit denote the planar dual of fi. Then there is a unique edge [vJ, v{] in fit which crosses [vo, vIJ. We choose the labels so that v} naturally corresponds to Aj, j = 0, 1. Set h(vj) := j, j = 0, 1. If we consider h as a function on fi, then it is discrete harmonic except at Vo and VI. This easily implies (see, e.g., [7] or, more explicitly, [4]) that there is a discrete harmonic conjugate k defined on the vertices of fit; that is, for every directed edge e = [u, v] in fi if {u, v} =f. {vo, VI}, then the discrete Cauchy-Riemann equation h(v) - h(u) = k(v t) - k(u t) holds, where [u t, vt] is the edge of fit intersecting e from right to left. In fact, k is harmonic in fit except at vJ and v{. The function k is unique, up to an additive constant. We choose the additive constant so that k(vJ) = O. Since h 2: 0, by considering the neighbors of Vo and the orientation, it follows that k(v{) 2: O. Consider a sequence Dn of such domains satisfying rado(Dn) 2: n, with arcs 'fI = 'fin and such harmonic functions hn , kn . Let Ln denote the maximum value of kn, which is the value of kn on v{. Since E > 0 is fixed, we can consider a subsequence of n ~ 00 such that the arcs Ao and A 1 converge to arcs Ao and A 1 of length at least E, and the distance between them is at least E. Let Ao and Ai denote the two components of au \ (Ao U AI), so that Ao, Ao, AI, Ai is the positive order along au of these arcs. We now separate the argument into two cases according to whether or not Ln > 1. Suppose that Ln > 1 for infinitely many n and take a further ~ubsequence of n such that Ln > 1 along that subsequence. Then kn I Ln and hnl Ln are both bounded by 1. It follows from Lemma 5.3 that after taking a further subsequence, if necessary, there are harmonic functions h and k on U
4.
981
990 such that L;; 1hn
G. F. LAWLER, O. SCHRAMM AND W. WERNER 0
0/;; --+ h and L;; 1kn 0/;; --+ k uniformly on compact subsets 0
of 1U (appropriately interpreted, since hn and kn are only defined on vertices and dual vertices, not on every point of Dn). Moreover, (5.4) shows that hand k are harmonic conjugates, because the discrete Cauchy-Riemann equations tend to the continuous Cauchy-Riemann equations. By Lemma 5.4, it follows that k is, respectively, equal to 0 and 1 in the relative interior of AD, Ai, and similarly h has boundary values 0 and l/i in Ao and AI, where i := limn - H1o Ln (where the limit is along the subsequence and must exist and be finite). By Schwarz reflection, say, this implies that hand k satisfy Neumann boundary conditions in AD U Ai and Ao U AI, respectively. It now easily follows (e.g., from the maximum principle) that h + ik is the (unique) conformal map taking 1U to the rectangle [0,1/ i] x [0, 1] which takes the four arcs Ao, AD, AI, Ai to the corresponding sides of the rectangle. The argument in the case where Ln :::: 1 for infinitely many n proceeds in the same manner, except that one should not divide hn 0 0/;; and kn 0 0/;; by Ln. It remains to remove the assumption that the distance between Ao and A 1 is at least E. Observe that the probabilistic description of h shows that it is monotone increasing in Al and monotone decreasing in Ao. Take E' > 0 much smaller than E. Then h(0) for the given configuration is bounded from above by the value of h(0) for the configuration where arcs of length E' are removed at the two ends of AI, and Al is adjusted accordingly. Similarly, h(O) is bounded from below by the value of h(O) for the configuration where such arcs are removed at the two ends of Ao. The difference between the value of h for the original versus any of the modified configurations goes to 0 as E' --+ 0, since h depends continuously on (AI, A2), as long as the length of Al U A2 is not O. Consequently, we get the proposition by applying the restricted version proved above with E' in place of E and by "sandwiching." D
6. Other lattices. For convenience and simplicity, the proofs up to now have been written for the loop-erased random walk and UST Peano curve on the square grid. The purpose of this section is to briefly indicate how to adapt the proofs to more general walks on more general grids. In order to keep this section short, we will not try to consider the most general cases. Let L be a (strictly two-dimensional) lattice in ]R2; that is, L is a discrete additive subgroup of]R2 that is not contained in a line. Discrete means that there is some neighborhood of 0 whose intersection with L is {O}. Suppose that G is a planar graph whose vertices are the elements of L, and G is invariant under translation by elements of L. That is, if u, VEL are neighbors in G and.e E L, then.e + u neighbors .e + v. It is not hard to verify that there is a linear map taking L to the triangular lattice such that neighbors in G are mapped to vertices at distance 1. In particular, as a graph, G is isomorphic to the triangular grid or to the square grid. 982
CONFORMAL INVARIANCE
991
Let N be the set of neighbors of 0 in G and let N' := {O} U N. Let X be an N'-valued random variable and let Xl, X2, ... be an i.i.d. sequence where each Xn has the same law as X. Consider the random walk n
Sn:= LXj j=l
on G. We are interested in the situation where the scaling limit of Sn is standard Brownian motion. For this purpose, we require that E[X] = 0 and that the covariance matrix of X is the identity matrix. (Note that if the covariance matrix of X is nondegenerate but not equal to the identity, we can always apply a linear transformation to the system to convert to the above situation. Therefore, what we say below also applies in that case, provided that we appropriately modify the linear complex structure on JR.2 .) Note that, under these assumptions, the Markov chain corresponding to the walk Sn does not need to be reversible. An interesting particular example the reader may wish to keep in mind is where P[X = exp(2nij /3)] = 1/3 for j = 0,1,2. THEOREM 6.1. walk Sn.
Theorem 1.1 applies to the loop erasure of the random
PROOF. An inspection of the proof of Theorem 1.1, including all the necessary lemmas, shows that only the generalization of the proof of Lemma 5.1 to the present framework requires special justification, which is given below. D LEMMA 6.2. Let Tr denote the first time n with ISnl ::: r. There exists a constant C, depending on X but not on r, such that, for all r ::: C, wEN and y EL,
Here, pw denotes the law of the Markov chain started from w; that is, the law of (Sn + w:n EN) under P = pO. This lemma is clearly sufficient to provide the necessary analog of Lemma 5.1 for Sn. PROOF OF LEMMA 6.2. There are various ways to prove the lemma (via coupling, for instance). We give here a proof based on the Green's functions, as in [25]. Without loss of generality, we assume that P[ X = 0] > 0 and that L is the minimal lattice containing {w EN: Pw > O}. Then the random walk is irreducible on L. The discrete Laplacian t1x associated with X is defined by t1xf(z):= E[f(z
+ X)] -
fez).
Let a be the potential kernel for the random walk, 00
a(z) := L(po[Sj = 0] - pO[Sj = -z]). j=O 983
992
G. F. LAWLER, O. SCHRAMM AND W. WERNER
It is known that the series converges and, in fact, a(z) = CI log Izl
(6.1)
+ C2 + O(lzl- I )
as Iz I ---+ 00, Z E L (where CI , C2 depend on the law of X). This is proved in [12] for the lattice:;z2 with arbitrary nondegenerate covariance matrix (with an appropriate dependence on the matrix), so the above follows for other L by applying a linear transformation. Since pO[Sj = -z] = PZ[Sj = 0], it follows that 1,
~xa(z) = / 0,
z=o, z#o.
Let G r denote the Green's function for the walk in L n rlU, that is, Gr(z, z') := LjEN pZ[j < fr, Sj = z']. Note that, for all z, W E L n rlU, (6.2)
a(z - w)
+ Gr(z, w) =
EZ[a(Srr - w)],
since for fixed w both sides are ~x-harmonic for z E L n rlU and equality holds for z E L \rlU. SetM:= max{lwl: wEN} and Z = {z E L :r/2 ~ Izl < M +r/2}. By (6.1) and (6.2), Gr(z, w) = CI log 2 + O(r-I) for z E Z and wEN'. The same argument applied to the rev~rse walk -Sj, which has potential kernel a(z) = a( -z) and Green's function Gr(z, w) = Gr(w, z), gives (6.3)
V wEN', Vz E Z,
Gr(w, z) = CI log 2 + O(r-I).
Assuming r > 4M, by considering the last vertex in Z visited by the walk before time fr, we obtain, for all wEN' and all y E L, PW[ Srr = y] =
L
Gr(w, Z)PZ[Srr = y, minU ~ 1: Sj E Z} > fr].
ZEZ
Together with (6.3), this completes the proof of the lemma.
D
Observe that Theorem 1.1 also holds for the simple random walk on the honeycomb grid, because two steps on the honeycomb lattice are the same as a single step on a triangular grid containing every other vertex on the honeycomb grid, and so Lemma 6.2 may be applied. We now tum our attention to spanning trees and the generalizations of Corollary 1.2 and Theorem 1.3. Suppose that X and -X have the same distribution, so that the walk S is reversible. For an edge e = [x, y], define Pe = P[X = y - x] = P[X = x - y]. In this case, it is easy to generalize the definition of UST to a measure on trees related to the law of X. This can be done either by using Wilson's algorithm or, equivalently, by giving to each tree T a probability that is proportional to the product of the transition probabilities along the edges of T. In other words, P[T] = Z-l fleET Pe, where Z is a normalizing constant. (The equivalence is proved in [44]; see also [26].) We call this the UST corresponding to the walk S (even if this probability measure is not uniform). Note that Lemma 4.1 also holds in the present setting because the probability P[T] is given in terms of a product. 984
CONFORMAL INVARIANCE
993
THEOREM 6.3. Assuming that -x has the same distribution as X (i.e., Sj is reversible), Corollary 1.2 and Theorem 1.3 hold for the UST corresponding to the walk S. PROOF. The proof of Corollary 1.2 holds in this generality. In the proof of Theorem 1.3, the only significant changes concern the discrete harmonic conjugate function, used in the proof of Proposition 4.2. Recall that there is an appropriate definition for the discrete harmonic conjugate for reversible walks on planar graphs, where the discrete Cauchy-Riemann equation is modified (see [7] or [19], Section 6.1). If G is graph isomorphic to the square grid, the same is true for the dual graph. If G is graph isomorphic to the triangular grid, then the dual is graph isomorphic to the honeycomb grid. As pointed out above, Lemma 6.2 may therefore be applied to the harmonic conjugate. The details are left to the reader. D
In the nonreversible setting, instead of a spanning tree, one should consider a spanning arborescence, which is an oriented tree with a root and the edges are oriented toward the root. Fix a finite Markov chain with state space V and a root 0 E V. Consider the measure on spanning arborescences of V with root 0, where the probability for T is proportional to the product of the transition probabilities along the directed edges of T. This is the analog of the UST in the nonreversible setting. Wilson's algorithm holds in this generality (see [44]); however, the choice of the root 0 clearly matters. If we consider a finite piece of the lattice L, and we wire part or all of the boundary, it is natural to pick the wired vertex as the root. With this convention, Corollary 1.2 holds for the wired tree. It would be interesting to see if the free tree with root chosen at E D is invariant under conformal maps preserving 0, say (in the nonreversible setting). Of course, one needs to choose a grid approximation of D where there is an oriented path from each vertex to the root 0. In the proof of Theorem 1.3, we have used reversibility in two places. The proofs of Theorems 10.7 and 11.1 of [38], which we quoted, currently require reversibility. However, these results were only used to improve the topology of convergence to SLE. More seriously, Section 5.4 uses the conjugate harmonic function, whose definition in the nonreversible setting is not clear. Notwithstanding the obstacles, it seems likely that these results can be proven in the nonreversible setting, too.
°
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[3] AIZENMAN, M., BURCHARD, A., NEWMAN, C. M. and WILSON, D. B. (1997). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319-367. [4] BENJAMINI, I. and SCHRAMM, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219-1238. [5] CARDY, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25 L201-L206. [6] COLLATZ, L. (1960). The Numerical Treatment of Differential Equations, 3rd ed. Springer, Berlin. [7] DEHN, M. (1903). iller die Zerlegung von Rechtecken in Rechtecke. Math. Ann. 57314--332. [8] DUBINS, L. E. (1968). On a theorem of Skorohod. Ann. Math. Statist. 392094--2097. [9] DUDLEY, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA. [10] DUPLANTIER, B. (1987). Critical exponents of Manhattan Hamiltonian walks in two dimensions, from Potts and o(n) models. J. Statist. Phys. 49411-431. [11] DUPLANTIER, B. (1992). Loop-erased self-avoiding walks in two dimensions: Exact critical exponents and winding numbers. Phys. A 191516-522. [12] FOMIN, S. (2001). Loop-erased walks and total positivity. Trans. Amer. Math. Soc. 353 3563-3583. [13] FUKAI, Y. and UCHIYAMA, K. (1996). Potential kernel for two-dimensional random walk. Ann. Probab. 24 1979-1992. [14] GUTTMANN, A. and BURSILL, R. (1990). Critical exponent for the loop-erased self-avoiding walk by Monte-Carlo methods. J. Statist. Phys. 59 1-9. [15] HAGGSTROM, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59267-275. [16] HE, Z.-X. and SCHRAMM, O. (1998). The Coo-convergence of hexagonal disk packings to the Riemann map. Acta Math. 180219-245. [17] KARATZAS, I. and SHREVE, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York. [18] KASTELEYN, P. W. (1963). A soluble self-avoiding walk problem. Physica 291329-1337. [19] KENYON, R. (1998). Tilings and discrete Dirichlet problems. Israel J. Math. 10561-84. [20] KENYON, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239-286. [21] KENYON, R. (2000). Long-range properties of spanning trees. J. Math. Phys. 411338-1363. [22] KENYON, R. (2000). Conformal invariance of domino tilings. Ann. Probab. 28 759-795. [23] KOZMA, G. (2002). Scaling limit of loop-erased random walks: A naive approach. Unpublished manuscript. [24] LAWLER, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47655-693. [25] LAWLER, G. F. (1991). Intersections of Random Walks. Birkhauser, Boston. [26] LAWLER, G. F. (1999). Loop-erased random walk. In Perplexing Problems in Probability 197-217. Birkhauser, Boston. [27] LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2001). Values of Brownian intersection exponents I. Half-plane exponents. Acta Math. 187 237-273. [28] LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2001). Values of Brownian intersection exponents II. Plane exponents. Acta Math. 187275-308. [29] LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179-201. [30] LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2002). One-arm exponent for critica12D percolation. Electron. J. Probab. 7 No.2. [31] LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2003). Conformal restriction properties: The chordal case. J. Amer. Math. Soc. 16917-955. 986
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[32] LYONS, R. (1998). A bird's-eye view of uniform spanning trees and forests. In Microsurveys in Discrete Probability 135-162. Amer. Math. Soc., Providence, RI. [33] MAJUMDAR, S. N. (1992). Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions. Phys. Rev. Lett. 682329-2331. [34] PEMANTLE, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 191559-1574. [35] POMMERENKE, CH. (1992). Boundary Behaviour of Conformal Maps. Springer, Berlin. [36] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin. [37] ROHDE, S. and SCHRAMM, O. (2001). Basic properties of SLE. Unpublished manuscript. [38] SCHRAMM, O. (2000). Scaling limits ofloop-erased random walks and uniform spanning trees. Israel J. Math. 118221-288. [39] SCHRAMM, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115-120. [40] SMIRNOV, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sir. I Math. 333 239-244. [41] SMIRNOV, S. (2001). Critical percolation in the plane. I. Conformal invariance and Cardy's formula. II. Continuum scaling limit. Preprint. [42] SMIRNOV, S. and WERNER, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8729-744. [43] STRASSEN, V. (1967). Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 315-343. Univ. California Press. [44] WILSON, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing 296-303. ACM, New York. G. LAWLER
O. SCHRAMM
DEPARTMENT OF MATHEMATICS
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987
Annals of Mathematics, 161 (2005), 883-924
Basic properties of SLE By
STEFFEN ROHDE*
and
ODED SCHRAMM
Dedicated to Christian Pommerenke on the occasion of his 70th birthday
Abstract SLEt<; is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed K,. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all K, -I- 8 the SLE trace is a path; for K, E [0,4] it is a simple path; for K, E (4,8) it is a self-intersecting path; and for K, > 8 it is space-filling. It is also shown that the Hausdorff dimension of the SLEt<; trace is almost surely (a.s.) at most 1 + K,/8 and that the expected number of disks of size E needed to cover it inside a bounded set is at least E-(1+t<;/8)+o(1) for K, E [0,8) along some sequence E '\. O. Similarly, for K, 2': 4, the Hausdorff dimension of the outer boundary of the SLEt<; hull is a.s. at most 1 + 2/ K" and the expected number of disks of radius E needed to cover it is at least E-(1+2/t<;)+o(1) for a sequence E '\. o. 1. Introduction
Stochastic Loewner Evolution (SLE) is a random process of growth of a set K t . The evolution of the set over time is described through the normalized conformal map gt = gt(z) from the complement of K t . The map gt is the solution of Loewner's differential equation with driving parameter a onedimensional Brownian motion. SLE, or SLEt<;, has one parameter K, 2': 0, which is the speed of the Brownian motion. A more complete definition appears in Section 2 below. The SLE process was introduced in [SchOO]. There, it was shown that under the assumption of the existence and conformal invariance of the scaling limit of loop-erased random walk, the scaling limit is SLE2 . (See Figure 9.1.) It was also stated there without proof that SLE6 is the scaling limit of the *Partially supported by NSF Grants DMS-0201435 and DMS-024440S.
I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_31, 989
884
STEFFEN ROHDE AND ODED SCHRAMM
Figure 1.1: The boundary of a percolation cluster in the upper half plane, with appropriate boundary conditions. It converges to the chordal SLE6 trace. boundaries of critical percolation clusters, assuming their conformal invariance. Smirnov [SmiOl] has recently proved the conformal invariance conjecture for critical percolation on the triangular grid and the claim that SLE6 describes the limit. (See Figure 1.1.) With the proper setup, the outer boundary of SLE6 is the same as the outer boundary of planar Brownian motion [LSW03] (see also [WerOl]). SLE8 has been conjectured [SchOO] to be the scaling limit of the uniform spanning tree Peano curve (see Figure 9.2), and there are various further conjectures for other parameters. Most of these conjectures are described in Section 9 below. Also related is the work of Carleson and Makarov [CMOl], which studies growth processes motivated by DLA via Loewner's equation. SLE is amenable to computations. In [SchOO] a few properties of SLE have been derived; in particular, the winding number variance. In the series of papers [LSWOla], [LSWOlb], [LSW02]' a number of other properties of SLE have been studied. The goal there was not to investigate SLE for its own sake, but rather to use SLE6 as a means for the determination of the Brownian motion intersection exponents. As the title suggests, the goal of the present paper is to study the fundamental properties of SLE. There are two main variants of SLE, chordal and radial. For simplicity, we concentrate on chordal SLE; however, all the main results of the paper carryover to radial SLE as well. In chordal SLE, the set K t , t 2': 0, called the SLE hull, is a subset of the closed upper half plane 1HI and gt : 1HI \ K t ---+ 1HI is the conformal uniformizing map, suitably normalized at infinity. We show that with the possible exception of /'i, = 8, a.s. there is a (unique) continuous path , : [0, 00) ---+ 1HI such that for each t > 0 the set K t is the union of ,[0, t] and the bounded connected components of 1HI \ ,[0, t]. The path , is called the SLE trace. It is shown that limt-HXl !r(t) I = 00 a.s. We also 990
BASIC PROPERTIES OF SLE
885
describe two phase transitions for the SLE process. In the range K, E [0,4], a.s. K t = ,[a, t] for every t 2 a and, is a simple path. For K, E (4,8) the path , is not a simple path and for every z E 1HI a.s. z tf:- ,[0,(0) but z E Ut>o K t . Finally, for K, > 8 we have 1HI = ,[0,(0) a.s. The reader may wish to examine Figures 9.1, 1.1 and 9.2, to get an idea of what the SLE/i; trace looks like for K, = 2, 6 and 8, respectively. We also discuss the expected number of disks needed to cover the SLE/i; trace and the outer boundary of K t . It is proved that the Hausdorff dimension of the trace is a.s. at most 1 + K,/8, and that the Hausdorff dimension of the outer boundary aKt is a.s. at most 1 + 2/ K, if K, 2 4. For K, E [0,8), we also show that the expected number of disks of size c needed to cover the trace inside a bounded set is at least c-(1+/i;/8)+o(1) along some sequence c '\. O. Similarly, for K, 2 4, the expected number of disks of radius c needed to cover the outer boundary is at least c-(1+2//i;)+o(1) for a sequence of c '\. o. Richard Kenyon has earlier made the conjecture that the Hausdorff dimension of the outer boundary is a.s. 1 + 2/ K,. These results offer strong support for this conjecture. It is interesting to compare our results to recent results for the deterministic Loewner evolution, i.e., the solutions to the Loewner equation with a deterministic driving function ((t). In [MR] it is shown that if ( is Holder continuous with exponent 1/2 and small norm, then K t is a simple path. On the other hand, there is a function Holder continuous with exponent 1/2 and having large norm, such that K t is not even locally connected, and therefore there is no continuous path, generating K t . In this example, K t spirals infinitely often around a disk D, accumulating on aD, and then spirals out again. It is easy to see that the disk D can be replaced by any compact connected subset of 1HI. Notice that according to the law of the iterated logarithm, a.s. Brownian motion is not Holder continuous with exponent 1/2. Therefore, it seems unlikely that the results of the present paper can be obtained from deterministic results. Our results are based on the computation and estimates of the distribution of 19~(z)1 where z E 1HI. Note that in [LSWOlb] the derivatives 9~(X) are studied for x E R The organization of the paper is as follows. Section 2 introduces the basic definitions and some fundamental properties. The goal of Section 3 is to obtain estimates for quantities related to E[19~(z)laJ, for various constants a (another result of this nature is Lemma 6.3), and to derive some resulting continuity properties of 9;:1. Section 4 proves a general criterion for the existence of a continuous trace, which does not involve randomness. The proof that the SLE/i; trace is continuous for K, i- 8 is then completed in Section 5. There, it is also proved that 9;:1 is a.s. Holder continuous when K, i- 4. Section 6 discusses the two phase transitions K, = 4 and K, = 8 for SLE/i;. Besides some quantitative
e,
991
886
STEFFEN ROHDE AND ODED SCHRAMM
properties, it is shown there that the trace is a.s. a simple path if and only if '" E [0,4], and that the trace is space-filling for", > 8. The trace is proved to be transient when", 0; 8 in Section 7. Estimates for the dimensions of the trace and the boundary of the hull are established in Section 8. Finally, a collection of open problems is presented in Section 9. Update. Since the completion and distribution of the first version of this paper, there has been some further progress. In [LSW] it was proven that the scaling limit of loop-erased random walk is SLE 2 and the scaling limit of the UST Peano path is SLEs . As a corollary of the convergence of the UST Peano path to SLEs , it was also established there that SLEs is generated by a continuous transient path, thus answering some of the issues left open in the current paper. However, it is quite natural to ask for a more direct analytic proof of these properties of SLEs. Recently, Vincent Beffara [Be~ has announced a proof that the Hausdorff dimension of the SLEII; trace is 1 + ",/8 when 4 0; '" ::; 8. The paper [SS] proves the convergence of the harmonic explorer to SLE4 .
2. Definitions and background 2.1. Chordal SLE. Let B t be Brownian motion on JR, started from Bo = O. For '" ~ 0 let ~(t) := yl"fiBt and for each Z E 1HI \ {O} let 9t(Z) be the solution of the ordinary differential equation
90(Z) = z.
(2.1)
The solution exists as long as 9t(Z) - ~(t) is bounded away from zero. We denote by T(Z) the first time T such that 0 is a limit point of 9t(Z) - ~(t) as t / T. Set H t :=
{z
E
1HI: T(Z)
> t},
K t :=
{z
E
1HI: T(Z)::;
t}.
It is immediate to verify that K t is compact and H t is open for all t. The parametrized collection of maps (9t : t ~ 0) is called chordal SLEII;' The sets K t are the hulls of the SLE. It is easy to verify that for every t ~ 0 the map 9t : H t ----+ 1HI is a conformal homeomorphism and that H t is the unbounded component of 1HI \ K t . The inverse of 9t is obtained by flowing backwards from any point w E 1HI according to the equation (2.1). (That is, the fact that 9t is invertible is a particular case of a general result on solutions of ODE's.) One only needs to note that in this backward flow, the imaginary part increases, hence the point cannot hit the singularity. It also cannot escape to infinity in finite time. The fact that 9t(Z) is analytic in Z is clear, since the right-hand side of (2.1) is analytic in 9t(Z). 992
BASIC PROPERTIES OF SLE
887
The map gt satisfies the so-called hydrodynamic normalization at infinity: lim gt (z) -
(2.2)
z---+oo
Z
= 0.
Note that this uniquely determines gt among conformal maps from H t onto 1HI. In fact, (2.1) implies that gt has the power series expansion 2t (2.3) gt(z) = Z + - + ... , Z ----+ 00. Z
The fact that gt i- gt' when t' > t implies that K t , i- K t . The relation K t , ::J K t is clear. Two important properties of chordal SLE are scale-invariance and a sort of stationarity. These are summarized in the following proposition. (A similar statement appeared in [LSW01a].) (i) SLEI< is scale-invariant in the following sense. Let K t be the hull of SLEI<' and let a > O. Then the process t f----+ a- 1 / 2 Kat has the same law as t f----+ K t . The process (t, z) f----+ a- 1 / 2 gat ( y'az) has the same law as the process (t, z) f----+ gt(z). (ii) Let to > O. Then the map (t, z) f----+ jit(z) := gHt o o g;;; 1 (z+~(to)) -~(to) has the same law as (t, z) f----+ gt(z); moreover, (jit)t>o is independent from (gt)o
The scaling property easily follows from the scaling property of Brownian motion, and the second property follows from the Markov property and translation invariance of Brownian motion. One just needs to write down the expression for atiit. We leave the details as an exercise to the reader. The following notations will be used throughout the paper. +
Jt
-1
:= gt
,
The trace, of SLE is defined by
,(t)
:= lim It(z) , z---+o
where z tends to 0 within 1HI. If the limit does not exist, let ,(t) denote the set of all limit points. We say that the SLE trace is a continuous path if the limit exists for every t and ,(t) is a continuous function of t. Radial SLE. Another version of SLEI< is called radial SLEI<' It is similar to chordal SLE but appropriate for the situation where there is a distinguished point in the interior of the domain. Radial SLEI< is defined as follows. Let B(t) be Brownian motion on the unit circle aU, started from a uniform-random point B(O), and set ~(t) := B(K,t). The conformal maps gt are defined as the solution of gt(z) + ~(t) atgt(z) = -gt(z) gt(z) _ ~(t) , go(z) = z, 2.2.
993
888
STEFFEN ROHDE AND ODED SCHRAMM
for Z E 1[J. The sets K t and H t are defined as for chordal SLE. Note that the scaling property 2.1.(i) fails for radial SLE. Mainly due to this reason, chordal SLE is easier to work with. However, appropriately stated, all the main results of this paper are valid also for radial SLE. This follows from [LSWOl b, Prop. 4.2], which says in a precise way that chordal and radial SLE are equivalent. 2.3. Local martingales and martingales. The purpose of this subsection is to present a slightly technical lemma giving a sufficient condition for a local martingale to be a martingale. Although we have not been able to find an appropriate reference, the lemma must be known (and is rather obvious to the experts). See, for example, [RY99 , §IV.l] for a discussion of the distinction between a local martingale and a martingale. While the stochastic calculus needed for the rest of the paper is not much more than familiarity with Ito's formula, this subsection does assume a bit more. LEMMA 2.2. Let B t be stardard one dimensional Brownian motion, and let at be a progressive real valued locally bounded process. Suppose that X t satisfies
Xt and that for every t >
a there
= i t asdBs ,
is a finite constant c( t) such that a; ::; c(t)X;
(2.4)
+ c(t)
for all s E [0, t] a.s. Then X is a martingale. Proof We know that X is a local martingale. Let M > a be large, and let T := inf{t : JXtJ 2:: M}. Then yt := Xt/\T is a martingale (where t 1\ T = min{t, T}). Let f(t) := E[Y?J. Ito's formula then gives f(t') Our assumption a; ::; c(t)X; (2.5)
=E
[i
+ c(t)
f(t') ::; c(t) t'
t'
a; ls
therefore implies that for t' E [0, t]
+ c(t) itl f(s) ds.
This implies f(s) < exp(2c(t)s) for all s E [O,t], since (2.5) shows that t' cannot be the least s E [O,t] where f(s) 2:: exp(2c(t)s). Thus,
E [(X, X)t/\T J = E [(Y, Y)tJ = E [~2J =
f(t) < exp(2 c(t) t) .
Taking M ----7 00, we get by monotone convergence E [(X, X)tJ ::; exp(2 c(t) t) < 00. Thus X is a martingale (for example, by [RY99 , IV. 1.25]). D 994
889
BASIC PROPERTIES OF SLE
3. Derivative expectation In this section, gt is the SLEt;; flow; that is, the solution of (2.1) where ~(t) := B(K,t), and B is standard Brownian motion on lR starting from o. Our only assumption on K, is K, > O. The goal of the section is to derive bounds on quantities related to E [Ig~ (z W]. Another result of this nature is Lemma 6.3, which is deferred to a later section.
3.1. Basic derivative expectation. We will need estimates for the moments of 11;1. In this subsection, we will describe a change of time and obtain derivative estimates for the changed time. For convenience, we take B to be two-sided Brownian motion. The equation (2.1) can also be solved for negative t, and gt is a conformal map from 1HI into a subset of 1HI when t < O. Notice that the scale invariance (Proposition 2.1.(i)) also holds for t < o. 3.1. For all fixed t E lR the map z bution as the map z f--t it(z) - ~(t). LEMMA
f--t
g-t (z) has the same distri-
Proof Fix tl E lR, and let
(3.1) Then
~h :
lR
--t
lR has the same law as
~
: lR
--t
R Let
fft(z) :=9h+togt;:1(z+e(td) -e(tl), and note that ffo(z)
= z and ff-t, (z) =
8 tgt = fft A
it, (z) -
2 + ~(td - ~(t + tl)
e(tl). Since 2 fft - edt) ,
the lemma follows from (2.1).
D
Note that (2.1) implies that Im(gt(z)) is monotone decreasing in t for every z E 1HI. For z E 1HI and u E lR set
(3.2) We claim that for all z E 1HI a.s. Tu
i- ±oo.
By (2.1),
18t gt (z)1 = 2Igt(z) - ~(t)l-l. Setting ~(t) := sup{I~(s)1 : s E [O,tn, the above implies that Igt(z)1 :::; Izl ~(t) + 20 for t < T(Z), since 8t lgt (z)1 :::; 2/(lgt(z)1 - ~(t)) whenever Igt(z)1 ~(t). Setting Yt := Imgt(z), we get from (2.1), -8dog Yt ~ 2 (Izl
+ 2 ~(t) + 20)-2 .
995
+ >
890
STEFFEN ROHDE AND ODED SCHRAMM
The law of iterated logarithms implies that the right-hand side is not integrable over [0,(0) nor over (-00,0]. Thus, ITul < 00 a.s. We will need the formula '( )1
at log 1gt
(3.3)
Z
= Re (azEh9t(Z)) gi(z) = Re ( gt'( z )-1 az gt(z) 2_ ~(t) ) = -2Re( (gt(z) - ~(t)r2) .
Set u = u(z, t) := log Imgt(z). Observe that (2.1) gives
(3.4) and (3.3) gives
(3.5)
Fix some
z = x + if) E H.
For every u E JR, let
and
x(u)
:=
Re(z(u)),
y(u)
:=
Im(z(u)) = exp(u).
3.2. Let z = x + if) E H as above. Assume that f) =I- 1, and set v := -sign(1ogf)). Let b E R Define a and oX by THEOREM
(3.6)
a:= 2b+ vK;b(l- b)/2,
oX :=
4b + v K;b (1 - 2b)/2.
Set Then
(3.7) Before we give the short proof of the theorem, a few remarks may be of help to motivate the formulation and the proof. Our goal was to find an expression for
E[lg~o(z)(z)la].
It turns out to be more convenient to consider
The obvious strategy is to find a differential equation which F must satisfy and search for a solution. The first part is not too difficult, and proceeds as follows. Let Fu denote the (J" - field generated by (~( t) : (t - Tu) v 2: 0). Note that the 996
891
BASIC PROPERTIES OF SLE
strong Markov property for and it := log y,
~
and the chain rule imply that for u between 0
(3.8) Hence, the right-hand side is a martingale. Observe that
dx =
(3.9)
2 x dt 2 2 X +y
-
d _ - 2 y dt y - 2 + 2' x y
d~,
4 y2 dt 2)2 x +y
dlog'ljJ =
(2
.
(The latter easily follows from (3.3) and (3.4).) We assume for now that F is smooth. Ito's formula may then be calculated for the right-hand side of (3.8). Since it is a martingale, the drift term of the Ito derivative must vanish; that is, where
(3.10) (The -v factor comes from the fact that t is monotone decreasing with respect to the filtration Fu if and only if v = 1.) Guessing a solution for the equation AG = 0 is not too difficult (after changing to coordinates where scale invariance is more apparent). It is easy to verify that
F(x
+ iy) =
Fb,>..(X + iy)
:=
(1 + (x/y)2) by>",
satisfies AF = O. Unfortunately, F does not satisfy the boundary values F = 1 for y = 1, which hold for F. Consequently, the theorem gives a formula for F, rather than for F. (Remark 3.4 concerns the problem of determining F.) Assuming that F is C 2 , the above derivation does apply to F, and shows that AF = O. However, we have not found a clean reference to the fact that FE C 2 . Fortunately, the proof below does not need to assume this. Proof of Theorem 3.2.
Note that by (3.4)
du =
-2Izl- 2 dt.
Let
Then
B is
a local martingale and
d(B) =
-2Izl- 2 dt = duo
Consequently, B(u) IS Brownian motion (with respect to u). 'ljJ(u)a F(z(u)). Ito's formula gives
dMu =
-
2M
2
bx
x +y
;n2 d~ = V 2 K, M
997
bx
Jx2 +y2
dB. A
Set Mu .-
892
STEFFEN ROHDE AND ODED SCHRAMM
Thus M is a local martingale. In fact, Lemma 2.2 then tells us that M is a martingale. Consequently, we have
(3.11) and the theorem clearly follows.
D
In Section 8 we will estimate the expected number of disks needed to cover the boundary of K t . To do this, we need the following lower bound on the expectation of the derivative.
3.3. Let", > 0 and b < ('" + 4)/(4",), and define a and.x by (3.6) 1. Then there is a constant c = c( "', b) > 0 such that
LEMMA
with
1/
=
E[Ij{(2W 1rm (il(z»:;:C] 2: c(1 + (x/y?)b yA-a holds for every
x
2 = x + iy E lHI
satisfying
121 :s; c.
Proof As before, let u := logy. Set v = v(u) := sinh-l(x/y); that is, Then Ito's formula gives
= ysinh(v).
The formula for dv in terms of Band du is ~
(3.12)
X
A
dv= y",/2dB- 41z1 (8+"'1/)du.
Recall that 'lj; (u) a F (z (u)) is a martingale, since F probability measure by
= F.
Define a new
for every event A. This is the so-called Doob-transform (or h-transform) corresponding to the martingale 'lj;(u)a F(z(u)). Recall that if a(w) is a positive martingale for a diffusion process dw = ql(W, t) dB(t)+q2(W, t) dt, t :s; tl, where B(t) is Brownian motion, then for t < tl, dw = ql(W, t) dB(t) + q2(W, t) dt + ql (w, t) 8w log a( w) ql (w, t) dt, where B is Brownian motion with respect to the probability measure weighted by a(w(tl)); that is, the Doob transform of a. This follows, for example, from Girsanov's Formula [Dur96, §2.12]. We apply this with w = (v, 'lj;) and u as the time parameter (in this case, ql and q2 are vectors and 8w loga(w) is a linear functional), and get by (3.12)
dv
=
~
-
y ",/2 dB -
1
x
-1-1 (8 + "') du + - '" 8v log F du , 4 z 2 998
893
BASIC PROPERTIES OF SLE
where B(u) is Brownian motion under
dv =
(3.13)
P.
This simplifies to
J 1'1,/2 dB - b tanh(v) du,
where
Thus, v(u) is a very simple diffusion process. When Ivl is large, tanh(v) is close to sign(v), and v has almost constant drift -bsign(v), pushing it towards o. At this point, do not assume 121 ::; c, but only 121 ::; 1. Let W : [il,O] ----t lR be the continuous function that is equal to 1 at 0, equal to lill + 1 at il, has slope - (b 1\ 2 + 1) /2 in the interval [il, il/2] and has constant slope in the interval [il/2,0], and let A be the event
A
:=
{\fu
E
[il,O],
Iv(u)l::; W(u)} .
Note that our assumption 121 ::; 1 implies that Ivl ::; lill + log 2. Since b > 1 it easily follows from (3.13) that there is a constant Cl = cl(b, 1'1,) > 0 such that P[A] > Cl. In particular, Cl does not depend on 2. This means
However, note that v(O) and hence x(O) are bounded on A. Therefore, there is some constant C2 > 0 such that
(3.14) We now estimate To on the event A. Recall that To ::; O. From (3.4) we have ouTu = -lzI 2 /2 and therefore on A
To = -
1°
1
> -~ _ ~
-
1°
y2 x2 -du-du= -y2 - -1 - -1 sinh(v)2e 2U du '112 '112 442'11 0
4
2
Juf
O 2 e \[1(u)+2u
du > -C3
-,
where C3 = c3(b,K,) < 00 is some constant. That is, we have To E [-C3,0] on A. On the event To 2': -C3, we clearly have 1m (g-C3 (2)) 2': 1 and also Ig~o (2) (2) by (3.3). Consequently, Lemma 3.1 and (3.14) give
I ::; Il-C3 I,
(3.15)
E [lj~3 (2) la lIm(!c3(z))~1] 2':
C4
(1 + (x/y)2) b yA-a .
This is almost the result that we need. However, we want to replace C3 by 1. For this, we apply scale invariance. In this procedure, the assumption 121 ::; 1 needs to be replaced by 121 ::; 1/.JC3. The proof of the lemma is now complete. D 999
894
STEFFEN ROHDE AND ODED SCHRAMM
°
It is not too hard to see that for every constant C > the statement of the lemma can be strengthened to allow c < 121 < C. The constant c must then also depend on C.
Remark 3.4. Suppose that we take
b=~+211 4
/'1,
and define a and A using (3.6). Define P as in the proof of the lemma. Then as the proof shows, v becomes Brownian motion times /'1,/2 under P, since b vanishes. This allows a detailed understanding of the distribution of Ig~o(2)1. For example, one can determine in this way the decay of P['lf'!(O) > 1/2] as f) '\. 0. It also follows that for such a, b, A, and every A C ~ one can write
v'
[Ig~o (2) la 1x(a)EA] , since for
down an explicit expression for E have
A((1 + (x/y)2)b yA
~
exp (va - v)2)
yllU
every Va E
~ we
= 0.
lI/'1,U
This equation is the PDE facet of the fact that v is Brownian motion times under P. However, these results will not be needed in the present paper.
.JKJ2
3.2. Derivative upper bounds at a fixed time t. From Theorem 3.2 it is not hard to obtain estimates for 11£1 : COROLLARY 3.5. Let bE [0,1 + 4/'1,-1], and define A and a by (3.6) with 1. There is a constant C(/'1" b), depending only on /'1, and b, such that the following estimate holds for all t E [0,1], y,8 E (0,1] and x E ~.
1I
=
p[I1£(x+iy)1 ~8y-1] ~C(/'1"b)(1+x2/y2)b(y/8)A1J(8,a-A),
(3.16)
where 8- S 1J(8,s)=
{
> 0, s = 0, s < 0.
s
~+IIog81
Proof Note that a ~ 0. We assume that 8 > y, for otherwise the right-hand side is at least C(/'1" b), and we take C(/'1" b) ~ 1. Take z = x + iy. By Lemma 3.1, jHz) has the same distribution as g'-t(z). Let U1 := log Im(g_t(x + iy)). Recall the notation Tu from (3.2). Note that Il-t(z)/g~u(z)(z)1 ~ exp(lu - u11),
(3.17)
la
since u log Ig~(z) II ~ 1, by (3.5). Moreover, it is clear that there is a constant c such that U1 ~ c, because t, y ~ 1. Consequently, a
L
p[ll-t(z)1 ~ 0(1)8y-1] ~ 0(1)
j=ilogyl
1000
P[lgh(z)(z)1 ~ 8y-1].
895
BASIC PROPERTIES OF SLE
The Schwarz lemma implies that ylg'(z)1 ::; Im(g(z)) if 9 : 1HI the above gives
p[lg~t(z)l::::: 0(1)8y- 1J ::; 0(1)
(3.18)
----+
1HI. Therefore,
o
L
P[lg~j(z)(z)l::::: 8y-1J.
j=ilog<51
By scale invariance, we have for all j E [logy, 0]
where F
= Fb is as in Theorem 3.2. Hence
P[lg~j(z)(z)1 ::::: 8y- 1J = P[lgh(z)(zWy a 8- a ::::: 1J ::; E[lgh(z)(zWy a 8- aJ ::; 8- a eja Fb(e-jz). Consequently, by (3.18) and Theorem 3.2,
o
L
p[lg~t(z)1 :::::0(1)8y- 1J ::;0(1)(1+x2/y2)b8- a yA
ej(a-A).
j=ilog<51
The corollary follows.
3.3.
D
Continuity of it(O).
In the deterministic example of nonlocally connected hulls described in the introduction, there is a time to for which limz-to (z) does not exist (the limit set is the outer boundary of the prescribed compact set). Even when the SLE trace is a continuous path, it is not always true that (z, t) f---+ it (z) extends continuously to 1HI x [0, (0) (this is only true for simple paths). The next theorem shows that it(O) = it(~(t)) exists as a radial limit and is continuous. Together with the result of Section 4 below, this is enough to show that the SLE trace is a path.
ito
THEOREM
3.6. Define
H(y, t) If'"
:=
y>O, tE[O,oo).
it(i y),
#- 8, then a.s. H(y, t) extends continuously to
[0,(0) x [0,(0).
Proof Fix", #- 8. By scale invariance, it is enough to show continuity of H on [0, (0) x [0,1). Given j, kEN, with k < 22j , let R(j, k) be the rectangle R(j, k)
:=
[Tj-\ Tj] x [k T 2j, (k + 1) T 2j],
and set
d(j, k)
:=
diamH(R(j, k)) .
Take b = (8 + ",)/(4",) and let a and A be given by (3.6) with v = 1. Note that A> 2. Set 0"0:= (A - 2)/max{a, A}, and let 0" E (0,0"0). Our objective is to 1001
896
STEFFEN ROHDE AND ODED SCHRAMM
prove 00
22j-1
L L
(3.19)
P[d(j,k) ~
Tja]
< 00.
j=O k=O
Fix some such pair (j, k). The idea of the proof is to decompose the time interval [k 2- 2j , (k + 1) 2- 2j ] into (random) sub-intervals using partition points iN, i N- 1, ... , io such that the change of on each interval is controlled. Then it is not hard to control it on each sub-interval. Now, the details. Set to = (k + 1)2-2j . Inductively, let
e
tn+1
:=
sup{t
< tn : le(t) - e(tn)1
= Tj},
and let N be the least n E N such that tN :S to - 2- 2j . Also set too .to - 2- 2j = k 2- 2j and
in
:=
max{ tn, too} .
Observe that the scaling property of Brownian motion shows that there is a constant p < 1, which does not depend on j or k, such that P[N > 1] = p. Moreover, the Markov property implies that P [N ~ m + 1 I N ~ m] :S p for every mEN. In particular, P[N > m] :S pm. For every s E [0,(0), the map is measurable with respect to the (T-field generated by ~(t) for t E [0, s], while in is determined by ~(t) for t ~ tn (i.e., in is a stopping time for the reversed time filtration). Therefore, by the strong Markov property, for every n E N, s E [too, to] and 8 > 0,
is
and consequently,
p[liiJiTj)1 > 8I in> too] :S
sup
sE[O,l]
p[IR(iTj)1 > 8].
This allows us to estimate, P[3n EN:
liiJiTj)1 > 8] 00
:S P [liLJi Tj)1 > 8] + L p[in > too] P [liiJi Tj)1 > 8 I in > too] :S E[N + 1]
n=O
sup P [Ii~(i Tj)1
sE[O,l]
By Corollary 3.5, and because (T
> 8] :S 0(1) sup P [Ii~(i Tj)1 > 8] . sE[O,l]
< (A - 2)/ max{a, A}, this gives
(3.20) for some c:
= C:(/'i;) > 0. Let S be the rectangle S:= {x+iy:
Ixl:S Ti+ 3 , 1002
y
E [Tj-1, Ti+ 3 ]} .
897
BASIC PROPERTIES OF SLE
We claim that N
(3.21 )
H(R(j, k))
C
U fiJ8) ,
n=O
and \In E N
(3.22)
Indeed, to prove (3.21), let t E [in+1' in] and y E [2- j -1, 2- j ]. Then we may write (3.23) We want to prove (3.21) by showing that 9in +1 (It(iy)) - ~(in+1) E 8. Set
cp(s)
:=
gs(lt(iy)) for s:S t. Then cp(t)
=
iy + ~(t) and by (2.1)
oscp(s) = 2 (cp(s) - ~(s)r1 . Note that osIm(cp(s)) < 0, and hence Im(cp(s)) ~ Im(cp(t)) ~ 2- j - 1. This then implies that loscp(s) I :S 2]+2, and therefore It - in+11 :S 2- 2j gives Icp(in+1) - cp(t) I :S 22- j . Since I~(t) - ~(in+1)1 :S 21- j , by (3.23) this gives ft(iy) EftAn+l (8) and verifies (3.21). We also have (3.22), because taking
t = in in the above gives ftA n (iy) EftAn+l (8). Since If;(z)I/Ii:(i2- j )1 is bounded by some constant if z E 8 (this follows from the Koebe distortion theorem, see [Pom92, §1.3]), we find that diam(!t(8)) :S 0(1) Tj 1i£(iTj)l. Therefore, the relations (3.21) and (3.22) give
(3.24)
d(j,k):S
N
N
n=O
n=O
2: diam (liJ8)) :S O(l)2-j 2:lfi)iTj)1
By (3.20), we get
P [d(j, k)
> TjO"] :S P[N > j2] + 0(1)2-2 j Tcj :S
rJ2 + 0(1)2-2 j Tcj
:S 0(1) T 2j Tcj
,
which proves (3.19). From (3.19) we conclude that a.s. there are at most finitely many pairs j, kEN with k :S 22j - 1 such that d(j, k) > 2- jO". Hence d(j, k) :S C(w) 2- j O" for all j, k, where C = C(w) is random (and the notation C(w) is meant to suggest that). Let (y', t') and (y", t") be points in (0,1)2. Let j1 be the smallest integer larger than min { - log2 y', -log2 y", - ~ log2 It' - t" I} . Note that a rectangle R(j1, k') that intersects the line t = t' is adjacent to a rectangle 1003
898
STEFFEN ROHDE AND ODED SCHRAMM
R(j1, k") that intersects the line t
= t". Consequently, IH (y', t') - H (y", t") I ::;
Lj~jl (d(j, kj)+d(j, k'j)) ::; 0(1) C(w) 2- aj1 , where R(j, kj) is a rectangle meet-
ing the line t = t' and R(j, k'j) is a rectangle meeting the line t = t". This shows that for every to E [0,1) the limit of H(y, t) as (y, t) ----t (0, to) exists, and thereby extends the definition of H to [0,00) x [0,1). Since H is clearly D continuous in (0,00) x [0, t), the proof is now complete. For /'i, = 8, we are unable to prove Theorem 3.6. However, a weaker result does follow easily, namely, a.s. for almost every t 2: the limit limy,,"o It(iy) exists.
°
Update. It follows from [LSW] and the results of the current paper that the Theorem holds also when /'i, = 8.
4. Reduction The following theorem provides a criterion for hulls to be generated by a continuous path. In this section we do not assume that ~ is a (time scaled) Brownian motion. THEOREM 4.1. Let ~ : [0,00) ----t lR be continuous, and let 9t be the corresponding solution of (2.1). Assume that f3(t) := limy,,"o 9i1(~(t) + iy) exists for all t E [0,00) and is continuous. Then gil extends continuously to 1HI and H t is the unbounded connected component of 1HI \ f3([0, t]), for every t E [0,00). In the proof, the following basic properties of conformal maps will be needed. Suppose that 9 : D ----t ill is a conformal homeomorphism. If a : [0,1) ----t D is a path such that the limit h = limV1 a(t) exists, then l2 = limt/1 goa exists, too. (However, if a : [0,1) ----t ill is a path such that the above limit exists, it does not follow that limt/1 9- 1 0 a(t) exists. In other words, it is essential that the image of 9 is a nice domain such as ill.) Moreover, limt/1 9- 1 (tl 2 ) exists and equals h. Consequently, if a: [0,1) ----t D is another path with limV1 a existing and with limV19 0 a(t) = limV1 go a(t), then limV1 a(t) = limV1 a(t). These statements are well known and easily established, for example with the notion of extremal length. See [Pom92, Prop. 2.14] for the first statement and [Ahl73 , Th. 3.5] implies the second claim.
Proof Let S(t) c 1HI be the set of limit points of 9i 1(z) as z ----t ~(t) in 1HI. Fix to 2: 0, and let Zo E S(to). We want to show that Zo E f3([0, to)), and hence Zo E f3([0, to]). Fix some c > 0. Let t' := sup{ t E [0, to] : K t n D(zo, c) = 1004
0} ,
BASIC PROPERTIES OF SLE
899
where D(zo, c:) is the open disk of radius c: about zoo We first show that
(3(t')
(4.1)
E
D(zo, c:).
Indeed, D(zo,c:) n Ht a # 0 since zo E 8(to). Let p E D(zo,c:) n H ta , and let p/ E Ktt n D(zo, c:). Let p" be the first point on the line segment from p to p/ which is in Ktt. We want to show that (3(t/) = p". Let L be the line segment [P,p"), and note that L c Htt. Hence gtt(L) is a curve in 1HI terminating at a point x E R If x # ((t/), then gt(L) terminates at points x(t) # ((t) for all t < t/ sufficiently close to t/. Because gT(P") has to hit the singularity ((7) at some time 7 ::; t', this implies p" E K t for t < t/ close to t/. This contradicts the definition of t/ and shows x = t;(t/). Now (3(t/) = p" follows because the conformal map g;;l of 1HI cannot have two different limits along two arcs with the same terminal point. Having established (4.1), since c: > was arbitrary, we conclude that zo E (3([0, to)) and hence zo E (3([0, to]). This gives 8(t) C (3([0, t]) for all
°
t 2: 0. We now show that H t is the unbounded component of 1HI \ UT::;t 8(7). First, H t is connected and disjoint from UT::;t 8(7). On the other hand, as the argument in the previous paragraph shows, 8Ht nlHI is contained in UT::;t 8(7). Therefore, H t is a connected component of 1HI \ UT
5. Continuity We have now established all the results needed to show that the SLE" trace is a continuous path a.s.
°
THEOREM 5.1 (Continuity). Let K, almost surely. For every t 2: the limit
'Y(t):= exists, "1 : [0,(0) of 1HI \ "1([0, t]).
----7
#
lim
z-+O,zElH!
8. The following statements hold It(z)
1HI is a continuous path, and H t is the unbounded component
We believe the theorem to be valid also for K, = 8. (This is stated as Conjecture 9.1.) Despite repeated efforts, the proof eluded us. Update.
This extension to
Proof of Theorem 5.1.
K,
= 8 is proved in [LSW].
By Theorem 3.6, a.s. limy,,"o It(iy) exists for all
t and is continuous. Therefore we can apply Theorem 4.1, and the theorem
follows.
D 1005
STEFFEN ROHDE AND ODED SCHRAMM
900
It follows from Theorem 5.1 that it extends continuously to 1HI a.s. The next result gives more information about the regularity of it on 1HI. It neither follows from Theorem 5.1, nor does it imply 5.1.
THEOREM 5.2 (Holder continuity). For every Ii; # 4 there is some h( Ii;) > 0 such that for every bounded set A C 1HI and every t > 0, a.s. it is Holder continuous with exponent h(li;) on A, for all
z, z' E A,
where C = C(w, t, A) is random and may depend on t and A. limA;","o h(li;) = ~ and limA;/,oo h(li;) = 1.
Since it(z) - z
---+
0 as z
---+ 00,
Moreover,
it easily follows that for every t a.s.
[it(z) - it(z')[ :S C(w, t) max([z - z'[, [z - Z'[h(A;»).
(5.1)
We do not believe that the theorem holds for a simple path which "almost" touches itself.
Ii;
= 4, for then the trace is
Update. For Ii; :S 4, the fact that 'Y is a simple path in 1HI, Theorem 6.1 below, implies h(li;) :S 1/2. Thus the estimate limA;","o h(li;) = ~ is best possible. On the other hand, this nonsmoothness of is localized at (0): Joan Lind (manuscript in preparation) has shown that the Holder exponent h(li;) of (}tC.jz))2 satisfies limA;","o h(li;) = 1.
it
A=
Proof Fix Ii; # 4 and t [-1,1] x (0,1]. Denote
> O. By scaling, we may assume 0 <
Zj,n=(j+i)T n ,
O:Sn
We will first show that there is an h (5.2)
it-l
=
h(li;)
_2 n <j<2n .
> 0 such that a.s.
[i; (Zj,n) [ :S C(w, t)2 n(1-h),
\fj,n.
Using Corollary 3.5 with 8 = 2- nh we have
Hence
2n
L L 00
p [[i;(Zj,n)[ 2:: 2n(1-h)]
<
00
n=Oj=-2 n
provided that or that
1 + 2b - (1 - h)>" < 0 1 + 2b - >.. + ah < 0 1006
and and
a - >..
:S 0,
a - >.. 2:: O.
t
:S 1 and
BASIC PROPERTIES OF SLE
901
If 0 < '" ::; 12, b = 1/4 + 1/", and h < ('" - 4)2/((", + 4)('" + 12)), the first condition is satisfied. For", > 12, b = 4/", and h < 1/2 - 4/", the second condition is satisfied, and (5.2) follows. To see that one can actually achieve h(",) ---+ 1/2 as '" "" 0, set b := -1/2+ Jl/2 + 2/", for 0 < '" < 4, and let h be smaller than but close to ("\-1-2b)/..\. To get liml
[j;(z)[ ::; 0(1) C(w, t) yh-l for all z E A. It is well-known and easy to see, by integrating [f'[ over the hyperbolic geodesic from z to z' (similarly to the end of the proof of Theorem 3.6), that this implies Holder continuity with exponent h on A. The theorem follows. D The following corollary is an immediate consequence of Theorem 5.2. In Section 8 below, we will present more precise estimates. COROLLARY 5.3. For", a. s. satisfies
# 4 and every t, the Hausdorff dimension of 8Kt dim8Kt < 2.
In particular, a.s. area8Kt = O. Proof By [JM95] (see also [KR97] for an easier proof), the boundary of the image of a disk under a Holder continuous conformal map has Hausdorff dimension bounded away from 2. Consider the conformal map T(z) = (z - i)/(z + i) from 1HI onto 1U. By (5.1), To it 0 T- 1 is a.s. Holder continuous in 1U. Since T preserves Hausdorff dimension, the corollary follows. D
6. Phases
In this section, we will investigate the topological behavior of SLE, and will identify three very different phases for the parameter "', namely, [0,4], (4,8), and [8,(0). The following result was conjectured in [SchOO]. There, it was proved that for '" > 4, a.s. K t is not a simple path. The proof was based on the calculation of the harmonic measure F(x) below, which we will repeat here for the convenience of the reader. THEOREM 6.1. In the range", path and "([0,00) c 1HI U {O}.
E
[0,4], the SLEI< trace "( is a.s. a simple
1007
902
STEFFEN ROHDE AND ODED SCHRAMM LEMMA
6.2. Let
K,
[0,4], and let "( be the
E
"([0,00)
C
SLEI<
trace. Then a.s.
1HI U {O}.
Set Yx(t) := gt(x) - ~(t), x E JR, t :::::: O. Then Yx(t)/ y'K, is a Bessel process of dimension 1 + 4/ K,. The basic theory of Bessel processes then tells us that a.s. Yx(t) i- 0 for t :::::: 0, x i- 0 and K, E [0,4]' which amounts to x i- K t . However, we will prove this for the convenience of the reader unfamiliar with Bessel processes.
Proof Let b> a > O. For x T
=
Tx
:=
E
inf{t :::::: 0 : Yx(t) t/'- (a, b)}.
Set A
F(x)
[a, b], let
:=
{X(I<-4)/1<
K,
i- 4,
log(x)
K,
= 4,
and
F(x)
:=
~(x) - ~(a)
. F(b) - F(a) T)) is a local martingale, and since F is
ItO's formula shows that F(Yx(t 1\ bounded in [a, b], this is a martingale. Consequently, the optional sampling theorem gives F(x) = E[F(Yx(T)J = P[Yx(T) = bJ. Note that F(x) ----7 1 when a '\. O. (That's where the assumption K, ::; 4 is crucial.) Hence, given x > 0, a.s. for all b > x, there is some s > 0 such that gt(x) is well defined for t E [0, s], inf{Yx(t) : t E [0, sJ} > 0 and Yx(s) = b. Note that the Ito derivative of Yx(t) (with respect to t) is
dYx = (2/Yx) dt + d~ . It follows easily that a.s. Yx(t) does not escape to 00 in finite time. Observe that if x' > x, then YX1(t) :::::: Yx(t). Therefore, a.s. for every x > 0 we have Yx(t) well defined and in (0,00) for all t :::::: O. This implies that a.s. for every x > 0 and every s > 0 there is some neighborhood N of x in
Proof of Theorem 6.1. Let t2 > tl > O. The theorem will be established by proving that ,,([O,tl] n"([t2,00) = 0. Let s E (tl,t2) be rational, and set §t (z) := gHs
0
g';- I
(Z + ~ (s ))
- ~ (s ) .
By Proposition 2.1 (§t : t :::::: 0) has the same distribution as (gt : t :::::: 0). Let i's(t) be the trace for the collection (§t : t :::::: 0); that is,
i's(t) = gs 0 gtJs(~(t + s)) - ~(s) = gs 0 "((t + s) - ~(s). 1008
903
BASIC PROPERTIES OF SLE
We know from Lemma 6.2 that a.s. for all rational s E [tl' t2],
'i'8[0, (0) clHIU{O}. Consequently, by applying g;l, we conclude that a.s. for every rational s > 0, 'Y[s, (0)
n (JR. U K = 8 )
b(s)} .
Since gt 2 ogi/ is not the identity, it follows from Theorem 5.1 that there is some s E (tl' t2) such that 'Y(s) ~ JR.UKt" and this holds for an open set of s E (tl' t2). In particular, there is some rational s E (tl' t2) with 'Y( s) ~ JR. U K t, . Since 'Y[O, tIl C Ktl U JR., it follows that 'Y[O, tIl n 'Y[t2, (0) = 0, proving the theorem. D
Recall the definition of the hypergeometric function in Izi < 1,
(6.1) where (1J)n := rr:i~t(1J + j). Let 2
2
2
G(x+iy)=Ga,,,.(x+iY):=2Fl(1JO,1Jl,1/2,x /(x +y )), A
A
where
LEMMA
6.3. Let
1'1,
>
°and z = x + iy
E
1HI. Then the limit
Z(z):= lim y Ig~(z)I/Im(gt(z)) t/T(Z)
exists a.s. We have Z(z) = 00 a.s. if 1'1, Moreover, for all a E JR. we have E[Z(z)aJ = {
:::::
Ga,,,.(z)/Ga,,,.(l)
° 00
8 and Z(z) <
00
a.s. if
1'1,
< 8.
a < 1 - 1'1,/8, 1'1, < 8, a::::: 1- 1'1,/8, a> 0, a
< 0,
1'1,:::::
8.
Before proving the lemma, let us discuss its geometric meaning. For 1HI and t < T(ZO) we have Zo E Ht . Set rt := dist(zo,8Ht ), and let ¢ : 1HI ----+ 1U be a conformal homeomorphism satisfying ¢(gt(zo)) = 0. Note
Zo
E
that I¢'(gt(zo)) I
= (2Imgt(zo)r l . Hence, the Schwarz lemma applied to
the map z f---+ ¢ 0 gt(zo + rt z) proves rt Ig~(zo) I :::; 2Imgt(zo). On the other hand, the Koebe 1/4 Theorem applied to the function gil 0 ¢-l implies that Imgt(zo):::; 2rt Ig~(zo)l. Therefore, since limt/T(zo)rt = dist(zo,'Y[O,oo) UJR.),
(6.2)
Z(zo)-l 1m zo/2 :::; dist(zo, 'Y[O, (0) U JR.) :::; 2 Z(zo)-l 1m Zo. 1009
904
STEFFEN ROHDE AND ODED SCHRAMM
In particular, the lemma tells us that ,[0,(0) is dense in H if and only if K, 2 8. (For K, = 8 we have not proven that, is a path. In that case, ,[0,(0) is defined as the union over t of all limit points of sequences Jt(Zj) where Zj ----t 0 in H. By an argument in the proof of Theorem 4.1, Ut~O 8Kt = ,[0,00 ).)
The way one finds the function G is explained within the proof below, and is similar to the situation in Theorem 3.2.
Proof Let G(z) = Ga,,,,(z) := E [Z(z)aJ. Fix some 2 = x + i Y E H, and abbreviate Z := Z(2). Let Zt = Xt + iYt := gt(2) - ~(t). From (3.9) we find that which implies that (6.3)
Z = exp ( 1
7(2)
o
4y; IZtl-4dt) = exp (
17(2) 0
-2 y2 2 t 2 dlogYt).
Xt
+ Yt
In particular, the limit in the definition of Z exists a.s. and G(2) = E[za] (possibly +(0). Let F t denote the iT-field generated by (~(s) : s::; t). A direct application of Ito's formula shows that
is a local martingale 1 . As before, let Ut = logYt. Note that w(u) = Wt := xt!Yt is a timehomogeneous diffusion process as a function of u. It is easy to see that 7(2) is a.s. the time at which u hits -00. (For example, see the argument following (3.2).) If tl < 7(2) is a stopping time, then by (6.3) we may write (6.4) Note that if s > 0 then the distribution of Z(2) is the same as that of Z(s 2), by scale invariance. It is clear that for every z' E H there is positive probability that there will be some first time tl < 7(2) such that Zt)IZt11 = z'/lz'l. Conditioned on tl < 7(2) and on F t1 , the second integral in (6.4) has the same distribution as log Z(z'), by the strong Markov property and scale invariance. Consequently, we find that P [Z (2) > sJ 2 c P [Z (z') > sJ holds for every s, where c = c(2, z', K,) > O. In fact, later we will need the following stronger lThe Markov property can be used to show that (fj [g~(z)[/Ytr G(zt) is a local martingale. One then assumes that G is smooth and applies Ito's formula, to obtain a PDE satisfied by G. Scale invariance can then be used to transform the PDE to the ODE (6.9) appearing below. Then one obtains G as a solution. That's how the function G was arrived at. 1010
905
BASIC PROPERTIES OF SLE
statement (which is not needed for the present lemma, but we find it convenient to give the argument here). Let w E (0, (0). Then assuming Iwol :::; w, there is a c = c(w, "') > 0 such that (6.5)
Vs
> 0,
P[Z(i) >
sJ 2: P[Z(2) > sJ 2: cP[Z(i) > 2sJ.
The right-hand inequality is proved by taking t1 to be the first time smaller than 7(Z) such that Rezt1 = 0 and the first integral in (6.4) is greater than log 2, on the positive probability event that there is such a time. The lefthand inequality is clear, for if IRegt(i)/Imgt(i)1 never hits Iwol, then we have Z(i) = 00, by (6.3). Note that G(i) = 1. Assume, for now, that G(i) < 00. We claim that in this case G(x+i) > 0 for each x E lR. Let So > 0, let T := inf{ t 2: 0 : IWtl = so}. Since w(u) is time-homogeneous, clearly, T < 7(2). If G(x + i) :::; 0 for some x E R, we may choose So := inf{x > 0 : G(x + i) = O}. Then MT = 0 if Iwol :::; So· Note that 1 :::; Y Ig~(2)I/YT < Z. Since Mt is a local martingale on t < 7(2), there is an increasing sequence of stopping times tn < 7(2) with limn tn = 7(2) a.s. such that Mt/\tn is a martingale. Set En := T 1\ tn. The optional sampling theorem then gives for 2 = i
1=
G(2)
=
Mo
=
E[Md
=
E[(y IgL (2)I/YtJ a G(ZtJJ :::; E[max{l, za} G(ZtJJ
.
Since G(Zt/\T) is bounded, limn--->oo G(ZtJ = 0 a.s., and E[za] < 00, we have limn--->oo E [max{l, za} G(ZtJ J = 0, contradicting the above inequality. Consequently, G(x + i) > 0 for every x E R. We maintain the assumption G(i) < 00. By (6.5), this implies G(z) < 00 for all Z E lliI. Let Sl, S2, ... be a sequence in (0,00) with Sn ----t 00 such that L := limn G(Sn + i) exists (allowing L = (0). Now set Tm := inf{ t 2: 0 : IWtl = sm}. The argument of the previous paragraph shows that Mo = E[MT",/\tnJ ----t E[MT",J, as n ----t 00. Therefore, m----t 00, unless L = 00, which happens if and only if G(2) = O. Moreover, L does not depend on the particular sequence Sn, and we may write G(l) instead of L. Thus, when G( i) < 00, (6.6)
G(2) = G(2)/G(1).
When a = 1 - ",/8, G simplifies to (y/lzl)(S-",)/"" with G(l) = 00 if", > 8 and G(l) = 0 if '" < 8. It follows that G(z) = 00 if a = 1 - ",/8 and", < 8. Clearly, G(z) = 00 also if a> 1- ",/8, and", < 8. Similarly, we have G(z) = 0 if '" > 8 and a = 1 - ",/8. This implies that Z = 00 a.s. when", > 8, which implies G = 0 when '" > 8 and a < o. There are various ways to argue that Z = 00 a.s. when", = 8. In particular, it is easy to see that G1 s(1) = 00 4 '
1011
906
STEFFEN ROHDE AND ODED SCHRAMM
(if an denotes the n- th coefficient of 2Fl (~, ~, ~, z), then (n + 1) a n+1 = nanb n with bn = (n + 1/4)2 /(n(n + 1/2)) and it follows from ITn bn > 0 that nan is bounded away from zero) so that the above argument applies. It remains to prove that G(i) < 00 when a < 1 - ",/8 and", < 8. Assume now that", < 8 and a < 1 - ",/8 and a is sufficiently close to 1 - ",/8 so that TJl > O. We prove below that in this range inf G(z)
(6.7)
zEIHI
> O.
Since G(2) = Mo = E[MtnJ when 2 = i, we immediately get G(i) < 00 from (6.7). Since G(i) < 00 for some a implies G(i) < 00 for any smaller a, it only remains to establish (6.7) in the above specified range. To prove (6.7), first note that TJo ::::; TJl, 0 < TJl < 1/2 and TJo + TJl = 1 - 4/", < 1/2. By [EMOT53, 2.1.3.(14)]' when TJ2 > TJl + TJo and TJ2 > TJl > 0, the right-hand side of (6.1) converges at z = 1 and is equal to
r(TJ2) r(TJ2 - TJl - TJo) r(TJ2 - TJd r(TJ2 - TJo) .
(6.8)
This gives G(l) > O. Now write
g(x) := Ga,,,,(x + i)/G 1_",/8,,,,(X + i) = (1
+ x 2)(8-",)/(2",) Ga,,,,(x + i).
G satisfies
Since (6.9) for y
= 1, it follows that
(6.10)
('" - 8 + 8a) 9 + 2 ('" - 4) x (1
+ x 2) g' + '" (1 + x 2)2 g" = O.
It is immediate that g(O) = 1, g'(O) = 0 and g"(O) > 0, e.g., by differentiation or by considering (6.9). Consequently, 9 > 1 for small Ixl > O. Suppose that there is some x i- 0 such that g(x) = 1. Then there must be some Xo i- 0 such that g(xo) > 1 and 9 has a local maximum at Xo. Then g'(xo) = 0 2: g"(xo). But this contradicts (6.10), giving 9 > 1 for x i- O. Together with G(l) > 0, this implies (6.7), and completes the proof. D
We say that z E 1HI is swallowed by the SLE if z ~ ,/[0,00) but z E K t for some t E (0,00). The smallest such t will be called the swallowing time of z.
'"
~
THEOREM 6.4. If'" (4,8), then a.s. no z LEMMA
6.5. Let z
E
E
E
(4,8) and z E 1HI \ {O}, then a.s. z is swallowed. If 1HI \ {O} is swallowed.
1HI \ {O}. If'" > 4, then P [T(Z) < 00 J = 1.
In fact, in this range", > 4, the stronger statement that a.s. for all z E 1HI we have T(Z) < 00 holds as well. This follows from Theorem 7.1 (and its extension to '" = 8 based on [LSW]). 1012
907
BASIC PROPERTIES OF SLE
Proof Let b:= 1- 4/11" and 0:= exp(i1r(l - b)/2). Set
h(z):= Im(Ozb),
z
E
1HI.
A direct calculation shows that the drift term in Ito's formula for (gt(z) -e(t)) b is zero. Consequently, h (gt (z) - ~ (t)) is a local martingale. Note that 11, > 4 implies that h(z) -+ 00 as z -+ 00 in 1HI, that h(z) > 0 in 1HI \ {O} and h(O) = O. Fix some Zo E 1HI and some E > O. For t 2: 0 set Zt := gt(zo) - ~(t). We shall prove that there is some T = T(zo, E) such that P [zo E KT J > 1 - E. Indeed, let R> 1 be sufficiently large so that E inf{h(z) : Izl = R} > 2h(zo). Let TR:= inf( {t E [O,T(ZO)): IZtl2: R} U {T(ZO)}). We claim that if T = T(zo, E) is sufficiently large, then P [TR > TJ < E/2. Let t 2: 0 and let itl := inf{k E Z : k 2: t}. Say that t is a sprint time if I~(t) - ~(ltl)1 = R and there is no s E [t, itl] such that I~(s) - ~(ltl)1 = R. Note that for every integer k 2: 0 there is positive probability that there is a sprint time in [k, k + 1] and this event is independent from (~(t) : t E [0, kJ). Therefore, there are infinitely many sprint times a.s. Let to < tl < t2 < . .. be the sequence of sprint times. Let Aj be the event that ~(ltj l) > ~(tj). Observe that for all j EN, (6.11) where F t denotes the O"-field generated by (~(s) : s E [0, tJ), because given tj and (~(t) : t E [0, tjJ), the distribution of (~(t) : t > tj) is the same as that of (2~(tj) - ~(t) : t > tj), by reflection symmetry. Let Xt := Re(zt), and note that Consequently, for s 2: tj, (6.12)
On Aj we have ~(s)-~(tj) > 0 for all s E [tj, itj l]. Hence, it follows from (6.12) that on Aj and assuming Xtj :::; 0, we have Xs :::; 0 for all s E [tj, itj l] and moreover Xitjl < -R. Similarly, if Xt j 2: 0 and ,Aj holds, then Xitjl > R. Hence, by (6.11) we have P [TR > itj lJ :::; 2- j , which implies that we may choose T such that P[TR > T] < E/2. Set T* := TR 1\ T, where T is as above. Since h(zt) is a local martingale, and h(ZtM*) is bounded, h(ZtM*) is a martingale. Therefore,
h(zo) = E[h(ZT*)] 2: inf{h(z) : Izi = R} P [IZT* 1= RJ . By our choice of R, this gives P [IZT* I = RJ < E/2. Since P[TR > T] < E/2, it follows that P[T(ZO) :::; T] 2: 1 - E, and the lemma is thereby established. D 1013
908
STEFFEN ROHDE AND ODED SCHRAMM
We now prove a variant of Cardy's formula. (See [Car] for a survey of Cardy's formula.) A proof of (a generalized) Cardy's formula for SLEr.; appears in [LSWOla]. However the following variant appears to be different, except in the special case K, = 6, in which they are equivalent due to the locality property of SLE 6 . (See [LSWOla] for an explanation of the locality property.) LEMMA
6.6. Fix
K,
E (4,8) and let
X := inf([I, (0)
n 'Y[O, (0)) .
Then for all s 2: 1, (6.13)
P X >s = 4(r.;-4)/r.; )7f2 F l(1 - 4/K" 2 - 8/K" 2 - 4/K" l/s) s(4-r.;)/r.; [ -] r(2-4/K,)r(4/K,-1/2)
The proof will be brief, since it is similar to the detailed proof of the more elaborate result in [LSWOla].
Proof Let s> 1, and set y; ._ gt(1) - ~(t) t .-
gt(s) -
~(t)
,
for t < T(I). It is not hard to see that limt/7(1) yt = 1 if X > sand limt/7(1) yt = 0 if X < s: Just notice that for X > sand t / T(I), the extremal distance within H t from [1,s] to [Re'Y(t),'Y(t)] tends to 00, whereas for X < s we have T(I) < T(S). Direct inspection shows that the right-hand side of (6.13) is zero when s = 00 and is 1 when s = 1. In particular, it is bounded for s E [1,(0). Moreover, Ito's formula shows that when we plug l/yt in place of s in the right-hand side of (6.13), we get a local martingale. Since Yo = 1/ sand yt E (0,1) for t < T(I), it therefore suffices to prove that limt/7(1) yt E {O, I} a.s. The only way this could fail is if X = s occurs with positive probability. However, scale invariance shows that P[X = Sf] 2: P[X = s] when Sf E (1, s). Hence P[X = s] = 0 for all s i- 1. D
Proof of Theorem 6.4. The case K, E [0,4] is covered by Theorem 6.1 and Theorem 5.1. Suppose that K, E (4,8) and z E 1HI. Lemma 6.5 shows that Z E Ut>o K t a.s. If Z E 1HI, then Lemma 6.3 and (6.2) show a.s. Z ~ 'Y[O, (0). Lemma 6.6 shows that 1 ~ 'Y[O, (0) a.s., and the same follows for every Z E ~ \ {O} by scale and reflection invariance. This covers the range K, E (4,8). Now suppose that Zo is swallowed, and let to be the swallowing time. If 8 > 0 is smaller than the distance from Zo to 'Y[O, to], then all the points within distance 8 from Zo are swallowed with Zo. Therefore, if the probability that some Zo E 1HI \ {O} is swallowed is positive, then by (6.2) there is some Zo such that P [Z(zo) < 00] > O. By Lemma 6.3, this implies that for K, 2: 8 a.s. there is no z E 1HI \ {O} which is swallowed. D 1014
909
BASIC PROPERTIES OF SLE
Exercise 6.7. Prove that P[T(Z) < ooJ =
°
when
t£
E (0,4] and
ZE
JH[.
7. Transience THEOREM 7.1 (Transience).
For all
t£
-I- 8
the SLE,,; trace ,( t) is tran-
sient a.s. That is, P [liminf b(t)1 t-+oo
LEMMA 7.2. Suppose that ,[0,(0).
---;-::--;-
t£
= 00] =
1.
:S 4, and let x E lR \ {o}. Then a.s. x
~
Proof By symmetry, we may assume that x > 0. Moreover, scaling shows that it is enough to prove the lemma in the case x = 1. So assume that x = 1. Let s E (0,1/4) and suppose that there is some first time to > Osuch that I,(to) - 11 = s. This implies that Igt o(1/2) - ~(to)1 = O(s). Indeed, Igto(1/2) - ~(to)1 is the limit as y ----+ 00 of y times the harmonic measure in JH[ \ ,[0, to] from i y of the union of [0,1/2] and the "right-hand side" of ,[0, to]. By the maximum principle, this harmonic measure is bounded above by the harmonic measure in JH[ of the interval [1, ,(to)], which is O(s/y). In the proof of Lemma 6.2, we have determined the probability that Yx(t) = gt(x) - ~(t) hits b before a, where x E [a, b] and < a < b. For t£ < 4, that probability goes to 1 as a '\,. 0, uniformly in b. Consequently, a.s. there is a random E > such that for all b > x, Y x (t) hits b before E. Since SUPtE[O,T] Yx(t) < 00 a.s. for all finite T, it follows that inft::::: o Yx(t) > a.s. Using this with x = 1/2, we conclude from the previous paragraph that a.s. 1 ~ ,[0,(0), at least when t£ < 4. Assume now that t£ = 4. Although the above proof does not apply (since Yx(t) is recurrent), there are various different ways to handle this case. Let < y < x < 00. Let 1(t) denote the complex conjugate of ,(t), and set Dt := C \ (,[0, t] u 1[0, t] u (-00, yJ). Using Schwarz reflection, gt may be conformally extended to Dt . Note that gt(D t ) = C \ (-oo,gt(y)J. By the Koebe 1/4 Theorem applied to gil, the distance from x to oD t is at least -,Hgt(x) - gt(y))/gi(x). Consequently, it suffices to show that a.s.
°
°
°
°
supg~(x)/(gt(x) - gt(Y)) t:::::O
< 00.
Set Q(t) = Qx,y(t) := loggi(x) -log(gt(x) - gt(Y)). Then the above reduces to verifying that SUPt Q(t) < 00. The time derivative of Q is
OtQ(t) = -2 Yx (t)-2
+ 2 Yx(t)-l Yy(t)-l .
In particular, Q(t) is increasing in t. Define 1
G(s) := log s 10g(1 + s) - "2log2(1 1015
+ s) + L1.2( -s),
910
STEFFEN ROHDE AND ODED SCHRAMM
where Li2(z) = Jzo s-llog(1 - s) ds. Note that G satisfies s (1 + s)2 G"(s) + s (1 + s) G'(s) = 1. A straightforward application of ItO's formula shows that Q(t) - G t is a local martingale, where
G
'= t .
G(gt(x) - gt(Y)) gt(Y) - ~(t) .
Consequently, there is an increasing sequence of stopping times tn tending to 00 a.s. such that E[Q(t n )] = E[Gd + Q(O) - Go. It is immediate to verify that sup{G(x): x > O} < 00. Hence, limsuPn---tooE[Q(tn)] < 00. Because Q is increasing in t, it follows that E[supt Q(t)] < 00 and hence SUPt Q(t) < 00 a.s., D which completes the proof. LEMMA 7.3. Suppose that K, > 4, K, 0; 8, and let t > O. Then a.s. there is some c > 0 such that K t ~ {z E 1HI: Izl < c}.
Proof Consider first the case K, E (4,8). We know that "((t) is a continuous path, by Theorem 5.1. Note that by scale and reflection invariance, Lemma 6.6 shows that for a < b < 0 there is a positive probability that the interval [a, b] is swallowed all at once; i.e., P [la, b] n "([0, (0) = > 0, but a.s. there is some t such that [a, b] C K t . When this happens, we have [a, b] n H t = 0. Let S be the set oflimit points of gT(l)(Z) as z ---+ 0 in HT(l)' (If 0 ¢:. HT(l), set S = 0.) Then S C (-00, ~(7(1))). Note that (gT(l)b(t + 7(1))) - ~(7(1)) : t 2: 0) has the same distribution as ("(( t) : t 2: 0). Consequently, by the above paragraph, there is positive probability that gT(l) b[7(1), (0)) is disjoint from S, but there is some t > 7(1) such that gT(l)(Kt \ K T(l») ~ S. This implies that with positive probability there is some t > 0 such that
0J
6:= P[O
¢:."1:lt] > O.
By scale invariance, 6 does not depend on t. Since these events are monotone in t, it follows that P [Vt
> 0, 0 ¢:. "1:lt] = 6.
By Blumenthal's 0-1 law it therefore follows that 6 = 1. This completes the proof in the case K, E (4, 8). Now suppose that K, > 8. The above argument implies that it is sufficient to show that there is some t > 0 such that with positive probability 0 ¢:. H t . Striving for a contradiction, we assume that for all t > 0 a.s. 0 E H t . Let Zo E 1HI be arbitrary. We know from Theorem 6.4 and Lemma 6.5 that a.s. Zo E "([0,00). Set to := 7(ZO)' Then Zo = "((to). Let t > O. Conditioned on F to , the set gto(Kt+to \KtJ has the same distribution as Kt+~(to). It therefore follows by our above assumption that a.s. ~ (to) E gt o(HHto)' Mapping by gt"a this implies that a.s. Zo E HHto; that is, Zo E 8Kt+to' We conclude that for
r,
1016
911
BASIC PROPERTIES OF SLE
every z E 1HI and every t > 0 a.s. z E 8KHT (z). Therefore, a.s. area( 8Kt ) =
1
1zE 8K, dx dy ;::
zE~
1
1t >T(z) dx dy ,
zE~
and Fubini implies that 8Kt has positive measure with positive probability for large t. However, this contradicts Corollary 5.3, completing the proof of the lemma. D
Proof of Theorem 7.1. First suppose that K, > 4. Then we know from Lemma 7.3 that there is a random but positive r > 0 such that K1 ::) {z E 1HI : Izl < r}. Let c: > 0, and let ro = ro(K" c:) > 0 be a constant such that P[r < ro] < c:. Then the scaling property of SLE shows that for all
t>O
This implies that P [3t'
> t : Ir(t') I < roVt] < c:.
Since c: > 0 was arbitrary, this clearly implies transience. Consider now the case K, E [0,4]. In this range, "( is a simple path. Then a.s. there are two limit points xo, Xl for gl (z) as z ----* 0 in H 1. Note that gl(,,([l,oo)) has the same distribution as "([0,00) translated by ~(1). Consequently, Lemma 7.2 shows that a.s. Xo and Xl are not in the closure of gl("([l,oo)). This means that 0 rt. ,,([1,00) a.s. The proof is then completed as in the case K, > 4 above. D COROLLARY
7.4. Suppose that
K,
> 8, then "([0,00) = 1HI a.s.
Proof From Lemma 6.3 and (6.2) we know that "([0,00) is a.s. dense in 1HI. Since "( is a.s. transient, "([0,00) is closed in te, and hence is equal to 1HI. D Update. Corollary 7.4 and Theorem 7.1 are true also for K, = 8. The proofs are based on the extension [LSW] to K, = 8 of Theorem 5.1, and are otherwise the same. 8. Dimensions
In this section we study the size of the SLE trace and the hull boundary. When K, ~ 4, these two sets are the same, and it is interesting to compare the methods and results. For sets A c te, denote by N(c:) = N(c:, A) the minimal number of disks of radius c: needed to coverA. 1017
912
STEFFEN ROHDE AND ODED SCHRAMM
8.1. The size of the trace. THEOREM
8.1. For every nonempty bounded open set A with A
. logE[N(c,/,[O,oo)nA)] hmsup c---+o I log c I
=
{1+/1,/8
c H,
/1,<8, /1, 2:: 8.
2
Proof We may assume /1, < 8 because /,[0,00) = H if /1, > 8, by Corollary 7.4, and when /1, = 8 we know that for every Z E H we have Z E /,[0,00) a.s., by Theorem 6.4 and Lemma 6.5. Fix c > smaller than the distance from A to JR, and denote those points Z E cZ 2 with dist(z, A) < c by Zj, 1 :S j :S n. Then n is comparable to 1/c2. Set N := N(c,/,[O, oo) n A). Obviously N :S #{Zj : dist(zj,/,[O,oo)) < c} :S lOON. From (6.2) and (6.5) we get that E[N] is comparable to
°
n
LP[Z(Zj)
> c- 1].
j=l
Write F(t) = P[Z(i) > t]. Using (6.5), the proof of the theorem reduces to showing that . logF(t) hmsup I = /1,/8 - 1. t---+oo ogt By (6.8), Ga ,,,(l) y0rr(4//1, - 1/2) lim a/l-,,/8 1 - /1,/8 - a 2r(4//1,) and since Ga,,,(i)
= 1 from Lemma 6.3 we obtain
C-1
<E[Z(i)a]
1
1 - /1,/8 - a -
-
1
1 - /1,/8 - a
for all a < 1- /1,/8 sufficiently close to 1- /1,/8. Hence, for small
C~ 2::
1D.
E [Z(i)1-,,/8-i5] 8 d8
=
1D.1
OO
~
=
~(/1,)
> 0,
(1 - /1,/8 - 8) C,,/8-i5 F(t) dt 8 d8.
From faD. C,,/8-i5 8 d8 2:: C C,,/8 / (log t)2 for t 2:: e (where ~ is treated as a constant), we obtain 00 C,,/8 F(t) e (logt)2 dt=O(l).
1
Using F(t)/F(2t)
= 0(1) from (6.5) this easily implies for large F(t) :S Ct,,/8-1 (logt)2.
There is a sequence tn
--+
00 with t,,/8-1
F( tn) 2:: (l~g t n )2 ' 1018
t
913
BASIC PROPERTIES OF SLE
for otherwise we would have E[Z(i)I-fi:/8] dicting Lemma 6.3.
= 0(1)
Jeoo t(I~;t)2
<
00,
contraD
An immediate consequence is COROLLARY 8.2. For 1'1, bounded above by 1 + 1'1,/8.
< 8, the Hausdorff dimension of ,[0, 00) is a.s.
8.2. The size of the hull boundary. When studying the size of the hull
boundary fJKt , it suffices to restrict to t = 1, by scaling. We will first estimate the size of fJH 1 n {y > h} for small h > 0 in terms of the convergence exponent of the Whitney decomposition of HI. A Whitney decomposition of HI is a covering by essentially disjoint closed squares Q C HI with sides parallel to the coordinate axes such that the side length d( Q) is comparable to the distance of Q from the boundary of HI. One can always arrange the side lengths to be integer powers of 2 and that
(8.1)
d(Q):S dist(Q,fJH1 ):S 8d(Q).
For example, one can take all squares of the form Q = 2n ([k, k + 1] x [j, j + 1J) such that Q CHI, d(Q) :S dist(Q, fJHl) , and Q is not strictly contained in any other Q' satisfying these conditions. One important feature of the Whitney decomposition is that the hyperbolic diameters of the tiles in the decomposition are bounded away from 0 and 00. (The hyperbolic metric on HI is the metric for which the conformal map gl : HI ---+ 1HI is an isometry, where 1HI is taken with the hyperbolic metric with length element y- 1 [dz[.) This follows from the well-known consequence of the Koebe 1/4 theorem saying that the hyperbolic length element in HI is comparable to dist(z, fJH1 )-1 [dz[. This description of the hyperbolic metric will be useful below. Fix a Whitney decomposition of HI, and denote by W the collection of those squares Q for which dist(Q,Kl) :S 1 and sup{Imz : z E Q} 2 h. For a> 0 let S(a) = Sh(a) := d(Q)a.
L
QEW
THEOREM
8.3. Set
8(1'1,) For 1'1,
:=
{I +
1'1,/8 K,:S 4, 1+2/1'1, 1'1,>4.
> 0, 1'1, i- 4, and all sufficiently small h E[S(a)]
{< =
00 00
E
(0,1) we have
a> 8(1'1,), a:S 8(1'1,).
For j, nEZ, let Xj,n := j 2- n , Yn := 2- n and the interval gl(fJK1 ) - ~(1); that is, j-l(fJKd. 1019
Zj,n
= Xj,n
+ i Yn.
Let I be
914
STEFFEN ROHDE AND ODED SCHRAMM
LEMMA 8.4. There are constants C = C(h), c = c(h) > 0 such that the following holds. Every Q E W is within distance C in the hyperbolic metric of HI from some 11 (Zj,n) with Xj,n E I and n 2: 0, and every 11 (Zj,n) such that Xj,n E I, n 2: 0 and 1m 11 (Zj,n) 2: h is within hyperbolic distance C from some Q E W. Moreover, in each of these cases,
Proof Observe that the diameter of Kl is bounded from below, because
the uniformizing map 9 : 1HI \ K ----t 1HI with the hydrodynamic normalization (2.2) tends to the identity as K tends to a point, and the uniformizing map gl for Kl satisfies gl(Z) = Z + 2/z + ... near 00, by (2.3). Let 2 E Q E W. It follows from the definition of Wand the fact that diam(KI} is bounded from zero that the harmonic measure of Kl from 2 in HI is bounded from zero. Therefore, the harmonic measure of I from /;1(2) in 1HI is bounded from zero. This implies that there is some Zj,n within bounded distance from /;1(2) in the hyperbolic metric of 1HI such that Xj,n E I. Then 11(Zj,n) is within bounded hyperbolic distance from Q. Because s~p{Imz : Z E K 1} = 0(1), we also have 1m2 = 0(1), and therefore Im!I(zj,n) = 0(1). Since Im!I(zj,n) 2: Imzj,n, it follows that max{O, -n} = 0(1). Then z' := Xj,n + i 2nAO is equal to Zj',n' for some j' E Z, n' E N, and satisfies the requirements in the first statement of the lemma. From the fact that diam(Kl) is bounded from zero it also follows that iIi, the length of I, is bounded from zero. Suppose that we have Xj,n E I, j E Z, n E N. Then the harmonic measure of I from Zj,n is bounded from zero. Consequently, the harmonic measure of Kl from !I (Zj,n) in HI is bounded from zero. Now suppose also that 1m 11 (Zj,n) 2: h. Since sup{Imz : Z E K 1} = 0(1) and 1m 11 (Zj,n) = 0(1), it follows that the Euclidean distance from 11 (Zj,n) to Kl is bounded from above. If dist (!1 (Zj,n), K 1) ::::; 1, then the Whitney tile Q containing 11 (Zj,n) is in W. Otherwise, let p be a point closest to f~ (Zj,n) among the points Z satisfying 1m Z 2: hand dist(z, KI} ::::; 1. Since h < 1 and dist(!1 (Zj,n), Kl) = 0(1), it follows that Ip- 11 (Zj,n) I = 0(1). Since the points in [p, 11 (Zj,n)] have distance at least h from 8H1, it follows that the hyperbolic distance from p to 11 (Zj,n) is bounded above. For the second statement of the lemma we may then take Q to be the Whitney tile containing p. The last statement holds because 11 is an isometry from the hyperbolic metric of 1HI to the hyperbolic metric of HI. D Proof of Theorem 8.3. Set 00
00
1020
BASIC PROPERTIES OF SLE
915
We now show that this quantity is comparable to S(a). Let C be the constant in the lemma. Since each square Q E W has a bounded diameter in the hyperbolic metric of HI, and the hyperbolic distance in 1HI between any two distinct Zj,nJs bounded from below, it follows that there are at most a bounded number of h(zj,n) within distance C, in the hyperbolic metric of HI, from any single Q E W. Similarly, there are at most a bounded number of Q E W within hyperbolic distance C from any point in HI. Moreover, if Z is within hyperbolic distance C from some Q E W, then 1m Z is bounded away from zero, because the hyperbolic distance in HI between any two points is an upper bound on the distance between them in the hyperbolic metric of 1HI (by the Schwarz lemma, for instance). Consequently, Lemma 8.4 implies (8.2) for some constant el, h', which depends only on hand a. Now, 00
E[Sh(a)] =
(8.3)
00
00
LL L
E[lif(Zj,n) Ynl a 1A(j,n,m)]
n=O m=O j=-oo
whereA(j,n,m) denotes the event {Xj,n E I,Imil(Zj,n) 2': h,m:::::: III < m+1}. We split this sum into two pieces, depending on whether m :::::: n or m > n. Clearly, 0 E I. It is easy to see that
III:::::: 0(1 + O~f~II~(t)l) , for example, by considering the differential equation (2.1) directly. This implies p [III 2': m] :::::: 0(1) e-C(~) m2, for some C(K,) > O. Since
lyj{(x+iy)l:::::: 0(1)Imil(x+iy):::::: 0(1) for 0
< y :::::: 1, we have the trivial estimate 00
E
[L
lif (Zj,n) Ynla1A(j,n,m)] : : : 0(1) m2n p [III 2': m]
j=-oo
:::::: 0(1) m 2n e-C(~)m2 for all m 2': O. This shows that in (8.3) the terms with m 2': n have finite sum. For m :::::: n, we use Theorem 3.2 together with (3.17) and obtain
E[ L 00
lif(zj,n) Ynla1A(j,n,m)] : : : C(h,K"b) m 2b+ I T n (A-2b-I),
j=-oo
where a, >., band K, are related by (3.6) with 1/ = 1 and b > O. For b between 1/2 and 2/K, we have>. - 2b - 1 > 0 and conclude that E[S(a)] < 00 in this range. By (8.2), E[S(a)] < 00 for K, < 4, a> 1+K,/8 and for K, > 4, a> 1+2/K,. Now consider a = 8(K,). Suppose that 2- n < h/4 and il(Zj,n) 2': h. We claim that the harmonic measure in HI of KI from il(Zj,n) is bounded from 1021
916
STEFFEN ROHDE AND ODED SCHRAMM
zero. Indeed, fix a constant h1 > 1 such that ht/2 2: sup{Im Z : Z E K1}. (It is easy to see that h1 = 4 works.) Consider the harmonic function Y(z) := Img1(z) on H1. Then Y(x + iy) - y ----+ 0 uniformly in x, as y ----+ 00, by the hydrodynamic normalization (2.2) of gl. Since 1HI + ~ h1 C H 1, the maximum principle applied to Y(z) - Imz implies that Y(x + i h 1) 2: ht/2 for all x E R Consequently, for every z E H1 with Imz < h1, the harmonic measure in H1 n {y < h 1} from z of the line y = h1 is at most 2Y(z)/h 1. However, the harmonic measure from z of y = h1 in the domain 1HI n {y < hd is Imz/h 1. Therefore, the harmonic measure from z of K1 is at least the difference between these two quantities, which is at least Imz/h1 - 2Y(z)/h1. Applying this to z := i1 (Zj,n) proves our claim that the harmonic measure of K1 from i1 (Zj,n) in H1 is bounded from zero. This implies that the harmonic measure of I from Zj,n in 1HI is bounded from zero, which implies that at bounded hyperbolic distance from Zj,n we can find a point Zj',n' with Xj',n' E I. Consequently, to prove that E [Sh (15(1\;))] = 00 for all sufficiently small h > 0, it suffices to prove that for all no E fir and all sufficiently small h > 0 00
j=-oo
L
n=no
1Im !1(Zj,n»h
IR (Zj,n) Ynl"(K)]
=
00.
This follows from Lemma 3.3 with b = 1/2 if I\; < 4 or b = 2/1\; if I\; > 4. An appeal to (8.2) gives E[S(i5(I\;))] = 00. Since E[S(a)] < 00 for a > 15(1\;), it also follows that E [S (a)] = 00 for a < 15 (1\;), and the proof is complete. D COROLLARY 8.5. FoT' all
I\;
> 0,
I\;
0; 4, and all sufficiently small h > 0
lim sup Ilog tf+h c"(K) E [N(c, 8K1 n {y > c:--+O
h})] =
Proof Denote by W(n) the number of Q E W with 2- n Then (8.1) immediately gives
(8.4)
W(n) S C N(Tn,8K1 n {y
The corollary now follows from S(a)
=
2:~=o
1
00.
< d(Q) S 2- n .
> h}).
W(n)2-an and Theorem 8.3. D
Finally we state an upper bound for the covering dimension. THEOREM 8.6. FoT' all 1\;, h
> 0 and a> 15(1\;), a.s.
limca N(c,8K 1 n {y > h})
c:--+O
= O.
Proof Suppose first that I\; 0; 4. By Theorem 5.2 we know that i1 is a.s. Holder continuous. There is no lower bound version of (8.4) for general domains. However, by results of Pommerenke [Pom92] and Makarov [Mak98] it is known that for Holder domains (i.e., the conformal map from lU into the 1022
917
BASIC PROPERTIES OF SLE
domain is Holder) the asymptotics of N can be recovered from the convergence exponent of the Whitney decomposition. Our proof follows this philosophy. We claim that there is a (random) constant C = C(w) > 0, depending on h and the Holder norm of iI, such that for every small E > and every point Zo E 8Kl n {y ~ 2h}
°
(8.5)
:3Q E W,
Q C D(Zo,E), E ~ d(Q)
~ CE/llogEI.
In fact, this follows from [KR97], but we include a short proof here, for the convenience of the reader. Let Xo E lR be a point such that il(XO) = ZOo Clearly, there is such an Xo, because il extends continuously to 1HI. Let {3 (s) := il (xo + is), s > 0, let Q (s ) be the set of Whitney cells Q E W which intersect {3 ((0, s)), and set
q(s):=
L
d(Q).
QEQ(s)
°
Since il is an isometry between the hyperbolic metric of 1HI and that of HI, it follows that for every s > the arc {3[s, 2s] meets a bounded number of Whitney cells, and each of these has Euclidean diameter comparable to the Euclidean diameter of {3[s, 2s]. Suppose that il is Holder with exponent a > 0. Then, d(Q) :::; 0(1) sa for Q E Q(s), and consequently also q(s) :::; 0(1) sa. Observe that there is a constant Co so that q(s)/sup{q(s') : s' < s} < Co, because there are constant upper bounds on the number of Whitney cells containing a point, and on the ratio d(Q)/d(Q') for neighboring Whitney cells Q and Q'. Let So := sup{s : q(s) < E} and Sl := sup{s : q(s) < cOl E/2}. The choice of Co ensures that q(so) - q(Sl) ~ E/2. Clearly Q(so) \ Q(Sl) has at most 0(1 + log(so/ Sl)) cells. Therefore, there is some Q E Q(so) \ Q(Sl) with d(Q) ~ E/0(1 + log(so/sr)). Since q(s) :::; 0(1) sa, it follows that 0(1) Sl ~ Ella, and we get 0(1) d(Q) ~ E/llogEI, proving (8.5). We know that for every set Fee the covering number N(E, F) is bounded by a constant times the maximum number of disjoint disks of radius E centered in F. Therefore, by (8.5), N(E, 8Kl n{y > 2 h}) is bounded by a constant times the number of Whitney cells in W with size between 0(1) E and E/O(llog EI); that is, n+O(logn) N(Tn,8Kl n {y > 2h}) :::; C(w) W(j),
L
j=n
which implies for all a' > 0, N(Tn,8Kl
n {y > 2 h}) :::; C(w) 2(n+O(logn»a' S(a') .
By Theorem 8.3, S(a') < 00 a.s. for a' > 8(A;). Therefore, the case A; Theorem 8.6 now follows by taking a' E (8 (A;) , a) . 1023
i-
4 in
918
STEFFEN ROHDE AND ODED SCHRAMM
The following argument covers not only the case I'\, = 4, but the whole range I'\, E (0,4]. For such 1'\" Theorem 8.1 can be used, since oK1 = ,,[0,1] for I'\, E (0,4]. Although Theorem 8.1 discusses the expected number of balls needed to cover ,,[0,(0) n A, where A is an arbitrary bounded set satisfying A c 1HI, one can refine the result to bound N(c,oKl n {y > H}), as follows. Given a point z = x + i y, y> h, we need to estimate the probability that z is close to Kl. We may certainly restrict attention to the case where y is bounded by some constant. There is a constant C 2: 1 such that the probability that z is close to Kl is bounded by the probability that there is some time tl < 1 such that ~(tl) is within distance C from x times the probability that Z(z) is large, conditioned on the existence of such a tl. The proof is then completed by applying the estimates in the proof of Theorem 8.1 and the Markov property. The details are left to the reader. D The upper bound for the Hausdorff dimension follows immediately: COROLLARY 8.7. For most 1 + 2/1'\,.
I'\,
> 4, the Hausdorff dimension of OKI is a.s. at
9. Further discussion and some open problems We believe Theorem 5.1 to hold also in the case
I'\,
= 8:
CONJECTURE 9.1. The SLE 8 trace is a continuous path a.s. Update. This is proved in [LSW]. CONJECTURE 9.2. The Hausdorff dimension of the SLE/i trace zs a.s. 1 + 1'\,/8 when I'\, < 8. Update. A proof of this result is announced in
[Be~
for
I'\,
i- 4.
Corollary 8.2 shows that the Hausdorff dimension cannot be larger than 1 + 1'\,/8. CONJECTURE 9.3 (Richard Kenyon). is a.s. 1 + 2/ I'\, when I'\, 2: 4.
The Hausdorff dimension of oK1
Corollary 8.7 shows that the Hausdorff dimension cannot be larger than 1+2/1'\,. Problem 9.4. Find an estimate for the modulus of continuity for the trace of SLE4 . Improve the estimate for the modulus of continuity for SLE/i for other 1'\,. What better modulus of continuity can be obtained with a different parametrization of the trace? 1024
BASIC PROPERTIES OF SLE
919
Figure 9.1: A sample of the loop-erased random walk; believed to converge to radial SLE2. Based on the conjectures for the dimensions [DupOO], Duplantier suggested that SLE 16 /,,; describes the boundary of the hull of SLE,,; when Ii, > 4. This has further support, some of which is discussed below. However, at this point there isn't even a precise version for this duality statement.
Problem 9.5. Understand the relation between the boundary ofthe SLE,,; hull and SLE 16 /,,; when Ii, > 4. Update. See [Dub] for recent progress on this problem. We now list some conjectures about discrete processes tending to SLE,,; for various Ii,. Let D be a bounded simply connected domain in C with smooth boundary, and let a and b be two distinct points in aD. Let ¢ be a conformal map from H to D such that ¢(O) = a and ¢( (0) = b. The SLE trace in D from a to b is defined as the image under ¢ of the trace of chordal SLE in H. Similarly, one defines radial SLE in arbitrary simply connected domains strictly contained in C. In [SchOO] it was shown that if the loop-erased random walk in simply connected domains in C has a conformally invariant scaling limit, then this must be radial SLE2. See Figure 9.1 for a sample of the loop-erased random walk in U. In [SchOO] it was also conjectured that the UST Peano curve, with appropriate boundary conditions (see Figure 9.2), tends to SLEs as the mesh goes to zero. This offers additional support to Duplantier's conjectured duality 9.5, Since the outer boundary of the hull of the UST Peano curve consists of two loop-erased random walks, one in the tree and one in the dual tree. See [SchOO] for additional details. 1025
920
STEFFEN ROHDE AND ODED SCHRAMM
Figure 9.2: A piece of the UST Peano curve; believed to converge to SLEs.
Update. The convergence of LERW to SLE2 and of the UST Peano path to SLEs is proved in [LSW]. CONJECTURE 9.6. Consider a square grid of mesh c in the unit disk U. Then the uniform measure on simple grid paths from 0 to on this grid converges weakly to the trace of radial SLEs as c '\.. 0, up to reparametrization. The uniform measure on simple paths joining two distinct fixed boundary points converges weakly to the trace of chordal SLEs , up to reparametrization.
au
We have not bothered to make this conjecture entirely precise, since there is more than one natural choice for the topology on path space. However, this does not seem to be very important at this point. Note that we do not require that the grid path be space filling (i.e., visit every vertex). Indeed, it is not hard to see that a uniformly selected path will be essentially filling, in the sense that the probability that a nonempty open set inside 1U is unvisited goes to zero as c '\.. o. Consider an n x n grid in the unit square, and identify the vertices of the right and bottom boundary arcs to a single vertex woo Let G n denote this graph, and note that the dual graph G~ is obtained by rotating G n 180 0 and shifting appropriately. See Figure 9.3. Let w be a sample from the critical random-cluster measure on G n with parameters q and p = p(q) = y'Q/(1 + y'Q) (which is the conjectured value of Pc = Pc(q)) and let w t be the set of edges in G~ that do not cross edges of w. Then the law of w t is the random-cluster measure on G~ with the same parameters as for w. Let f3 be the set of points in the unit square that are at equal distance from the component of w containing Wo and the component of wt containing the wired vertex of Gt. 1026
BAS IC PROPERTIES OF SLE
921
Figure 9.3: A partially wired grid and its dual. 9.7. If q E ( 0 , 4 ), then as n --+ 00, the law of f3 converges weakly to the trace of SL£It , up to reparamei1ization, where "" = cos I t':Jii/2)' CONJECTU RE
The a bove value of K is derived by applying the predictions [SD87] of the dimensions of t he perimeters of clusters in the random cluster model. Consider a random ~ uniform domino tiling of the unit square, such as in the top right of Figure 9.4. Taking the union with another independent domino tiling with the two lower cornel' squares removed, one gets a path joining t he two removed squares; see Figure 9.4 . This model was considered by [RHA97]. Richard Kenyon proposed t he following Problem 9.8. Does th is double-domino model converge weakly (up to reparametrization) to SL£ 4 as t he mesh of t he grid tends to zero?
Kenyon also produced some calculations for this model that agree with the corresponding calcula t ions for 5L£4. Let a E JR, c > 0, and consider the following random walk S on the grid c Z2. Set So = 0, and inductively, we construct SII+1 given { So, ... , SrI} ' If Sn fJ. 1IJ, t hen stop. Otherwise, let h n : c Z2 --+ [0, 1] be the function which is 1 outside of 1IJ, on {So, .. . , Sn}, and discrete-harmonic elsewhere. Now choose 8 11 + 1 among t he neighbors v of SII such t hat h{v) > 0, with probability proportional to h(v)u. When a = 1, this is called Laplacian random walk, and is known to be equivalent to loop-erased random walk. When a = 0 this walk is believed to be closely related to t he bounda ry of percolation clusters. The following problem came up in discussions with Richard Kenyon .
°
Problem 9.9 . For what values of a does this process converge weakly to radial SLE? \Vhat is the c01'l'csponcience K = K(a)? 1027
922
ST EFFEN ROHDE AND ODED SC HRAi\It.l
Figure 9.4: The double domino path. IVlotivated in part by Duplantier 's duality conjecture 9.5, in [LSW03] it is conjectured t hat chordal 8L£8/3 is t he scaling limi t of t he self-avoiding walk in a half-plane . (That is, the limit as J '\. 0 of the limit as n / 00 of the law of a random-uniform n-step simple path from 0 in SZ2 n IHI.) Some strong support to this conjecture is provided there, which in turn can be viewed as
fur ther support for the duali ty in 9.5. Many of the processes believed to converge to chordal SLE have the following revers ibility property. Up to reparametrization, the path from a to b in D (a, bEaD) has the same distribut ion as t he pat h from b to a. It is t hen reasonable to conject ure: C ONJECTU RE 9.10.
The chordal SLE trace is reversible for
Ii
E
[0,8J .
It is not too hard to see that this fails for fi > 8. In particular, when fi > 8 the chordal SLE path is more likely to visit 1 before visit ing i . Since inversion in the unit circle preserves 1 and i, the probabili ty for visiting i before 1 would have to be 1/ 2 if one assumes reversibility. To apply this argument one has to note that a.s. t he path visits 1 and i exactly once.
Update. The above conjecture is known to hold when fi = 0, 2,8/3, 6, 8. The cases Ii = 2,8 follow from [LSW) (though some additional work is needed in the case where Ii = 2, since the convergence of the LERW is to radial SLE), the case fi = 6 foll ows from Smil·nov's Theorem [SmiDl], and the case fi = 8/3 follows from the characte rizat ion in [LSWD3] of SLEsj3 as the restrict ion measure with exponent 5/8. 1028
BASIC PROPERTIES OF SLE
923
Some conjectures and simulations concerning the winding numbers of SLE's are described in [WW].
Acknowledgments. The authors thank Richard Kenyon for useful discuss ions and Jeff Steiff for numerous comments on a previous version of this paper. UNIVERSITY OF WASHINGTON, SEATTLE, WA
E-mail address: [email protected] Web page: HTTP://WWW.MATH.WASHINGTON.EDU/~ROHDE/
MICROSOFT RESEARCH CORPORATION, REDMOND, WA
E-mail address:[email protected] Web page: HTTP:RESEARCH.MICROSOFT.COM/~SCHRAMM/
REFERENCES
V. AHLFORS, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973.
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J. CARDY,
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B. DUPLANTIER,
The dimension of the SLE curves; arXiv:math.PR/0211322.
BEFFARA,
Conformal invariance and percolation; arXiv:math-ph/0103018.
and N. MAKAROV, Aggregation in the plane and Loewner's equation, Comm. Math. Phys. 216 (2001), 583-607. SLE(I\;, p) martingales and duality; arXiv:math.PR/0303128. Conformally invariant fractals and potential theory, Phys. Rev.
Lett., 84 (2000), 1363-1367. Stochastic Calculus: A Practical Introduction, CRC Press, Boca Raton, FL, 1996.
[Dur96]
R. DURRETT,
[EMOT53]
A. ERDELYI, W. MAGNUS, F. OBERHETTINGER, and F. G. TRICOMI, Higher Transcendental Functions Vol. I. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York, 1953.
[JM95]
P. W. JONES
[KR97]
P. KOSKELA
and N. G. MAKAROV, Density properties of harmonic measure, Ann. of Math. 142(1995), 427-455.
and S. ROHDE, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593-609.
[LSW01a] G. F. LAWLER, O. SCHRAMM, and W. WERNER, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), 237-273. [LSWOlb] - - - , Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001), 275-308. [LSW02]
- - - , Values of Brownian intersection exponents. III. Two-sided exponents, Ann. Inst. H. Poincare Probab. Statist. 38 (2002), 109-123.
[LSW03]
- - - , Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), 917-955 (electronic).
[LSW]
- - - , Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), 939-995.
[Mak98]
N. G. 1-62.
MAKAROV,
Fine structure of harmonic measure, Algebra i Analiz 10 (1998),
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STEFFEN ROHDE AND ODED SCHRAMM
D. MARSHALL
and
CH. POMMERENKE,
S. ROHDE,
The Loewner differential equation, preprint.
Boundary Behaviour of Conformal Maps, Springer-Verlag, New
York,1992. [RHA97]
R. RAGHAVAN, C. L. HENLEY, and S. L. AROUH, New two-color dimer models with critical ground states, J. Statist. Phys. 86 (1997), 517-550.
[RY99]
D. REVUZ and M. YOR, Continuous martingales and Brownian motion, Third edition, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences 293, Springer-Verlag, New York, 1999.
[SchOO]
O. SCHRAMM, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221-288.
[SD87]
H. SALEUR and B. DUPLANTIER, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58 (1987), 2325-2328.
[SmiOl]
S. SMIRNOV, Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), 239-244.
[SS]
O. SCHRAMM and S. SHEFFIELD, The harmonic explorer and its convergence to SLE( 4); arXiv:math.PR/0310210.
[Wer01]
W. WERNER,
[WW]
B. WEILAND and D. B. WILSON, Winding angle variance of Fortuin-Kasteleyn contours, Phys. Rev. E, to appear.
Critical exponents, conformal invariance and planar Brownian motion, in European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math. 202, 87-103, Birkhiiuser, Basel, 200l.
(Received August 8, 2001)
1030
Acta Math., 202 (2009), 21-137 DOl: 10.1007/s11511-009-0034-y © 2009 by Institut Mittag-Leffler. All rights reserved
Contour lines of the two-dimensional discrete Gaussian free field by ODED SCHRAMM
SCOTT SHEFFIELD
Microsoft Research Redmond, WA, U.S.A.
New York University New York, NY, U.S.A.
Contents 1. Introduction . . . . . . . . . 1.1. Main result . . . . . . 1.2. Conditional expectation and the height gap 1.3. GFF definition and background . . . . . . . . 1.4. SLE background and prior convergence results. 1.5. Precise statement of main result 1.6. Outline. 1. 7. Sequel . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . 2.1. A few general properties of the DGFF . 2.2. Some assumptions and notation. 2.3. Simple random walk background 3. The height gap in the discrete setting 3.1. A priori estimates .. . 3.2. Near independence .. . 3.3. Narrows and obstacles. 3.4. Barriers . . . . . . . . . 3.5. Meeting of random walk and interface . 3.6. Coupling and limit . . . . . . . . . 3.7. Boundary values of the interface .. . 4. Recognizing the driving term . . . . . . . . 4.1. About the definition of SLE(x; [iI, [i2) 4.2. A coordinate change . . . . . . . . . . . 4.3. The Loewner evolution of the DGFF interface . 4.4. Approximate diffusions . . . . . . . 4.5. Back to chordal . . . . . . . . . . . . 4.6. Loewner driving term convergence
22 22 25
29 32 35 38
39 40 40 41 42 44 44 55 59 65 71 90
93 103 104
105 106 111 124 126
During the revision process of this article, Oded Schramm unexpectedly died. I am deeply indebted for all I learned working with him, for his profound personal warmth, for his legendary vision and skill. There was never a better colleague, never a better friend. He will be dearly missed. (Scott Sheffield)
I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_32, 1031
22
O. SCHRAMM AND S. SHEFFIELD
4.7. Caratheodory convergence . . . . . . 4.8. Improving the convergence topology 5. Other lattices References . . . . . . . . . . . . . . . . . . . . .
128 129 133 135
1. Introduction 1.1. Main result
The 2-dimensional massless Gaussian free field (GFF) is a 2-dimensional-time analog of Brownian motion. Just as Brownian motion is a scaling limit of simple random walks and various other 1-dimensional systems, the GFF is a scaling limit of several discrete models for random surfaces. Among these is the discrete Gaussian free field (DGFF), also called the harmonic crystal. We presently discuss the basic definitions and describe the main results of the current work, postponing an overview of the history and general context to §1.3. Let G = (V, E) be a finite graph and let Va C V be some non-empty set of vertices. Let
n be the set of functions h: V --+ m. that are zero on Va. Clearly, n may be identified with m.V\ Va. The DGFF on G with zero boundary values on Va is the probability measure on n whose density with respect to the Lebesgue measure on m.V\ Va is proportional to exp (
L
-~(h(V)-h(U))2).
(1.1)
{u,v}EE
Note that under the DGFF measure, h is a multi-dimensional Gaussian random variable. Moreover, the DGFF is a rather natural discrete model for a random field: the term -~(h(v)-h(u))2 corresponding to each edge {u,v} penalizes functions h which have a large gradient along the edge. Now fix some function ha: Va--+m., and let nha denote the set of functions h: v--+m. that agree with ha on Va. The probability measure on nha whose density with respect to the Lebesgue measure on m.v\Va is proportional to (1.1) is the DGFF with boundary values given by ha. Let TG be the usual triangular grid in the complex plane, i.e., the graph whose vertex set is the integer span of 1 and ei7r/3=~(1+iV3), with straight edges joining v and w whenever Iv-wl=1. A TG-domain Dcm.2~C is a domain whose boundary is a simple closed curve comprised of edges and vertices in TG. Let V = VD be the set of TGvertices in the closure of D, let G=G D be the induced subgraph of TG with vertex set
VD , and write Va=8DnVD . While introducing our main results, we will focus on graphs G D and boundary sets Va of this form (though analogous results hold if we replace TG with another doubly periodic planar graph; see §1.5). 1032
23
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
We may assume that any function f: V ---+lR is interpolated to a continuous function on the closure of D which is affine on each triangle of TG. We often interpret
f
as a
surface embedded in 3 dimensions and refer to f (v) as the height of the surface at v. Let oD=o+Uo_ be a partition of the boundary of a TG-domain D into two disjoint arcs whose endpoints are midpoints of two distinct TG-edges in oD. Fix two constants
a, b>O. Let h be an instance of the DGFF on (CD, Va), with boundary function ha equal to -a on the vertices in 0_ and equal to b on the vertices in 0+. Then h (linearly interpolated on triangles) almost surely assumes the value zero on a unique piecewise linear path Ih connecting the two boundary edges containing endpoints of 0+. In §1.4, we will briefly review the definition of SLE(4) (a particular type of random chordal path connecting a pair of boundary points of D whose randomness comes from a I-dimensional Brownian motion), along with the variants of SLE(4) denoted SLE(4;
(]l, (]2).
THEOREM
Our main result, roughly stated, is the following. 1.1. Let D be a TG-domain, oD=o+Uo_ and let hand Ih be as above.
There is a constant A>O such that if a=b=A, then as the triangular mesh gets finer, the random path Ih converges in distribution to SLE(4). If a, b:;?::A are not assumed to equal A, then the convergence is to SLE(4;ajA-l,bjA-l). See §1.5 for a more precise version, which describes the topology under which the convergence is attained. As explained there, we can also prove convergence in a weaker form when the conditions a, b:;?::A are relaxed. We will elaborate on the role of the constant A in §1.2. This constant depends only
vr;
on the lattice used. Although we do not prove it in this paper, for the triangular grid the value of A is ATG:=3- l / 4
(see §1.7).
Figure 1.1 illustrates a dual perspective on an instance of Ih. Here, each vertex in the closure of a rhombus-shaped TG-domain D is replaced with a hexagon in the honeycomb lattice. Call hexagons positive or negative according to the sign of h. Then there is a cluster of positive hexagons that includes the positive boundary hexagons, a similar cluster of negative hexagons, and a path 1 forming the boundary between these two clusters. Figure 1.1 depicts a computer generated instance of the DGFFwith ±A boundary conditions-and the corresponding I. Followed from bottom to top, the interface 1 turns right when it hits a negative hexagon, left when it hits a positive hexagon. It closely tracks the boundary-hitting zero contour line Ih in the following sense: the edges in 1 are the duals of the edges of TG that are crossed by Ih. This is because h is almost surely non-zero at each vertex in V, so whenever a zero contour line contains a point on an edge of TG, h must be positive on one endpoint of that edge and negative on the other; hence the dual of that edge separates a positive hexagon from a 1033
24
0,
,,
,
.
\.
,"
•• • . " •
•~,
J'
•
,"
011
' ;, I
I ,"
'J
• •• ,, I , i
Figure 1.1. (a) OG F F
AND S. SHEFFIELD
. .,
•
C
SCHRA~ I I\!
,
,
(.
".• •
I' .
"
,
j
,• ,.
"
"
,, . I
j
"
, ,•
,
• ,
,..
... ,.. .-
,
~I,
'(
,
•,
" •
~
..
I,
,
'~ '
a 9Ox90 hexagon array with boundary values >. on the right and
- >. on the left; faces shaded by height. (b) Surface plot of DGFF .
1034
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
25
negative hexagon. In the fine mesh limit, there will be no difference between 1 and 1h. Thus (by Theorem 1.1) the path in Figures 1.1 (a) and 1.2 (a) approximates SLE(4), while the path of Figure 1.3 (a) approximates SLE(4; 2, 2). We will state and prove most of our results in terms of the dual perspective displayed in the figures.
1.2. Conditional expectation and the height gap
We derive the following well-known facts as a warm-up in §2.1 (see also, e.g., [Gil).
Boundary influence: The law of the DGFF with boundary conditions ha: Va--+R is the same as that of the DGFF with boundary conditions 0 plus a deterministic function
ha: V --+R which is the unique discrete-harmonic interpolation of ha to V. (By discreteharmonic we mean that for each VEV\ Va, the value h(v) is equal to the average value of h(w) over w adjacent to v.) In particular, the expected value of h(v) is discrete-harmonic in V\ Va.
Markov property: Let h: V --+R be a random function whose law is the DGFF on C with some boundary values ha on Va. Then, given the values of h on a superset Vo~Va, the conditional law of h is that of a DGFF on C with boundary set Vo and with boundary values equal to the given values. From these facts it follows that conditioned on the path 1 described in the previous section and on the values of h on the hexagons adjacent to 1, the expected value of h is discrete-harmonic in the remainder of CD. Figures 1.2 and 1.3 illustrate the expected value of h conditioned on the values of h on the hexagons adjacent to 1. The reader may observe in Figure 1.2 that although the expected value of h given the values along 1 varies a great deal among hexagons close to 1, the expected value at five or ten lattice spacings away from 1 appears to be roughly constant along either side of 1. On the other hand, in Figure 1.3, away from 1, the expected height appears to be a smooth but non-constant function. In a sense we make precise in §3 (see Theorem 3.28), the values -..\. and..\. describe the expected value of h, conditioned on 1, at the vertices near (but not microscopically near) the left and right sides of 1; in the fine mesh limit there is thus an "expected height gap" of 2"\' between the two sides of 1. In Figure 1.2 the height expectation appears constant away from 1, because the boundary values of
±..\. are the same as the expected values near (but not microscopically near) 1. Once we have established the height gap result, the proof of Theorem 1.1 (at least for the simplest case that the boundary conditions are -..\. and ..\.) is similar to the proof that the harmonic explorer converges to SLE(4), as given by the present authors in [SS], 1035
26
O . SCHRAhl 1>[ AND S. SHEFFIEL D
Figure 1.2. (a ) EXI>e<::tation of OCFF with boundary val ues bord ering the interface. (b) Surface plo t of the above.
1036
±..\
given its \'8lucs at hexagons
CONTOUR LI NES OF TilE TWO-D1l\lENS10NA L. DISCRET E GAUSSIAN FREE FIEL.D
F'igure 1.3. (a) Expectation of DOFF' given its values a t hexagons bordering the interface; exterior boundary values are - 3A on left, and 3A on right. (b) Surface plot of t he above.
1037
27
28
o.
SCHRAMM AND S. SHEFFIELD
which, in turn, follows the same strategy as the proof of convergence of the loop-erased random walk to SLE(2) and the uniform spanning tree Peano curve to SLE(8) in [LSW4]. We will now briefly describe some of the key ideas in the proof of the height gap result. The main step is to show that if one samples a vertex z on "( according to discreteharmonic measure viewed from a typical point far away from ,,(, then the absolute value of
h(z) is close to independent of the values of h (and the geometry of "() at points that are not microscopically close to z. In other words, if we start a random walk S at a typical point in the interior of D and stop the first time it hits a vertex z which either belongs to Va or corresponds to a hexagon incident to ,,(, then (conditioned on z tf:. Va) the random variable Ih(z)1 (and in particular its conditional expectation) is close to independent of the behavior of"( and h at vertices far away from z. To prove this, we will actually prove something stronger, namely that (up to multiplication by -1) the collection of all of the values of h (and the geometry of "() in a microscopic neighborhood of z is essentially independent of the values of h (and the geometry of "() at points that are not microscopically close to z. One consequence of our analysis is Theorem 3.21, which states that if one takes z to be the origin of a new coordinate system and conditions on the behavior of "( and S outside of a ball of radius R centered at z and S starts outside that ball, then as R tends to infinity the conditional law of the interface "( has a weak limit (which is independent of the sequence of boundary conditions chosen), which is the law of a random infinite path "( on the honeycomb grid TG* (almost surely containing an edge adjacent to the hexagon centered at the origin
z=O). We will define a function of such infinite paths "( which (in a certain precise sense) describes the expected value of Ih( z) I conditioned on "(; the value A is the expectation of this function when "( is chosen according to the limiting measure described above. We remark that many important problems in statistical physics involve classifying the measures that can arise as weak limits of Gibbs measures on finite systems. In such problems, showing the uniqueness of the limiting measure often involves proving that properties of a random system near the origin are approximately independent of the properties of the system far away from the origin. In our case, we need to prove that in some sense the behavior of the triple (h, ,,(, S) near the the origin (i.e., the first point S hits "() is close to independent of the behavior of (h, ,,(, S) far from the origin. Very roughly speaking, our strategy will be to describe the joint law of (h, ,,(, S) near the origin and (h, ,,(, S) far from the origin by considering a different measure in which the two are independent and weighting it by the probability that the inside and outside configurations properly "hook up" with one another. To get a handle on these "hook up" probabilities, we will need to develop various techniques to control the probabilities (conditioned on the values of h on certain sets) that certain zero-height level lines hook 1038
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
29
up with one another, as well as the probabilities that these level lines avoid certain regions. We will also need bounds on the probability that there exist clusters of positive or negative hexagons crossing certain regions; these are roughly in the spirit of the RussoSeymour-Welsh theorems for percolation, but the proofs are entirely different. All of the height gap related results are proved in §3.
1.3. GFF definition and background To help put our DGFF theorems in context and provide further intuition, we now briefly recall the definition of the (continuum) GFF and mention some basic facts described, e.g., in [Sh]. Let Hs(D) be the set of smooth functions supported on compact subsets of a planar domain D, and let H(D) be its Hilbert space completion under the Dirichlet inner product (f,g)\1= \If·\lgdx, where dx refers to area measure. We define an instance of the Gaussian free field to be the formal sum
JD
00
h= LOOjfj, j=1
where the ooj's are independent identically distributed I-dimensional standard (unit variance, zero mean) Gaussians and the fj's are an orthonormal basis for H(D). Although the sum does not converge pointwise or in H(D), it does converge in the space of distributions [Sh]. In particular, the sum 00
(h,gh:= LOOj(fj,g)\1 j=1
is almost surely convergent for every gEHs(D). It is worthwhile to take a moment to compare with the situation where D is 1dimensional. If D were a bounded open interval in JR, then the partial sums of 00
h= LOOjiJ j=1
would almost surely converge uniformly to a limit, whose law is that of the Brownian bridge, having the value zero at the interval's endpoints. If D were the interval (0,00), then the partial sums would converge (uniformly on compact sets) to a function whose law is that of ordinary Brownian motion B t , indexed by tE[O, 00), with Bo=O [Sh]. Let g be a conformal (i.e., bijective analytic) map from D to another planar domain D'. When g is a rotation, dilation, or translation, it is obvious that
1039
30
for any
O. SCHRAMM AND S. SHEFFIELD
iI, hEHs(D), and an elementary change of variables calculation gives this equal-
ity for any conformal g. Taking the completion to H(D), we see that the Dirichlet inner product-and hence the 2-dimensional GFF-is invariant under conformal transformations of D. Up to a constant, the DGFF on a TG-domain D can be realized as a projection of the GFF on D onto the subspace of H(D) consisting of functions which are continuous and are affine on each triangle of D [Sh]. Note that if f is such a function, then
(j, f)\1
=
~ 2)lf(k)- f(jW+lf(l)- f(jW+lf(l)- f(kW),
where the sum is over all triangles (j, k, l) in VD . This is because the area of each triangle is ~ v'3 and the norm of the gradient squared in the triangle is
Since each interior edge of D is contained in two triangles, for such
Ilfll~ = ~
L
If(k)- f(j)12+
{j,k}EEr
2~
L
f,
If(k)- f(j)12,
(1.2)
{j,k}EEa
where E J and Ea are the interior and boundary (undirected) edges of TG in
15. We will
refer to the sum 2:Er If(k) - f(jW as the discrete Dirichlet energy of f. It is equivalentup to the constant factor 3- 1 / 2 and an additive term depending only on the boundary values of f-to the Dirichlet energy (j, fh of the piecewise affine interpolation of f to D. The above analysis suggests a natural coupling between the GFF and a sequence of DGFF approximations to the GFF (obtained by taking finer mesh approximations of the same domain). The GFF can also be obtained as a scaling limit of other discrete random surface models (e.g., solid-on-solid, dimer-height-function, and 'Vcp-interface models) [Ke] [NS], [Sp]. Its Laplacian is a scaling limit of some Coulomb gas models, which describe random electrostatic charge densities in 2-dimensional domains [F], [FS], [Ko], [KT], [Sp]. Physicists often use heuristic connections to the GFF to predict properties of 2dimensional statistical physics models that are not obviously random surfaces or Coulomb gases (e.g., Ising and Potts models, O(n) loop models) [dN]' [DMS], [D], [Ka], [KN], [NI], [N2]. As a model for the field theory of non-interacting massless bosons, the GFF is a starting point for many constructions in quantum field theory, conformal field theory, and string theory [BPZ], [DMS], [Ga], [GJ]. Because ofthe conformal invariance ofthe GFF, physicists and mathematicians have hypothesized that discrete random surface models that are believed or known to converge 1040
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31
to the Gaussian free field (e.g., the discrete Gaussian free field, the height function of the oriented O(n) loop model with n=2, height functions for domino and lozenge tilings) have level sets with conformally invariant scaling limits [Co], [DSl], [DS2], [SD], [KDH], [HK], [Ke], [KH], [KHS], [Nl]. Our results confirm this hypothesis for the discrete Gaussian free field. Various properties of the DGFF contour lines (such as winding exponents and the fact that the fractal dimension is ~) have been predicted correctly in the physics literature [Co], [DS2], [DSl], [SD], [HK], [KDH], [KH], [KHS], [Nl]. The techniques used to make these predictions are also described in detail in the survey papers [D], [KN], [N2]. Analogous results about winding exponents and fractal dimension have now been proved rigorously for SLE [Sch], [RS], [B]. The study of level lines of the DGFF and related random surfaces is also related to the study of equipotential lines of random charge distributions in statistical physics. The so-called 2-dimensional Coulomb gas is a model for electrostatics in which the force between charged particles is inversely proportional to the distance between them. In this model, a continuous function f E Hs (D) is the Coulomb gas electrostatic potential function ("grounded" at the boundary of D) of -fj.f, when fj.f is interpreted as a charge density function. The value (j, f)v is then the total potential energy-also called the energy of assembly of the charge distribution -fj.f. In the Coulomb gas model, this is the amount of energy required to move from a configuration in which the charge density is zero throughout D to a configuration in which the charge density is given by -fj.f. In statistical physics, it is often natural to consider a probability distribution on configurations in which the probability of a configuration with potential energy H is proportional to e- H . If (} is a smooth charge distribution, then its energy of assembly is given by (-fj. -1 (}, -fj. -1 (})v = ((}, -fj. -1 (}); if we define (} to be the standard Gaussian in fj.H(D) determined by this quadratic form, then (} is the Laplacian of the Gaussian free
field (which, like the GFF itself, is well defined as a random distribution but not as a function). In other words, the Laplacian of a Gaussian free field is a random distribution that we may interpret as a model for random charge density in a statistical physical Coulomb gas. However, we stress that when physicists refer to the Coulomb gas method for O(n) model computations, they typically have in mind a more complicated Coulomb gas model in which the charges are required to be discrete (i.e., (} is required to be a sum of unit positive and negative point masses) and hard core constraints may be enforced. The surveys [BEF], [Gi], [Sp] contain additional references on lattice spin models that have the GFF as a scaling limit and Coulomb gas models that have its Laplacian as a scaling limit-for example, the harmonic crystal (also known as the discrete Gaussian free 1041
32
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SCHRAMM AND S. SHEFFIELD
field) with quadratic nearest-neighbor potential, the more general anharmonic crystal,
the discrete-height Gaussian (where h is a function on a lattice, with values restricted to integers), the Villain gas (where h is a function on a lattice and the values ofits discrete Laplacian p=-~h are restricted to integers), and the hard core Coulomb gas (where h is a function on a lattice and its discrete Laplacian p=-~h is ±1 valued). The physics literature on applications of the GFF to field theory and statistical physics is large, and the authors themselves are only familiar with parts of it. Outside of these areas, there is a body of experimental and computational research on contour lines of random topographical surfaces, such as the surface of the earth. Mandelbrot's famous How Long Is the Coast of Britain? [M], which prefigured the notion of "fractal" introduced by Mandelbrot years later, is an early example. The results about contour lines in these studies (including fractal dimension computations) are less detailed than the ones provided here and are not all mathematically rigorous. However, some of the models are similar in spirit to the GFF, involving functions whose Fourier coefficients are independent Gaussians. An eclectic overview of this literature appears in [I].
1.4. SLE background and prior convergence results
We now give a brief definition of (chordal) SLE(x) for x>o. See also the surveys [W], [KN], [L4], [Cal or [L3]. The discussion below along with further discussion of the special properties of SLE(4) appears in another paper by the current authors [SS]. That paper shows that SLE( 4) is the scaling limit of a random interface called the harmonic explorer (designed in part to be a toy model for the DGFF contour line addressed here). Let T>O. Suppose that "(: [0, T]--+iHi is a continuous simple path in the closed upper half-plane iHi which satisfies "([O,T]nlR=b(O)}={O}. For every tE[O,T], there is a unique conformal homeomorphism gt: 1HI\ "([0, t] which satisfies the so-called hydrodynamic normalization at infinity lim (gt (Z ) - z) = 0.
z-+oo
The limit
.
capoo ("([0, t]) := hm
z-+oo
z(gt(z)-z) 2
is real and monotone increasing in t. It is called the (half-plane) capacity of "([0, t] from 00, or just capacity, for short. Since capoo("([O, t]) is also continuous in t, it is natural to reparameterize,,( so that capoo("([O, t])=t. Loewner's theorem states that in this case the
maps gt satisfy his differential equation
(1.3)
1042
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
33
where Wt=gt(-r(t)). (Since "((t) is not in the domain of definition of gt, the expression
gt(-r(t)) should be interpreted as a limit of gt(z) as z-+"((t) inside lHI\,,([O, t]. This limit does exist.) The function tr--+ W t is continuous in t, and is called the driving parameter for "(. One may also try to reverse the above procedure. Consider the Loewner evolution defined by the ordinary differential equation (ODE) (1.3), where W t is a continuous, real-valued function. For a fixed z, the evolution defines gt(z) as long as Igt(z) - Wtl is bounded away from zero. For zEH let Tz be the first time t;?O in which gt(z) and W t collide, or set Tz=OO if they never collide. Then gt(z) is well defined on {zEH:Tz;?t}. The set Kt:={ZEH:Tz~t} is sometimes called the evolving hull of the evolution. In the case discussed above where the evolution is generated by a simple path "( parameterized by capacity and satisfying "((t)ElHI for t>O, we have Kt="([O,t]. The path of the evolution is defined as "((t)=limz-twt g;l(Z), where z tends to W t from within the upper half-plane 1HI, provided that the limit exists and is continuous. However, this is not always the case. The process (chordal) SLE(x) in the upper halfplane, beginning at 0 and ending at 00, is the path "((t) when W t is BtJX, where Bt=B(t) is a standard I-dimensional Brownian motion. (Standard means B(O)=O and
E[B(t?]=t, t;?O. Since (Btv'k:t;?O) has the same distribution as (Bxt:t;?O), taking Wt=B xt is equivalent.) In this case, almost surely "((t) does exist and is a continuous path. See [RS] (x#8) and [LSW4] (x=8). We now define the processes SLE(x; el, e2). Given a Loewner evolution defined by a continuous Wt, we let Xt and Yt be defined by Xt:=sup{gt(x):x
8tYt =
2 W Yt- t
(1.4)
for all t such that Xt<Wt
2dt dYt= Yt- W' t
(1.5)
noting that existence and uniqueness of solutions to this SDE (at least from the initial time r until the first s>r for which either xs=Ws or Ws=Ys) follow easily from standard results in [RY]. (The Xt and Yt are called force points because they apply a "force" affecting the drift of the process W t by an amount inversely proportional to their distance from
Wd 1043
34
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SCHRAMM AND S. SHEFFIELD
Some subtlety is involved in extending the definition of SLE(x; /?1, /?2) beyond times when W t hits the force points, and in starting the process from the natural initial values
xo=Wo=Yo=O. This is closely related to the issues which come up when defining the Bessel processes of dimension less than 2 and will be discussed in more detail in §4. Although many random self-avoiding lattice paths from the statistical physics literature are conjectured to have forms of SLE as scaling limits, rigorous proofs have thus far appeared only for a few cases: site percolation cluster boundaries on the hexagonal lattice (SLE(6), [Sm]; see also [eN]), branches (loop-erased random walk) and outer boundaries (random Peano curves) of uniform spanning trees (forms of SLE(2) and SLE(8), respectively, [LSW4]), the harmonic explorer (SLE(4), [SS]), and boundaries of simple random walks (forms of SLE(~), [LSW3]). In the latter case, conformal invariance properties follow almost immediately from the conformal invariance of 2-dimensional Brownian motion. In each of the other cases listed above, the initial step of the proof is to show that a certain function of the partially generated paths ,([0, t]), which is a martingale in t when, is SLE(x) for the appropriate x, has a discrete analog which is (approximately or exactly) a martingale for the discrete paths and is approximately equivalent to the continuous version in the fine mesh limit. For loop-erased random walk, harmonic explorer, and uniform spanning tree Peano curves, this initial step is the easy part of the argument; it follows almost immediately from the fact that simple random walk converges to Brownian motion. The analogous step for site percolation on the hexagonal lattice, as given by [Sm] , is an ingenious but nonetheless short and simple argument. By contrast, the analogous step in this paper (which requires the proof of the height gap lemma, as given in §3) is quite involved; it is the most technically challenging part of the current work and includes many new techniques and lemmas about the geometry of DGFF contours that we hope are interesting for their own sake. Another way in which the DGFF differs from percolation, the harmonic explorer, and the uniform spanning tree is that it has a natural continuum analog (the GFF) which can be easily rigorously constructed without any reference to SLE, and which is itself (like Brownian motion) an object of great significance. It becomes natural to ask whether the DGFF results enable us to define the "contour lines" of the continuum GFF in a canonical way; we plan to answer this question (affirmatively) in a subsequent work (see §1.7). A final difference is that, for the DGFF, there is a continuum of choices for left and right boundary conditions (a and b) which are equally natural a priori, so we are led to consider a family of paths SLE(4; a/>.-l, b/>.-l) instead of simply SLE(4). (The case a=b=O is particularly natural; see Figure 1.4.) In these processes, the driving 1044
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
35
parameters W t are generally no longer Brownian motions (rather, they are continuous semimartingales with constant quadratic variation and a drift term that can become singular on a fractal set). Proving driving parameter convergence to these processes requires some rather general convergence infrastructure (§4.4), which we hope will be useful in other settings as well.
1.5. Precise statement of main result
Let 1HI be the upper half-plane. Let D be any TG-domain and let aD=a+U/Y- be a partition of the boundary of D into two disjoint arcs whose endpoints are midpoints of two TG-edges contained in aD. As before, let V denote the vertices of TG in
15. Let
ha=-a on a_nV and ha=b on a+nv, where a and b are positive constants.
Let h: V ---+lR be an instance of the DGFF with boundary conditions ha. Let rPD be any conformal map from D to 1HI that maps a+ bijectively onto the positive real ray (0, (0). (Note that rPD is unique up to positive scaling.) There is almost surely a unique interface 'YcD between hexagons in the dual grid TG* containing TG-vertices where h is positive and such hexagons where h is negative, such that the endpoints of "I are on aD. In fact, the endpoints of "I are the same as the endpoints of a+. (As mentioned above, this interface "I stays within a bounded distance from the zero-height contour line 'YD of the affine interpolation of h.) Now, rPD0'Y is a random path on 1HI connecting 0 to 00. We will show that this path converges to a form of SLE( 4). Rather than considering a fixed domain jj and a sequence of discrete domains
Dn approximating D, with the mesh tending to 0, we will employ a setup that is more general in which the mesh is fixed (the triangular lattice will not be rescaled), and we consider domains D that become "larger". The correct sense of "large" is measured by rD = rD,:= rad (D), r/(i)
where radx(D) denotes the radius of D viewed from x, i.e., infy9!D Ix-YI. Of course, if rP"i}(i) is at a bounded distance from aD, then the image of the triangular grid under
rPD is not fine near i, and there is no hope for approximating SLE by rPD0'Y. We have chosen to use 1HI as our canonical domain (mapping all other paths into 1HI), because it is the most convenient domain in which to define chordal SLE. However, to make the completion of 1HI a compact metric space, we will endow 1HI with the metric it inherits from its conformal map onto the unit disk 1U. Namely, we let d* ( . , . ) be the metric on iHiu{ oo} given by d*(z, w) = 1\Ii(z) - \Ii (w) I, where \Ii(z) :=(z-i)/(z+i) maps iHiu{ oo} onto D. If zEiHi, then d*(zn,z)---+O is equivalent to IZn-zl---+O, and d*(zn,oo)---+O is equivalent to IZnl---+oo. 1045
36
O. SCHRAMM AND S. SHEFFIELD
If '/'1 and 12 are distinct unparameterized simple paths in H, then we define dubl' 12)
to be the infimum over all pairs (T/l, T/2) of parameterizations of 11 and 12 in [0,1] (i.e., T/j: [0, 1]-tH is a simple path satisfying T/j([O, 1])=Ij for j=l, 2) of the uniform distance suP{d*(T/l(t),T/2(t)):tE[0, I]} with respect to the metric d*. Our strongest result is in the case where a, b?;)". We prove the following result. THEOREM
1.2. There is a constant )..>0 such that if a, b?;).., then, as rD-too, the
random paths ¢Dol described above converge in distribution to SLE(4;a/)..-I,b/)..-I) with respect to the metric du . In other words, for every E>O there is some R=R(E) such that if rD>R, then there is a coupling of ¢Dol and a path ISLE whose law is that of SLE( 4; a/ ),,-1, b/ ),,-1) such that P[dU(¢DoI,ISLE) >E] <E.
We first comment that it follows that when rD is large, I is "close" to ¢r}olsLE. For example, if rD-too and rr} D tends to a bounded domain jj whose boundary is a simple closed path in such a way that the boundaries of the domains may be parameterized to give uniform convergence of parameterized paths and if rr/o+ converges, then rr}I converges in law to the corresponding SLE in D. To prove this from Theorem 1.2, we only need to note that in this case the maps rr}¢r} converge uniformly in HU{oo} (see, e.g., [P, Proposition 2.3]). When we relax the assumption a, b?;)" to a, b>O, we still prove some sort of convergence to SLE(4;a/)..-I,b/)..-I), but with respect to a weaker topology. In fact, we can allow a and b to be zero or even slightly negative, but in this case we need to appropriately adjust the above definition of the interface I' Say that a hexagon in the hexagonal grid TG* dual to TG is positive if either the center v of the hexagon is in D and h(v»O, or VEo+. Likewise, say that the hexagon is negative if vED and h(v)
1.3. For every constant A>O there is a constant Ao=Ao(A»O such that
if a, bE [-Ao, A] and I is the DGFF interface defined above, then as rD-too the Loewner driving term W t of ¢DoI, parameterized by capacity from 00, converges in law to the driving term W t of SLE(4; a/)..-I, b/)..-I) with respect to the topology of locally uniform ~
convergence. That is, for every T,E>O there is some R>O such that if rD>R, then I and SLE(4;a/)..-I,b/)..-I) may be coupled so that with probability at least l-E,
sup{IWt - Wtl: t E [0, T]} < E. 1046
CONTOUR LINES OF T i lE TWO- DI MENSIONAL DISCRETE GA USSIAN FREE FI ELD
37
Figure lA. Zero-height interfaces starting and ending on the boundary, shown for the discrete GF F on a 150x 150 hexagonal array with zero boundary. Interior white hexagons have height greater than zero; interior black hexagons have heig ht less than zero; boundary hexagons (of height zero) a re black. Hexagons that are not inciden t to a zero-height interface that reaches t he boundary are grey.
Some (essentially well-known) geometric consequences of this kind of convergence are proved in §4.7. A particularly interesting case of T heorem 1.3 is t he case a=b=O, corresponding to the DCFF with zero boundary values. In this case, when h is interpolated linearly to triangles, its zero level set will almost surely include a finite number of piecewise linear arcs in D whose endpoints on aD are vertices of TG. A dual representation of this set of arcs is shown in Figure 1.4 . For any fixed choice of endpoints on t he boundary, the interface connecting those endpoints will converge to SLE(4; - 1, - 1). The limit of the complete set of arcs in Figure 1.4 is in some sense a coupl ing ofSLE(4; -1, - 1) processes, one for each pair of boundary points. Finally, we d iscuss the generalizaLions: replace T G wi th an arbi trary weighted doubly periodic planar lattice-i.e. , a connected planar graph iB CIR 2 invariant under two linearly independent translations, T, and T2 , such that every compact subset of IR 2 meets only finitely many venices and edges, together with a map w from the edges of i.5 to the posi tive reals, which is invariant under T, and T 2 . A iB -domain D CIR 2 is a domain whose boundary is a simple closed curve comprised of edges and vertices in i.5. Let V = Vli be the set of i.5-vertices in the closure of D, let G= Go =(V, E) be the induced subgraph of i.5 with vertex set Vo, and write VD= aD n Vo . Given a boundary value function h{J: V{J;IR., the edge weighted DGFF on G has a density 1047
38
O. SCHRAMM AND S. SHEFFIELD
with respect to the Lebesgue measure on IRV\Va which is proportional to exp (
L
-~w( {u, v} )(h(V)-h(u))2).
{u,v}EE
If every face of ® has three edges, then every vertex in the dual graph is an endpoint of
exactly three edges, and the boundary between positive and negative faces can be defined as a simple path in this dual lattice, similar to the one shown in Figure 1.1 (a). If not every face of ® has three edges, then we may "triangulate" ® by adding additional edges to ®, while maintaining the invariance under Tl and T 2 , to make this the case (and set w to zero on these edges so that their presence does not affect the law of the DGFF).
We define the weighted random walk on ® to be the Markov chain with transition probability w({u,v})/L:vIW({u,v'}) from u to v, where we take w({u,v})=O unless u and v are neighbors in ®. It is well known and easy to prove that such a walk on the rescaled lattice c® converges to a linear transformation of time-scaled 2-dimensional Brownian motion when c tends to zero (but since we could not find a reference, we very briefly explain this in §5). It is convenient to replace the embedding of ® into IR2 described above with a linear transformation of that embedding that causes this limit to be standard Brownian motion. THEOREM 1.4. Both Theorems 1.2 and 1.3 continue to hold if TG is replaced by a general weighted doubly periodic planar lattice ®, as described above, provided that ® is embedded in IR2 in such a way that the weighted random walk converges to Brownian motion. If ® is the grid 71}, then one natural way to triangulate ® is to add all the edges of
the form {(x, y), (x+1, y+1)}. Another would be to add the edges {(x, y), (x+1, y-1)}. The above theorem implies, perhaps surprisingly, that the limiting law of the zero-height interface is the same in either case, with no need for a linear change of coordinates.
1.6. Outline In §2 we introduce the basic notation and assumptions that are necessary for the height gap results proved in §3. In §§3.1-3.4 we develop bounds and estimates related to the geometry of zero-height interfaces. The random walk S comes into the picture in §3.5, where we develop results about the near-independence of the triple (h, 'Y, S) on microscopic and macroscopic scales. In §3.6 we apply these results to prove uniqueness of the limiting measure, namely Theorem 3.21, and in §3.7 we apply this to prove our main height gap result, Theorem 3.28. 1048
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
39
Based on Theorem 3.28, the convergence in the case where the boundary values are
±>. is not too hard, using the method from [LSW4]. However, to prove the convergence to SLE(4;a/>.-1,b/>.-1) in §4, we need to contend with a few other issues which stem from the fact that the driving parameter of SLE(4; a/>.-l, b/>.-l) is the solution to an SDE with a drift term that blows up to infinity on a fractal set of times. To overcome these difficulties, we change coordinates to a coordinate system in which the drift terms stay bounded. In §4.4 we define and study approximate diffusions. These are random processes that are not necessarily Markov, but satisfy an approximate discrete version of an SDE. The Loewner driving term of the DGFF interface (before going to the scaling limit) is the approximate diffusion we are interested in. The main point is that an approximate diffusion of an SDE is shown to be close to the corresponding true diffusion satisfying the same SDE, under appropriate regularity conditions. This is how the convergence of the driving term of the DGFF interface to the driving term of the corresponding SLE(x; f21, Q2) is established. In §4.7 and §4.8 more geometric convergence results are deduced from the convergence of the Loewner driving term of the interface. Finally, the rather brief §5 then describes the (very minor) modifications required for the generalization to other lattices, Theorem 1.4.
1.7. Sequel
This paper is actually the first of two papers the current authors are writing about this subject. In the second paper we will make sense of the "contour lines" of the continuum GFF. An instance h of the continuum GFF is a random distribution, not a random function; however, given an instance of the GFF on a domain D, we can project h onto the space of functions which are piecewise linear on a triangulation of D to yield an instance of the DGFF which is, in some sense, a piecewise linear approximation to h. We can then define the level lines of the GFF to be the limits of the level lines of its piecewise linear approximations (after proving that these limits exist). We will also characterize these random paths directly-without reference to discrete approximations-by showing that they are the unique path-valued functions of h which satisfy a simple Markov property. Similar techniques allow us to describe the contour lines of h that form loops (instead of starting and ending at points on the boundary of D). The determination of the value of >. for a given lattice is not too hard, but fits better with the general spirit of our next paper on the subject, in which we will prove, in particular, that >'TG=3- 1 / 4
jf;. If the DGFF is scaled so that its fine mesh limit is
the ordinary GFF, we have >.=jf;.
1049
40
O. SCHRAMM AND S. SHEFFIELD
Acknowledgments. We wish to thank John Cardy, Julien Dubedat, Bertrand Duplantier, Richard Kenyon, Jane Kondev, and David Wilson for inspiring and useful conversations, and thank Alan Hammond and an anonymous referee for numerous remarks on an earlier version of this paper.
2. Preliminaries 2.1. A few general properties of the DGFF In this subsection, we recall a few well-known properties of the DGFF that are valid on any finite graph. Let (V, E) be a finite graph, and let VaCV be a non-empty set of vertices. Let h denote a sample from the DGFF with boundary values given by some function ha: Va ---+ R. When J is a function on V, the discrete gradient
V' J is
the function on the set
of ordered pairs (v,u) such that {V,U}EE defined by V'J((u,v))=J(v)-J(u). When defining the norm of the gradient we sum over undirected edges, i.e., we write
IIV' J(v)112 =
L
{u,v}EE
(j(v)- J(U))2.
(2.1)
Thus, the probability density of h is proportional to e- ll \7h(v)11 2 /2. Therefore, when ha=O on Va, h is a standard Gaussian with respect to the norm IIV'hll on Dirichlet inner product that defines this norm can be written
(j,g)\7 =
L
(j(v)- J(u))(g(v)-g(u)).
n.
The (discrete)
(2.2)
{u,v}EE Now write b..J(V)=Lu~v(j(U)- J(v)), where the sum is over all neighbors u of v. By expanding and rearranging the summands in (2.2), we find
(j, g)\7 = -(b..J, g).
(2.3)
Let VOCVD. We claim that the vector space of functions J: V ---+R that are zero on V\ Vo, and the vector space of functions J: V ---+R that are discrete-harmonic on Vo (i.e., b..J =0 on Vo) are orthogonal to each other with respect to the inner product (." )\7, and together they span RV. This basic observation will be used frequently below. Indeed, that they are orthogonal follows immediately from (2.3), and a dimension count now shows that the two spaces together span R v . The following consequence of this orthogonality property will be used below. Let VocV satisfy Vo:lVa and let ho denote the function that is discrete-harmonic in V\ Vo 1050
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
and equal to h in Vo. Then h-ho and ho are independent random variables, because h=ho+(h-h o) is the corresponding orthogonal decomposition of h. It also follows that h-ho is the DGFF in V\ Vo with zero boundary values on Vo.
(2.4)
Observe that the Markov property and the effect of boundary conditions that were mentioned in §1.2 are immediate consequences of (2.4). An additional useful consequence is that the law of ho is proportional to e-(ho,ho)~/2 times the Lebesgue measure on ]R.vo. We now derive a useful well-known expression for the expectation of h(v)h(u): E[h(v)h(u)]-E[h(v)]E[h(u)] =
~~;(:},
(2.5)
where G(u, v) is the expected number of visits to v by a simple random walk started at u before it hits Va and deg( v) is the degree of v; that is, the number of edges incident with it. (The function G is known as the Green function.) As we have noted in the introduction, h is the sum of the discrete-harmonic extension of ha and a DGFF with zero boundary values. It therefore suffices to prove (2.5) in the case where ha=O on Va. In this case, E[h(v)]=O=E[h(u)]. Setting Gv(u)=G(u,v), we observe (or recall) that !:lGv(u)=- deg(v)lv(u). Thus, h(v) = (h, Iv) = _ (h, !:lGv) (:;;) (h, Gv)v. deg(v) deg(v) If X is a standard Gaussian in ]R.n, and x,yE]R.n, then E[(X·x)(X·y)]=x·y.
Conse-
quently, when ha=O, we have E[h(v)h(u)] = E[(h, Gvhl(h, Gu)-g] = (Gv , Guhl deg(v) deg(u) deg(v) deg(u) -(Gv , !:lGu) G(u, v) (G v ,lu) deg(v) deg(u) deg(v) deg(v) .
This proves (2.5).
2.2. Some assumptions and notation We will make frequent use of the following notation and assumptions: (h) A bounded domain (non-empty, open, connected set) Dcll~? whose boundary aD is a subgraph of TG is fixed. The set of vertices of TG in D is denoted by VD and Va denotes the set of vertices in aD. A constant
A>O is fixed, as well as a function
ha: Va-+lR satisfying Ilhalloo~A. The DGFF on D with boundary values given by ha is denoted by h. Also set V=Vn=VDUVa. 1051
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O. SCHRAMM AND S. SHEFFIELD
We denote by TG * the hexagonal grid which is dual to the triangular lattice TG-so that each hexagonal face of TG * is centered at a vertex of TG. Generally, a TG *-hexagon will mean a closed hexagonal face of TG *. Denote by IE R the union of all TG *-hexagons that intersect the ball B(O, R). Sometimes, in addition to (h) we will need to assume: (D) The domain D is simply connected, and (to avoid minor but annoying triviali-
oD is a simple closed curve. We fix two distinct midpoints of TG-edges Xa and Ya oD. Let the counterclockwise (respectively, clockwise) arc of oD from Xa to Ya be
ties) on
denoted by 0+ (respectively, 0_). If H cD is a TG* -hexagon, we write h(H) as a shorthand for the value of h on the
center of H (which is a vertex of TG). Assuming (h) and (D), let f)+ denote the union of all TG *-hexagons contained in D where h is positive together with the intersection of 15 with TG* -hexagons centered at vertices in 0+. Let f)- be the closure of 15\f)+ (which almost surely consists of TG* -hexagons in D where h
2.3. Simple random walk background We need to recall a very useful property of the discrete-harmonic measure of simple random walk. LEMMA
2.1. (Hit near) Let v be a vertex of the grid TG, and let H be a connected
subgraph of TG. Set d=diamH.
The probability that a simple random walk on TG
started from v exits the ball B(v, d) before hitting H is at most c(dist(v, H)/d)'\ where c and (1 E (0,1) are absolute constants.
Likewise, the same bound applies to the probability that a simple random walk started at some vertex outside B(v,d) will hit B(v,dist(v,H)) before H.
=!.
In fact, we may take (1 The continuous version of this statement is known as the Beurling projection theorem (the extremal case is when H is a line segment). The above statement can probably be deduced from the discrete Beurling theorem as given 1052
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
in [L1, Theorem 2.5.2], though the setting there is slightly different. In any case, since we do not require any particular value for (1, the lemma is rather easily proved directly (see [Sch, Lemma 2.1]). We will also use the (well-known) discrete Harnack principle, in the following form. LEMMA 2.2. (Harnack principle) Let D, V, Va and V D be as in (h), let v, UEVD ,
and let f: V ---+ffi. be discrete-harmonic in VD. (1) If v and u are neighbors, then Ilflloo
If( v) - f( u) 1:( 0(1) dist( v, aD) . (2) If we assume that minimal distance to aD is
f~O
e,
and that there is a path of length C from v to u whose
then
The proof of statement (1) can be obtained by noting that f( v) is the expected value of f evaluated at the first hitting point on Va of a random walk started at v, and observing that a random walk started at v and a random walk started at u may be coupled so that they meet before reaching a distance of r:=dist(v,aD) with probability 1-0(ljr) and walk together after they meet.
(See also the more general [LSW4, Lemma 6.2], for
example.) To prove statement (2), let W:={ wEV:f(w) ~f(v)}, and observe that the maximum principle implies that
W contains a path from v to aD.
If we assume that
C:( !e, then a
random walk started at u has probability bounded away from zero to hit W before aD, which by the optional sampling theorem implies (2) in this case. The case
C>!e now
follows by induction on f2Cj el· We also need the following well-known estimate on the Green function G(u,v).
LEMMA 2.3. Let D, VD and V be as in (h), let u, VEVD, and let c>O. Set r:= dist( u, aD), and suppose that within distance r j c from u there is a connected component
of aD of diameter at least cr. If lu-vl:(~r, say, then GD(v,u) (the expected number of visits to u by a simple random walk starting at v before hitting aD) satisfies GD(v,u) =exp(Oc(l)) log
1
r+1
u-v
+1
1
The probability HD(v,u) that a random walk started from v hits u before aD can be expressed as Gdv, u)jGD(u, u) and hence by the lemma
HD(v,u) =exp(Oc(1)) ( 1-
1053
10g(lu-vl + 1)) log(r+1) .
(2.6)
44
o.
SCHRAMM AND S. SHEFFIELD
Let PR denote the probability that a simple random walk started at 0 does not return to 0 before exiting the ball B(O, R). We now show that the lemma follows from the well-known estimate exp( 0(1)) (2.7) PR = log(R+2) . In the setting of the lemma, consider some vertex wE VD such that Iw - u I> r. It is easy to see that with probability bounded away from zero (by a function of E), a simple random walk started from w will hit aD before u. Hence, q:=min{l-HD(w,U):WEVD \B(u,r)} is bounded away from 0 by a positive function of E. Clearly,
This proves the case u=v of the lemma from (2.7). Now start a simple random walk at a vertex vi=u satisfying Iv-ul:::::;~r, and let the walk stop when it hits aD. It is easy to see that the probability that the walk visits v after exiting the ball B(v, ~Iv-ul), say, is within a bounded factor from H D (v, u). Therefore, the expected number of visits to v after exiting B(v, ~Iv-ul) is within a bounded factor of Gn(v,u). But the former is the same as G D (V,v)-G B (v,lv-ul/4)(V,V). The lemma follows. We have not found a reference proving (2.7) in a way that generalizes to the setting of Theorem 1.4, though the result is well known. In fact, it easily follows from Rayleigh's method, as explained in [DoSn, §2.2]. In that book, the goal is to show that PR-+O, as R-+oo, for lattices in the plane but not in JR3. However, the method easily yields the quantitative bounds (2.7).
3. The height gap in the discrete setting 3.1. A priori estimates This subsection contains some technical (and uninspiring) estimates that are necessary to carry out the technical (but hopefully interesting) coupling argument of the later parts of the section. Suppose that
f3 is some path in the hexagonal grid TG*, which has a positive proba-
bility to be a subset of a zero-height interface h. We will need to understand rather well the behavior of h conditioned on f3 being a subset of a contour line. This conditioning amounts to conditioning h to be positive on vertices adjacent to f3 on one side and negative on vertices adjacent to f3 on the other side. (Here and in the following, a vertex v of TG is adjacent to f3 if f3 intersects the interior of one of the six boundary edges of the TG *-hexagon centered at v.) Thus, the following lemma will be rather useful. 1054
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45
3.1. (Expectation bounds) There is a finite c=c(A»O such that the following holds. Assume (h). Let V+ and V_ be non-empty disjoint subsets of VD, and set LEMMA
U:=VaUV+UV_. Suppose that every vertex in V+ has a neighbor in V_UVa and every vertex in V_ has a neighbor in V+ U Va. Let lC be the event that h > 0 on V+ and h < 0 on V_. Then for every v E V+ U V_ , E[e1h(v)1 IlC] and
< c,
(3.1)
- < E[lh(v)11 lC] < c.
1 c
(3.2)
E[lh(v)I-1/21 lC] < c.
(3.3)
Moreover,
Let BcD be a disk whose radius r is smaller than its distance to U and assume that BnVD #0. Then
E[max{lli(v)I : v E VDnB} I lC] < c, where
Ii
(3.4)
denotes the discrete-harmonic extension of the restriction of h to U (which is
also the conditional expectation of h given its restriction to U). Moreover, if s>O and U has a connected component whose distance from B is R and whose diameter is at least sR, then
(3.5) where c'=c'(s). Proof. The proof of (3.1) is a maximum principle type argument. Suppose that VI maximizes E[e1h(v)lllC] among VEU, and let M:=E[e1h(Vl)lllC]. Clearly, M <00. Assume
first that VI EV+. Set V:=U\ {VI}, and for VEV let Pv be the probability that the simple random walk starting at VI first hits V in v. We may write h(vd=X+Y,
where Y:= LPvh(v)
(3.6)
vEV
and X is a centered Gaussian independent of (h(v):VEV). Moreover, by (2.5), E[X2] is times the expected number of visits to VI by a simple random walk starting from VI until it first hits V. Set a:=E[X2]. Since VI has a neighbor in V, it follows that a=O(1).
i
1055
46
O. SCHRAMM AND S. SHEFFIELD
Also, clearly, a is bounded away from zero, since this random walk has at least one visit to
Vl.
For every yEffi., we have that
E[eh(vd [y = y, lC] = E[e x + y [Y = y, lC] = eYE [ex [X +y > 0] = eY
JCXl exp(x-x 2 /2a) dx
-:to
L y exp( -x 2 /2a) dx
.
If y>O, the right-hand side is bounded by O( eY). If y~O, then both integrals on the right are comparable to their value when the upper integration limit is reduced to -y+ 1. But then the ratio between the corresponding integrands is bounded by e l - y , which implies the same for the ratio of the integrals. Consequently, the right-hand side is 0(1) when y~O.
We take expectations conditioned on lC and get
M = O(l)E[e Y [lC]+O(l). Repeated use of Holder's inequality shows that for non-negative random variables
(3.7) Xl, ... ,
Xn and non-negative constants Pl, ... ,Pn such that L7=lPj~1 we have
Thus, (3.6) and (3.7) give
M ~ 0(1)
II E[eh(v) [lC]Pv +0(1). vEV
Clearly,
I,
E[eh(v) [lC] ~ {
lv!, e A,
if v E V_, if v E V+, if v EVa.
Setting P+:= LVEvnv+ Pv, we therefore obtain
M ~ O(l)MP+ e (l-p+)A+O(l). Since
Vl
(3.8)
has a neighbor in U\ V+, we have P+ ~ ~ and M ~O(l)eA follows. A symmetric
argument applies if VlEV_. If VlEVa, then obviously M~eA. Thus (3.1) holds with
c=O(e A ). The right-hand inequality in (3.2) is an immediate consequence of (3.1) (possibly with a different c). We now prove (3.4). Given VEVD and UEU, let H(v,u) denote the probability that a simple random walk started at v first hits U at u. Suppose that v, v' EBnVD and uEU. The discrete Harnack principle (Lemma 2.2) then gives H(v',u)~H(v,u)O(l). Thus,
[h(v')[ =
iL
uEU
H(v', u)h(u)i
~L
uEU
1056
O(l)H(v, u)[h(u)[.
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
47
The right-hand side therefore bounds max{lh(v')I:v'EVDnB}. Now, (3.4) follows from the right-hand inequality of (3.2) and L,uEU H(v, u)=l. We now prove (3.3). Consider some VEV+. We first claim that if w neighbors with
v and wtl-U, then E[h(w?IK]=Ox(1). To see this, first note that h(w) (where as above
h is the discrete-harmonic extension of the values of h on U) is a linear combination of the values h(v) for vEU (which are the same as the values of h there); since each ofthese values has mean and variance which are Ox(1) (by (3.1)), the mean and variance of h(w) are also Ox(1). By (2.5), conditioned on h, the value h(w) is Gaussian with variance fJ times the expected number of times a random walk started at w visits w before hitting U, which is 0(1) because U contains a neighbor of w.
Let Z denote the average of h on the neighbors of v, and let Z':=h(v)-Z. The above implies that E[Z2IK]=Ox(1). Since Z' is a centered Gaussian with variance for every ZElR,
E[h(v)-1/21 K Z = z] = E[(Z' +Z)-1/21 Z' +Z > 0] = ,
fJ'
];00 x-1/2e-6(x-z)2/2 dx "--'o'-------:-;=------:------:--::-c-_ _
Jooo e- 6(x-z)2/2 dx
If z~-2, then this is clearly bounded. Assume therefore that z<-2. It is easy to verify
that the integrals in the numerator and denominator are comparable to the same integrals restricted to the range XE[O,
Izl- 1 ].
But in this range, the maximum value of
is comparable to the minimum value of the same quantity. Consequently, when z< -2,
which gives
E[h(v)-1/21 K] = 0(1)+0(1)E[IZI 1/ 2 1K]. Since E[Z 2IK]=Ox(1), we certainly have E[IZI 1/ 2IK]=Ox(1). This proves (3.3). Now the left-hand inequality in (3.2) is an immediate consequence. We now prove (3.5). Consider two vertices v, uEBnVD. Assume that within distance R from B there is a connected component of U of diameter at least ER. Since
E[h(v) IK, h] 1057
=
h(v),
48
O. SCHRAMM AND S. SHEFFIELD
we have that
E[h(v)h(u) [K] =E[(h(v)-h(v))(h(u)-h(u)) [K]+E[h(v)h(u) [K].
(3.9)
Now, h(v) is just a weighted average of h(w) with wEU (according to the discreteharmonic measure from v). Consequently,
E[h(v)2 [K] ~ maxE[h(w)2 [K] ~ max2E[e 1h (w)1 [K] ~ 2c wEU
wEU
and the Cauchy-Schwarz inequality implies that the last summand in (3.9) is also bounded by 2c. Since h-h is the Gaussian free field with zero boundary values on U and is independent of the restriction of h to U, by (2.5) we have that
E[(h(v)-h(v))(h(u)-h(u)) [K] is fs times the expected number of visits to u by a random walker started at v which stops when it hits U. Thus
L
E[(h(v)-h(v))(h(u)-h(u)) [K]
uEBnVD
is fs times the expected number of steps that the walker spends in B, which is
by Lemma 2.3. The estimate (3.5) is now obtained by averaging (3.9) over all v, uE BnVD
D
.
The following lemma provides a variant of the right-hand bound in (3.2) in the case where instead oflooking for a zero-height interface of h, we consider instead a zero-height interface of h - g, for some fixed function g. LEMMA 3.2. (Further expectation bounds) In the setting of Lemma 3.1, suppose that g: D-+lR is zero on Va and Lipschitz in D. Let Kg be the event that h>g on V+ and h
(3.10) where c>O is a universal constant and q= [[g[[oo/[[Vg[[oo. 1058
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49
Proof. The proof is similar to the proof of Lemma 3.1. Suppose that VI maximizes E[h(v)-g(v)IJC g ] among VEV+, and let M:=E[h(VI)-g(VI)IJC g ]. By symmetry, it is
enough to get a bound on M. We define V, X, Y and Pv as in the proof of Lemma 3.1. For every yElR we have
where (x)+ :=max{O, x}. Consequently, M:(; 0(1)+E[(Y -g(VI))+ IJC g ]
:(; 0(1)+
L
PvE[(h(v)-g(VI))+ I JC g ]
vEV
:(; 0(1)+A+M
L
Pv+
vEV+
L
Pvlg(V)-g(VI)I·
vEV
Consider a simple random walk started at VI and stopped when it first hits V. Denote the vertex where it first hits has a neighbor in
V,
V by w.
(Then, P[v=w]=pv.) Set r:=qjlog(q+2). Since VI
we have P[lv-vII>r]:(;0(1)jlog(r+2), by standard random walk
estimates. When IV-VII:(; r, we have Ig( v) - g( VI) I:(;0(1)rllV'glloo. Therefore, "
Ilglloo
Ilglloo
~ Pvlg(V)-g(VI)I:(; 0(1)rllV'gII00+0(1)log(r+2) :(; 0(1)log(q+2)'
vEV
This gives M
" Pv:(; 0(1)+A+0(1) Ilglloo . ~ log(q+2) vEV\V+
Because L:vEV\ V+ Pv is bounded away from zero, the proof is now complete.
D
Our next result establishes the continuity of the conditional distribution of h in the specified data. More precisely, the following proposition holds. PROPOSITION
3.3. (Heights interface continuity) For every c>O there is some R=
R(c,A»1jc such that the following holds. Let D, V D , Va, ha, V+, V_, U and JC be as in Lemma 3.1, and let D, V D , Va, ha, V+, V_, fj and f( be another such system, which is also assumed to satisfy Ilhalloo:(;A. Let hK be a DGFF in D with boundary values given by ha conditioned on JC, and let hR be a DGFF in D with boundary values given by ha conditioned on f(. Suppose that within s.B R, the two systems are the same; that , v+ns.B R = v+ns.B R and V_ ns.B R = V_ ns.B R . Further suppose that OEU. Then there is a coupling of hK and hR such that for every vertex
is, Dns.BR=Dns.B R , halSE R =haISE R
vEs.B I / E we have E[lhK(v)-hR(v)I]
50
O. SCHRAMM AND S. SHEFFIELD
The following lemma will be needed in the proof. LEMMA
3.4. Let X be a I-dimensional Gaussian of zero mean and unit variance.
Let x, XER, let Z be a random variable whose distribution is the same as that of X +x conditioned on X +x>O, and let Z be a random variable whose distribution is the same as that of X +x conditioned on X +X>O. Then there is a coupling of Z and Z such that [Z-Z[<[x-x[ almost surely if xix. Moreover, there is a continuous function 8(x,x) satisfying 8(x, x) < 1 such that E[[Z - Z[] ~8(x, x) [x-xl under this coupling. ~
The coupling that we use is what is known as the quantile coupling of Z and Z.
Proof. Let F(s)=P[X <s], and let G=F- 1 . Set t:=F( -x) and t:=F( -x). Let p be a random variable uniformly distributed in [0, 1]. Then
Z(t) :=x+G(t+p(l-t)) =G(t+p-tp)-G(t) has the same distribution as Z. Therefore, (Z(t), Z(i)) is a coupling of Z and Z. Consequently, to verify the first claim it is sufficient to show that [8t Z(t)[<8t G(t). In fact, we will prove the stronger statement, -8t G(t) <8t Z(t) <0 for all pE (0,1), which is equivalent to
0< 8 t G(t+p-tp) < 8 t G(t). The left-hand inequality is immediate, because G'>O on (0, 1). The right-hand inequality translates to (l-p)G'(t+p-tp)
(1-(t+p-tp))G'(t+p-tp) < (l-t)G'(t). This is equivalent to (l-tp)jF'(G(tp))«I-t)jF'(G(t)), where tp:=t+p-tp>t. Now, note that
I-t _ F'(G(t)) -
-1=
r:
ex p(-s2j2)ds exp(-x 2j2) -
-x
e
(x 2 _s 2 )/2
_ ds-
(=
io
e
xs-s 2 /2
ds
is strictly decreasing in t, because x is strictly decreasing in t. This proves the first claim. The second claim follows with
8(x,x):= (I
io
Z(t)-~(t) x-x
and
8(x, x):=
dp
(I Z'(t)
io
when xix
-G'(t) dp.
1060
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
51
Proof of Proposition 3.3. Fix a coupling of hK and hj( that minimizes
L
E[lhdv)-hj((v)ll·
VEvDnvD
Standard continuity and compactness arguments show that there is such a coupling. Set f( v) :=E[lhd v) -hj(( v) Il for vertices vE (VDUVa) n(VD UVa). First, we claim that f is discrete-subharmonic on vertices in (]3R\U. Indeed, fix a
vertex wE (]3 R \ U. The conditional distribution of hdw) given the value of hK at every vertex but w is that of x+AX, where x is the average of hK on the neighbors of w, X is a standard Gaussian, and A is the lattice-dependent constant 1/V6 (since each vertex has six neighbors in TG). Similarly, hj((w)=x+AX. By the choice of coupling, when we fix the values of hK and hj( off of w, the corresponding conditioned coupling of hdw) and hj((w) minimizes the conditioned expectation of Ihdw)-hj((w)l. But one such conditioned coupling is obtained by taking X=X. Thus, f(w)~E[lx-xl]' which implies that f is discrete-subharmonic at w, since
L Ihdu)-hj((u)l;? I L hdu)- L hj((u)l, u~w
u~w
(3.11)
u~w
where the sums are over the neighbors of w. Next, consider any vertex vEV+n(]3R. As before, we may write hK(v)=x+AX, where x is the average of hK on the neighbors of v and X is a random variable whose conditionallaw given x is that of a standard Gaussian conditional on x+AX>O. Lemma 3.4 applied with x/A instead of x and x/A instead of x implies that f is also discretesubharmonic at v. We claim that there is a constant b=b(A,c»O such that ~f(v);? b,
where
~
if f(v);? ~c,
(3.12)
denotes the discrete Laplacian on TG. Indeed, the optimality of the coupling
gives E[lhdv)-hj((v)11 x, xl
~ 8(~, ~) lx-xl,
where 8 ( . , . ) < 1 is as in the lemma. Thus,
By (3.11) with v in place of w, we have
~f(v) ;? E[lx-xll- f(v) ;? E [( 1-8 (~, ~)) lx-xl]. 1061
(3.13)
52
O. SCHRAMM AND S. SHEFFIELD
For every neighbor u of v we have, by (3.5), that E[hdu?l
bO(E,A»O such that E[lxI1Al<-kE for every event A satisfying P[Albll < ~bo. The same inequalities will hold with and hR in place of x and hK . Therefore,
x
E[lx-xll =E[lx-xI1Ixlvlxl~btl+E[lx-xI1Ixlvlxl>btl :( E[lx-xI1IxIVlxl~btl +E[(lxl + IxJ) 1 IxIVlxl>btl
:( E[lx-xI1Ixlvlxl~btl+~E. Thus, if we assume that f(V)?::~E, then also E[lx-xll?::~E, and therefore
Therefore, (3.13) gives (3.12) with
Clearly,
f
is also discrete-subharmonic on V_nlER and (3.12) also holds for vEV_nlER
and (trivially) for v E Va n lE R. Next, we prove that for all vertices WElER we have (3.14) Fix such a w, and assume that wt}.U. We may decompose hdw) as a sum hdw)=y+Y, where y is the value at w of the discrete-harmonic extension of the restriction of hK to U, and Y is a centered Gaussian whose variance is times the expected number of visits to w by a simple random walk started at w that is stopped when it hits U. A simple random walk on TG started at w has probability at least a positive constant times
i
1/log R to reach distance R from w before returning to w, and once it does reach this distance, it has probability bounded away from zero to hit 0 before returning to w. Since OE U, it follows that E[y2l =O(log R). Thus, E[IYIJ =O(
Vlog R).
As y is the average of the value of hK on U with respect to the discrete-harmonic measure from w, it follows
from (3.2) that E[lylJ=OX(1). Thus, we have E[lhdw)ll:(Ox( VlogR). This certainly also holds if WEU, and a similar estimate holds for hR(w). Now (3.14) follows, since
f(w):(E[Jhdw)ll+E[lhR(w)ll· We now show that the established properties of f imply that f:(E on lEI/E if R is sufficiently large. Fix some vertex wElE I / E , and let St be a simple random walk on TG 1062
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
53
started at w. Let tl be the first time t such that [St [> ~ R or St =0. Since f is discretesubharmonic on VDnSB R , we have that tJ--'t f(St/\h) is a submartingale. The optional sampling theorem implies that f(w)~E[f(StJ]. By standard random walk estimates,
P [[Stl [> ~RJ ~O(l)[logE[/log R. (We assume, with no loss of generality, that E<~, say.) Consequently, [log E[
[log E[
f(w) ~ E[J(StJ] ~ f(O)+O(l)-l R max{f(u): u E VDnSB R} ~ f(O)+Ox(1) ;;-:::-::-no ~
v~R
This proves that f(W)~E if f(O)~~E and R is sufficiently large. Now assume that f(O)~~E. Let St be a simple random walk starting at O. Let
and let ns be the number of tE {O, ... , s-l} such that St =0. By (3.12) and our assumption that f(O) ~ ~E, we have that tJ--'t f(St/\tJ -bnt/\t* is a submartingale. Thus,
o~ f(O) ~ E[J(StJ]-bE[nt*] ~ max{f(u): u E VDnSBR}-bE[nt.l ~ Ox (1) Vlog R- bE[nt.l. Now note that as R-+oo, while 10 is fixed, E[nt.l grows at least as fast as a positive constant times log R, because the probability for St not to return to 0 after any specific visit to 0 is bounded by 0(1/10g R). Thus, the above rules out the possibility that
D
f(O) ~ ~E if R is sufficiently large. This completes the proof.
As a corollary of the proposition, we now show that the correlation in the values of h at two vertices in U decays when the distance between them tends to infinity. COROLLARY 3.5. (Correlation decay) For every 10>0 there is some R=R(E, A) such
that the following holds. Let D, VD, Va, ha, V+, V_, U and let Vl,V2EU satisfy [VI-V2[>R. Then
J(
be as in Lemma 3.1, and
Proof. Suppose, without loss of generality, that V2EV+.
Fix some a>O and let
X:=1{o
We may apply Proposition 3.3 to our present setup and to the setup
where the value of h(v2) is fixed at some constant yE(O,a] and V2EOD.
Thus, the
proposition would apply, provided that A is replaced by AV a. Consequently, we find that there is an R'=R'(E,A,a) such that if [VI-V2[>R', then
1063
o.
54
SCHRAMM AND S. SHEFFIELD
Since h(V2)X ~a, X 2=X and h(V2)X is h(v2)-measurable, this gives ~c ~ IE[h(Vl) I h(V2), JC]h(V2)X -E[h(Vl) I JC]h(V2)XI
= IE[h(Vl)h(V2)X I h(V2), JC]-E[h(Vl) I JC]h(V2)XI· Taking expectations conditioned on JC now gives
(3.15) Since
(1-X)h(v2)
2
~
(1-X)lh(v2W a
6e 1h (v 2 )1 a
~--
and h(vl)2~2elhCvt)I, if C denotes the constant satisfying (3.1), then the Cauchy-Schwarz inequality gives
Similarly,
Consequently, if a is chosen sufficiently large then
and The corollary now follows from (3.15).
D
Next, we provide a simple lemma which bounds the amount in which adding a function to h affects its distribution. LEMMA
3.6. (DGFF distortion) Assume (h). Let f: VDUVa--+1R satisfy f=O on Va.
Let JL be the law of h, and let JLj be the law of h:=h+ f. Then, for every event A,
Proof. Suppose that X is a standard Gaussian in IR n , yElR n is some fixed vector, and AclRn is measurable. Then P[X +y E A] =
C
nIn nIn
=C
1A exp (
_IIX~YI12) dx
1Aexp(x.y_IIY}2) exp ( _"~"2) dx,
1064
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
55
where C;:;-I = IIRn exp( -llxl1 2/2) dx and the integrals are with respect to the Lebesgue measure in lR.n . (This is the Cameron-Martin formula.) We may think of the right-hand side as the inner product of 1A and exp(x,y-IIYI12 /2) with respect to the Gaussian measure. Hence, the Cauchy-Schwarz inequality gives P[X +y E A] ::;; P[X E AV/ 2 E[exp(2X ·y_llyI12)V/ 2 = P[X E AV/ 2 exp (1Iy}2).
Let h denote the discrete-harmonic extension of ha. Then h-h is the DGFF with zero boundary values, and hence is a standard Gaussian on lR.VD with respect to the norm gf--tII'VgI1 2. The lemma follows.
D
3.2. Near independence
In this subsection we build on the infrastructure developed above to prove that under appropriate assumptions the shape of an interface inside a ball does not depend too strongly on the shape of an interface outside a slightly larger ball. More precisely, we have the following result. 3.7. (Near independence) Let C>l and let R>10 3 C. Assume (h) and Pi 5R CD. Let R I ,R2,R3 E[R,5R] satisfy PROPOSITION
R and R 3 + C < 5R.
Let V; and V! be disjoint sets of vertices in D\PiR3 and let V; and V! be disjoint sets of vertices in Pi Rl' Suppose that every vertex of V; neighbors with a vertex in V!, every vertex of V! neighbors with a vertex in V;, and similarly for V! and V;. Also suppose that a random walk started at 0 has probability at least l/C to hit V!UV; before exiting Pi 5R . Let Kl be the event that h>O on V; and that h
Proof. For j=O, ... ,8 set rj=R I +~j(R2-Rl)' Then r8=R2 and rj+l~rj+~C-l R. We set Wj:=(VDUVa)nPirj=VDnPirj, Wj:=(VDUVa)\Wj and Wr=wjnw k . Let h denote the discrete-harmonic extension of the restriction of h to WI UW7 . We may identify h with a point in lR.W1UW7; namely, its restriction to W I UW7 . As we have noted 1065
56
o.
SCHRAMM AND S. SHEFFIELD
after (2.4), the probability density of h with respect to the Lebesgue measure on IR WiUW7 is proportional to exp(-IIVhI12j2). Hence, (3.16) where the integrals are with respect to the Lebesgue measure on IR Wi UW7 • Let hI be the function that agrees with h on wI, is discrete-harmonic on
wI,
and is zero in W7 . Let
h3 be the function that agrees with h on W7, is discrete-harmonic on
wI,
and is zero in
WI. Clearly, h=hl +h3.
We claim that (3.17) where;::::: means equivalence up to multiplicative constants depending on A and C. Let V+:=V;UV; and V_:=V!UV!. Fix some VEW~. Let h denote the discrete-harmonic extension of the restriction of h to V+UV_. Then h(v)-h(v) and h(v)-h(v) are indepen-
dent Gaussian random variables, and both are independent of
h (by the orthogonality
i
property noted in §2.1). By (2.5), the variance of h(v)-h(v) is times the expected number of visits to v by a random walk started at v, which is stopped when it hits W I UW7, and the variance of h(v)-h(v) is times the expected number of visits to v by the same random walk stopped when it hits V+UV_. Consequently, the variance of h( v) - h( v) is times the expected number of visits to v after the first hit of WI UW7
i
i
and before the first hit of V+UV_. Note that the probability to hit v by a random walk started in WI UW7 before exiting 1l3 5R is O(1jlog R) and conditioned on hitting v before exiting 1l3 5R the number of visits to v prior to exiting 1l3 5R is
o (log R).
Our assumption
on the probability to hit V!UV; therefore easily implies that E[(h(v)-h(v))2J=Oc(1) and hence E[lh(v)-h(v)IJ=Oc(1). Since K I nK 3 is determined by h, it is independent of h(v)-h(v) and, consequently, E[lh(v)-h(v)IIKI' K3J=Oc(1). Now, by (3.4), we have E[lh(v) IIK I , K3J =OX(1). Combining these estimates, we get E[lh(v)IIK I , K3J =Oc,x(1). We will now apply the argument used to prove (3.4) in order to establish
(3.18) Indeed, let A denote the set of vertices in W~ neighboring with some vertex outside W~, and let H(v, u) denote the probability that a simple random walk started at v first hits A in u. As in the proof of (3.4), H(v, u)~Oc(1)H(v', u) for v, v' EWg. Now (3.18) follows as in the proof of (3.4). 1066
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
Next, we want to show that (3.18) holds with hl and h3 replacing h; that is, (3.19) where M:=max{lh j (v)I:VEWg,j=1,3}. Let Vl be the vertex VEW~ where Ih1(v)1 is maximized, and let V3 be the vertex VEW~ where Ih3(V)1 is maximized. Then M= max{lh 1(Vl)I,lh3(V3)1}. Assume, for now, that M=lh3(V3)1. The maximum principle for discrete-harmonic functions implies that V3 neighbors with a vertex outside ~r5 and Vl neighbors with a vertex in
~r3.
Let p be the probability that a simple random walk
started at V3 exits ~ R3 before hitting a vertex neighboring with a vertex in ~r3. Then p is bounded away from 0 by a function of C. Since hl composed with a simple random walk is a martingale while the walk stays in
wi, we get Ih 1(V3)1 ~ (l-p) Ih1(Vl)1 ~(l-p)M.
As
h=hl+h3, we get Ih(V3)1~lh3(V3)1-lhl(V3)I=M-lhl(V3)I~pM. The case M=lh1(Vl)1 is similarly treated. Using (3.18), we then get (3.19).
Next, we want to prove that (3.20) Since hl is discrete-harmonic in
wi, if vEwi we have
where the sum is over the neighbors of v. This is also true for VEW 1 , since h3 is zero there. Consequently, L
O.
(3.21 )
L(h3(U)-h3(V))hl(U) = O.
(3.22)
L(h 1 (v)-h 1 (u))h 3(v)
=
vEW4u~v
Similarly, we find that L
UEW4V~U
Set 8W 4 :={(v,U)EW 4 XW4 :U rv v}. By considering the contribution of each edge [v,u] to '\lh 1 . '\lh3, we compare the sum of the left-hand sides of (3.21) and (3.22) to '\lh 1 . '\lh3 and conclude that '\lh 1 · '\lh3 =
L
(h 3(v)h 1 (u)-h 1 (v)h 3(u))
(v,u)E8W4
L
((hl (u) -hl (v) )h3( u)+ (h3( v) -h3( u) )hl (u)).
(3.23)
(v,u)E8W4
The number of summands is clearly O(R). Note that for every VEW 4 neighboring with a vertex UEW4 , there is a disk of radius proportional to RIC such that all the vertices 1067
o.
58
SCHRAMM AND S. SHEFFIELD
in that disk are in W~. Consequently, the discrete Harnack principle (Lemma 2.2) gives
[hl(u)-hl(v)[=O(MjR) and [h3(U)-h3(V)[=O(MjR). Hence, (3.23) gives (3.20). Now, (3.19) implies that the expectation of ['Vh l ·'Vh3[1/2 conditioned on JC I nJC 3 is bounded by a function of C and A. In particular, there is a constant CI=CI(A,C»O such that
In terms of Lebesgue measure, this may be written as
Since h=hl +h3, this implies that
which gives one side of (3.17). The other direction is proved in essentially the same way. Under the probability measure weighted by exp( -~([['Vhl[[2+[['Vh3[[2)) (with respect to the Lebesgue measure on lRW1UW7), hI restricted to WI has the law of the DGFF with zero boundary values on
W7 restricted to WI. Similarly, with this weighting, h3 restricted to W7 has the law of the DGFF with zero boundary values on WI and with boundary values given by ha on aD, restricted to W7 . Moreover, under this measure, hI and h3 are clearly independent. The above arguments show that under this measure (3.19) holds (where JC I refers to hI while JC 3 refers to h3). Since (3.20) is still valid, the opposite inequality in (3.17) is then easily established. We may also apply (3.17) in the case where measure. Since
V;=V!=0, and hence JC I has full
from (3.16) we get
which completes the proof.
D 1068
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
59
3.3. Narrows and obstacles The present subsection and the next will use the infrastructure developed in §3.1 to prove some bounds on the probabilities that contour lines cross certain regions in specified ways. This is roughly in the spirit of the Russo-Seymour-Welsh theorem for percolation, though the proofs are entirely different. The following lemma is an estimate for having a crossing by TG* -hexagons where h is negative between two arcs on the boundary of some subset of the domain, conditioned on some zero-height interface paths. The statement below is slightly complicated, because we need to keep the geometric assumptions quite general. In percolation, boundary values do not playa role, of course. But in our case we need the crossing estimate in the case where one boundary arc of the domain is conditioned to be an interface. LEMMA 3.8. (Narrows) For every c->O there is a 6=6(1\, c-»O such that the following crossing estimate holds. Assume (h) and (D). Let K be the event that a fixed
collection h1, ... ,I'd of oriented paths in TG * are contained in oriented zero-height interfaces of h, and suppose that P[K] >0. Let aCD\ h1 U... Ul'k) be a simple path that has both its endpoints on the right-hand side (positive side) of 1'1. Let A be the domain bounded by a and a subarc of 1'1, and assume that A does not meet the left side of 1'1 and that Anh2UI'3U ... Ul'kUaD)=0. Let a1, a2 and a ' be three disjoint subarcs of a, where a1 contains one endpoint of a, and a2 contains the other endpoint of a. (See Figure 3.1.) Suppose that each point in a ' is contained in a TG* -hexagon whose center is outside A. Set d 1 :=suPzEa' dist(z, 1'1). Let C be the event that there is a path crossing from a1 to a2 in A inside hexagons where h is negative. Let d* be the infimum diameter of any path connecting a ' to 1'1 U... l'kUaD which does not contain a subpath connecting a\(a1Ua2) to 1'1 in A. If
d 1 +1 ~ 6 min{ d*, dist(a1, a/), dist(a2,
a/), diama /},
(3.24)
then
P[CIK]
l
Proof. Set N:= ~ diam a '
J.
Assume that (3.24) holds. Note that
diam 1'1 ~ diam a ' - 2d 1 · 1069
60
O. SCHRAMM AND S. SHEFFIELD
/1
Figure 3.1. Setup in the narrows lemma.
Choose some point zoEo:'. For j=l, ... , N let Zj be a point in 0:' at distance j from Zoo Now for each Zj we let 8j be some center of a hexagon that contains Zj satisfying Then 18j-8j'I~lj-j'I-O(1). Let U be the union of Va and the set of vertices adjacent to anyone of the paths /'1, ... , /'k. (These are precisely the vertices v where h takes boundary values, or the sign of h( v) is determined by K.) Set 8j~A.
b:=E[XIK] and fix some s'>O. We first claim that E[(X _b)21 K]
< s'
(3.25)
if 8 = 8 (s' , A) > 0 is sufficiently small. Let hu denote the discrete-harmonic extension of the restriction of h to U and set X U :=N- 1 Lf=l h U(8j). Note that E[X -Xu IK, hu]=O, and hence E[Xu IK]=b. For each uEU and j E{l, ... , N} let p(j, u) denote the probability that a simple random walk started from 8j first hits U at u. Also set p(u):=N- 1 Lf=l p(j, u).
Then Xu= LUEU p(u)h(u). Consequently, E[(XU- b)21 K]
=
L
p(u)p(u')(E[h(u)h(u') I K]-E[h(u) I K]E[h(u') I K]).
u,u'EU
Let Z (u, u') denote the term in parentheses corresponding to the summand involving u and u'. Then E[(Xu-b?IK] is just the average of the conditioned covariances Z(u,u') 1070
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
61
weighted by p(u)p(u'). We know from (3.1) that E[h(u)2IlC] is bounded by a constant depending only on
A.
It then follows by the Cauchy-Schwarz inequality that the same
is true for Z(u,u'). Consequently, to prove that E[(XU-b)2IlC] is small, it suffices to show that when (u, u') is chosen with probability p( u )p( u') it is very likely that 1Z (u, u') 1 is small. Suppose that we select j from {I, ... , N} uniformly at random and given j select uEU with probability p(j,u). Independently, we also select (j',u') with the same distribution. It suffices to show that 1 Z (u, u') 1 is likely to be small, and by Corollary 3.5 it suffices to show that the distance between u and u' is likely to be large. Since ISj-sj/l=
Ij - j'I+O(I) is unlikely to be much smaller than diama', which is larger than 8- l (d l +1), it follows from Lemma 2.1 (hit near) that for any fixed R the probability that lu-u'I
E[(Xu _b)21 lC] < ~s', provided that 8 is sufficiently small. Set Xj:=h(sj)-hu(sj). Recall from §2.1 that, given the restriction of h to U, the function h-h u is the DGFF on VD \U with zero boundary values on U. Therefore, by (2.5), E[XiXj IlC, hu] =iG(Si' Sj), where G(v, u) is the expected number of visits to u by a random walker started at v and stopped when it hits U. From Lemmas 2.1 and 2.3,
Since ISj-sj/l:?lj-/I-O(I) and (lE(O, 1), these estimates give
Now (3.25) follows for sufficiently small 8=8(s',A»0, since X-Xu is independent of
Xu and they are also independent given lC. We now claim that b:?O if 8 is sufficiently small. Let c be the constant given by Lemma 3.1. If u is a fixed vertex adjacent to "11 on the right, then E[h( u) lC] > 1/ c, by (3.2). On the other hand, E[lh(u)lllC]
is likely to first hit U at a vertex adjacent to the right-hand side of "11. Thus, when 8 is small, we have b=E[XullC]>O. Also, clearly, b~c. Now set a=b+1. Let Q denote the union of the closed hexagons in TG* for which
h(H) E [0, a] and let Q' denote the union of the edges in TG* that are on the common boundary of two hexagons HI and H2 satisfying h(Hl) <0 and a
62
o. SCHnA~Ih[ AND S. SHEFFIELD
Figure 3.2. A portion of the sequence of hexagons adjacent to 8A' \ 8A.
the connected component of (QuQ')nA that contains 1'1 n DA. Let Q denote the event
Qon(a\ (al U(2»=0. If C holds, then the corresponding crossing by hexagons where h is negative separates l ineA from a \ (OjU0'2) in A , and hence Q holds as well. Thus, Ce Q. On the event Q, let A' be t he connected component of
.4\Qo that contains a' , and
let UQ denote the set of centers of hexagons H such t hat H n A1 n Qo#0. Clearly, h(v)
that E [X I!C , Q_ I is negative and bounded away from zero when 1072
o=o(€', 11.»0
is small.
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
63
Since E[(X -b)2IK] <10' and b~O, we conclude that by choosing 10'>0 small it can be guaranteed that P[Q_IK] < ~E. On Q+, we clearly have h(u»a~b+1 on every uEUQ' Thus, as above, it follows that when 8 is small E[XIK, Q+]>b+~. This again implies that P[Q+IK] can be made smaller than ~E. Since C c Q+ U Q_, this completes the proof. D Next, we formulate an analogous lemma for crossings near the boundary of the domain. LEMMA 3.9. (Domain boundary narrows) There is a constant Ao=Ao(A»O such
that for every 10 > 0 there is a 8=8 (A, E) > 0 such that the following crossing estimate holds. Assume (h) and (D), assume that 0+ is a simple path contained in oD and that ha~-Ao on o+nVa. Set o_:=oD\o+. Let K be the event that a fixed collection hI, ... , ')'k} of oriented paths in TG* are contained in oriented zero-height interfaces of h, and suppose that P[K] >0. Let acD\ hI U... U')'k) be a simple path that has both its endpoints on 0+. Let A be the domain bounded by a and a subarc of 0+, and assume that Anhlu')'2u ... U')'kUO_)=0. Let a'Ca be a subarc. Suppose that each point in a' is contained in a hexagon whose center is outside A. Set d 1 :=suPzEa,dist(z,0+). Let al be a subarc of a that contains one of the endpoints of a, and let a2 be a subarc of a that contains the other endpoint of a. Let C be the event that there is a path crossing from al to a2 in A inside hexagons where h is negative. Let d* be the infimum diameter of any path connecting a' to ')'1 U ... ')'k UoD which does not contain a subpath connecting a\(alUa2) to 0+ in A. If d 1 +1 ~ 8 min{ d*, dist(al, a'), dist(a2, a'), diama'},
(3.26)
then P[CIK]<E.
Proof. The proof is slightly simpler but essentially the same as that of Lemma 3.8 (narrows). We use the same notation as in that lemma, and only indicate the few differences in the proof. In the present setting b=E[XIK] can be made larger than -2Ao by taking 8>0 small. Here, we define Qo as the connected component of (QUQ'uo+)nA that contains o+noA. Observe that UQnoD=0 on Q_nK. It follows that E[XIK, Q_] is negative and bounded away from zero (by a function of A) when 8>0 is small. By taking Ao>O sufficiently small, we can make sure that E[XIK, Q-]<-3Ao. But since E[(X-b)2IK] is arbitrarily small and b~-2Ao, this makes P[Q_IK] small. The rest of the argument is essentially the same. D The previous lemmas will help us control the behavior of the continuation of contours near existing contours or the boundary of the domain. The next lemma will help us control the behavior in the interior away from existing contours. 1073
64
O. SCHRAMM AND S. SHEFFIELD
LEMMA 3.10. (Obstacle) For every E>O there is some constant c=c(E,A»O such that the following estimate holds. Assume (h) and (D). Let lC be the event that a fixed collection
fYl, ... ,I'd
of oriented paths in TG* are contained in oriented zero-height
interfaces of h, and suppose that P[lC] >0. Let U be the union of Va and the vertices of TG adjacent to ;'Y:=l'lU ... Ul'k' Let g be a function defined on the vertices of TG that is 0 on U. Let 19 denote the union of the interfaces of h+g that contain anyone of the paths 1'1, ... , I'k. Let B(zo, r) be a disk of radius r that is centered at some vertex Zo satisfying Ig(zo)I~~llglloo. Let d>O and suppose that at distance at most c 1d from Zo there is a connected component of ;yU&D whose diameter is at least Ed. Also assume that Ilglloo/IIVglloo>cr>c. Then P[1gnB(Zo, r)
=f. 01 lC] ~ cllgll~2log~. r
Proof. With no loss of generality, we assume that g(zo»O. Set q:=llglloo/IIVgll oo and r1:= loq. Since between any two vertices z and z' in TG there is a path in TG whose length is at most 2Iz-z'l,
As g=O on U, it follows that E-1d~r1' Since we are assuming that q>cr, and we may assume that c is a large constant which may depend on E, it follows that d/r> 100, say. Thus, we also assume, with no loss of generality, that Ilglloo~y'c, since the required inequality is trivial otherwise. Let X denote the average value of h on the vertices in B(zo, r). The inequality (3.5) and d/r> 100 give
(3.27) If 1'1 is not a closed path, we start exploring the interface of h+g containing 1'1
starting from one of the endpoints of 1'1 until that interface is completed or B(zo, r) is hit, whichever occurs first. (This may entail going through several of the interfaces I'j, j> 1.) If that interface is completed before we hit B(zo, r), we continue and explore the
interface of h+g containing 1'2, and so forth, until finally either all of 19 is explored or B(zo, r) is hit. Let Q denote the event that B(zo, r) is hit, and let (3 be the interfaces explored up to the time when the exploration terminates. Let U' be the union of U with the vertices adjacent to (3. Since we are assuming that q>cr, r>l and Ilglloo~y'c, and since c may be chosen arbitrarily large, Lemma 3.2 shows that for every vertex VEU', we have that E[Jh(v)+g(v)11 lC, Q,(3]
1074
~ II~~;.
(3.28)
65
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
(Note that conditioning on Q and (3 amounts to conditioning that h+g>O on vertices adjacent to the right-hand side of (3 and h+g
Since we are assuming that q ~ cr, we may assume that the right-hand side is less than
110 •
Recall that g~~llgIICXl inside B(zO,r1). Outside B(zO,r1), the trivial estimate g~-llgIICXl applies. When these estimates are applied to (3.29), one gets
E[h(x)IK,Q,(3]_II~~;:::;-
L
Pug(u)-
uEU'nB(zO,Td
L
Pug(u)
uEU'\B(zo,Tl)
We may take expectation with respect to (3 and average with respect to x to conclude that E[XIK, Q]:::;-~llgIICXl' which implies, by Jensen's inequality, that
E[X2 I K, Q] Since
~ II~I~~ .
E[X2IK] P[QIK]:::; E[X21 Q,K]'
the lemma now follows from the above and (3.27).
D
3.4. Barriers
In this subsection we apply Lemmas 3.8, 3.10 and 3.6 to get a flexible (though slightly complicated) criterion giving lower bounds for the probability that contours avoid certain sets. The complications arise from the need to handle pre-existing contours that are highly non-smooth on large scales. 1075
66
O. SC lillAlI l l>I AND S. SIIEFF'IELD
A,
y
Figure 3.3. T he domain A . in a situation where condition (3) fails. (T he figure does not show detail on the scale of the lattice; that is, D("Y) appears as D\'Y.)
The following relative notion of distance will sometimes be used below: dist(A, B; X) := inf{diam a: 0: is a path in X connecting A and B}. If
(3 .30)
'Y is a collection of paths in Ii , let D{)') denote the complement in D of the union of
the closed triangles of T G meeting
1'.
We now define the notion of barrier. Assume (h) . Let I II 1'2, "' 1Ik be disjoint simple paths or simple closed paths in D. Set -Y:=·"'I'lU .. . Ul'k. Let T C D(i) be a pat h , which is contained in D(,),), except possibly for its two endpoints. Fix some R >O and c> O. We call
r
an (e, R)-bamer for the configuration (D, ')') if the follow ing conditions hold: (1) €R< d iam T :;;;R; (2) within distance
€- 1 R
from T there is a connected com ponent of ,?UoD whose
diameter is at least €R ; (3) if ZEo D(,?) is an endpoint of Y and XZ is the connected component of oB(z, £ R) n D(,?) first encountered when traversing T from Z (which exists by (1», then the connected component Az of D(,?)\Xz that contains points of T arbitrarily close to z satisfies (a) tlAz consists of Xz and a simple path contained in tlD(,?) and (b) T n A ~n oB( z,r) consists of a single point for every rE(O,€R); (4) for every point wE T such that dist(w , oD( '?))~ i- € R there is an endpoint zE
Y ntlD(,?) such that wEAz and Iw-zl<~€ R (roughly, T does not get close to oD(,?) except near its endpoints on oDe '?»~. One example where condition (3) fails is given in Figure 3.3. If we remove the strand of 1'2 from t hat fi gure, we get an example that illustrates that Azc B{z,€R) does not 1076
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
67
follow from the above conditions. Note that it may happen that 8Az \Xz, which is a simple path, by (3), consists of an arc in 8D together with one or two arcs in;Y that have endpoints in 8D. We now use barriers to "manipulate" contours of the DGFF. THEOREM 3.11. (Barriers) For every E>O, mEN+ and A>O there is a p=p(E, A,m) > 0 such that the following estimate holds. Assume (h) and (D). Let lC be the event that a fixed collection hI'"'' Id of oriented paths in TG * is contained in the oriented zeroheight interfaces of h, and suppose that P[lC] >0. Set k
;Y:=
U
Ij'
j=l
Let V+ (respectively, V_) be the set of vertices adjacent to ;Y on the right-hand (respectively, left-hand) side. Let R>O and let Y=Y+UY_ be a collection of m (E,R)-bamers for the configuration (D,;y). Assume that the endpoints of these barriers are not on 8D and that for every YEY+ (respectively, YEY_) and every endpoint zEYn8D(;y), the vertices in Az(Y)n8D(;Y) are in V+ (respectively, V_), where Az(Y) is as in condition (3). Also assume that dist(U Y+, U Y_; D(;Y)) ~2ER. In the situation where ER=O(l), we also need to assume that there is no hexagon in TG * meeting both V+ U (U Y+) and V_ U (U Y_) and there is no hexagon meeting
UY
and 8D. Let
9 denote
the union of the zero-height
interfaces of h which contain anyone of the arcs 11, ... , Ik. Then
P[(9\ ;y)n(U Y)
= 01
lC] > p.
The basic idea of the proof is as follows. We define a function g that is large (positive) near U Y+ away from 8D(;Y) and is negative and large in absolute value near U y_ away from 8D(;Y). The obstacle lemma (Lemma 3.10) will then imply that 9g , as defined there, is unlikely to hit Y, except near endpoints of barriers. The narrows lemma (Lemma 3.8) will be used to show that
9g
is also unlikely to hit U Y near endpoints. Finally, the
distortion lemma (Lemma 3.6) will be used to conclude that with probability bounded away from zero,
9 will not
hit Y.
Proof. Let
.til:= U{Az(Y)nB(z, ER): Y
E Y+ and z E Yn;y},
.ti!:= {z E D(;Y): dist(z, U Y+; D(;Y)) ~ 110ER}, 1077
68
o.
SCHRAMM AND S. SHEFFIELD
and A+:=A~UA!. Similarly, define A_, with y_ replacing Y+. We fix constants co>O, large, and 8>0 much smaller than E, and set
c08-1 R- 1 min{ 8R, dist(z, lR 2 \A+)}, { g(z):= -c0 8- 1 R- 1 min{ 8R, dist(z, lR2 \A_)},
if z EA+, if z E A_, otherwise.
0,
Note that A+nA_=0 and Ilglloo=co. Let Z:={ZElR 2 :lg(z)l=co}. Let cl>l be some large constant and set r:=8R/cl. Let W be a maximal collection of vertices in {v: Ig( v) I? ~co} such that the distance between any two distinct vertices in W is at least ~r, and for aEW let Ba denote the disk ofradius r centered at a. Then the disks B a , aEW, cover Z, assuming that r>l and cl>100, say. Note that
Fix some aEW. We wish to invoke Lemma 3.10 to get a good upper bound on P[9gnBa#0IK]. We now verify the assumptions of the lemma. We note that in our case Ilglloo=co and IIVgll oo =co/8R. Thus, q:=llglloo/IIVgll oo =8R. Consequently, we set Cl to be the maximum of 100 and twice the constant c in the lemma, and the assumption q> cr is satisfied. The assumption r> 1 will hold once R is large enough, which we assume for now (we promise that 8 will be a constant depending only on E, m and A). For the d in the lemma we may take d=R. Thus, log(d/r)=log(cI/8), and the lemma gives
Since UaEwBa~Z and IWI=O(m)(cI/8)2, we conclude that
Although we have not specified 8 yet, we choose cO=CO(Cl, 8, E, m, A) so that (3.31 ) Now that we have established that it is unlikely that 9g intersects Z, we need to worry
about the case in which 9g circumvents Z but hits UY\Z. This can only happen near endpoints of barriers. Let us fix some YEY+ that has an endpoint, say Zl, on oD(;;Y). Let XZ 1 be the arc oA Z1 (Y) \Xz 1 • (It is an are, by condition (3) of the definition of barrier.) We will now prepare the geometric setup that will enable the use of Lemma 3.8 to prove
that p[9gnA z1 (Y)nY\Z#0IK] is small. 1078
CONTOUR LINES OF THE TWO-DI~IENSIONAL DISCRETE GAUSSI AN FRE E FIELD
('
69
T
A
S'
Figure 3.4. T he set S, and the paths 00 and Cia . The shaded region is Q.
Let Q be the set of all hexagons of the grid TG* whose distance from &D(;;Y) is at most 2SR. Lei S be the connected com ponent of
A" nB(z" ~' R) \ (QuB(z" ~'R) ) that intersects I. (See Figure 3.4 .) Condition (4) in the definition of barriers and our assump tion that S«c: guarantees that there is a unique such component S. \Ve have ScZ, and so 9g is unlikely to hit S. Let S' and S be the two connected components of S\ T . Consider the connected component All. of oBGzlcR)\XZl that intersects T. Let Q:] be the arc in All. \S that has one endpoint in S' and the other in .\ZI . Likewise, let M2 be the connected component of &B(iZ]cR) \XZl that intersects T , and let a2 be the arc in M2 \S that has one endpoint in S' and the other in XZ1. Let a denote the union of a] U02 with the arc of OS' \ T that connects the endpoint of a] with the endpoint of a2. Let ACAz1 (I') be t he domain whose boundary consists of at Ua2, an arc of as' connecting at and a2 and an arc of
Xz,
connecting at and a2. Let ao be t he unique arc of aA\(aB(~ztc:R)UaB(~z,c:R)) connecting oB(~z] c R) with oB(~z, c R). Finally, let a' be any subarc of ao whose diameter is /GcR that intersects the circle aB(z" ~cR) . We use A, a" a2 and a' in Lemma 3.8. In the present situation, the value of d, of that lemma is d, =2SR+O(1) . If we have a path connecting a' to 8D u;;y, it must either exit B(z" ~cR)\B(z" ~cR), hit Y\B( Z11 ~cR) (whose distance from ;;Y is at least kcR) or connect ao to Xz, inside A. Consequently, the minimum on the right.-hand side in (3.24) is presently at least /6cR. The lemma now implies that if we choose our current J=J(c:, m, A»O sufficiently small 1079
70
O. SCHRAMM AND S. SHEFFIELD
Figure 3.5. No way to penetrate.
and make sure that R>1/8, then
where C is the event that there is a crossing of hexagons satisfying h
e,
when we replace S' by Likewise, we may define a1, a2, a', A and 1. paragraph. The same argument shows that P[elJ(l~
/om-
S in the above
Condition (3) of the definition of a barrier implies that SUAUAUXZl separates
YnA Z1 (Y) from all the endpoints of the strands 1'1, ... , I'k. Consequently, in order for 19 to hit YnA Z1 (Y), we must have 19n5i-0 or Cue. See Figure 3.5 (also compare Figure 3.4). A similar argument applies for every other endpoint of a barrier. Since
S C Z, we conclude that (3.32) We are really more interested in 1 than in 19. To do the translation, we will appeal to Lemma 3.6. For this purpose, note that
and that 9 is supported in a union of m sets of diameter O(R). Consequently, we have
IIV'gI12=Oc:,A,m(1). Let U be the set of vertices in 8D('Y) , and let hu denote the restriction 1080
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
71
of h to U. By (3.32), even if we further condition on hu, the left-hand side stays bounded away from one on an event whose probability is bounded away from zero, namely,
Because g=O on U, we may apply Lemma 3.6 and conclude that for hu such that the inner inequality above holds, 0E,A,m(1)p[;:Yn(U Y) = 01 K, hul? 1.
lo'
Since this set of hu has conditioned probability at least the theorem follows. It remains to remove the assumption that R is larger than some fixed constant Ro=Ro(E:, m, A). Assume now that R is bounded. It is not too hard to see that the event
H that h(H)E(O, 1) for every hexagon meeting UY+ and hE(-1,O) for every hexagon meeting UY_ has probability bounded below by (a rather small) positive constant. This is proved by considering these O(mR2) hexagons one by one. On the event H, we have ;:Yn(UY)=0. This completes the proof. D Remark 3.12. There is a corresponding analog of Theorem 3.11 in the case where the endpoints of the barriers are permitted to land on aD. In that case, it is necessary to assume that ha?-Ao (respectively, ha~Ao) on aAz(Y)nVa if YEY+ (respectively, Y_) and zEaDnY, where Ao=Ao(A»O is the constant given by Lemma 3.9. We refrain from stating a complete formulation of this variant, though it will be useful. The proof is the same, except that Lemma 3.9 is used to deal with the narrows near aD, instead of the narrows lemma (Lemma 3.8).
3.5. Meeting of random walk and interface
We now need to further develop the basic setup and introduce some more notation. If a is a path in the hexagonal grid TG*, we let V(a) denote the set of TG-vertices adjacent to it. If a is an arc of an oriented zero-height interface of h, let V+(a) denote the vertices adjacent to it on its right-hand side, and let V_(a) denote the vertices adjacent to it on its left-hand side. In addition to our previous assumptions (h) and (D), we will use the following setup: (S) Let 'Y denote the interface of h from Xa to Ya. Let Vo be some vertex of TG in D, and let S be a simple random walk on the vertices of TG started at Vo that is independent of h. Let T be the first time t such that St E aD h). The point ST will playa special role. Essentially, we will be interested in the configuration "as viewed from ST"; that is, in the coordinate system where ST is translated to 0. 1081
72
O. SCHRAMM AND S. SHEFFIELD
In order to eliminate too much additional notation, it will be convenient to consider the event ST =0 instead. Let TO be the first t such that St =0, and let S denote the reversed walk St:=STo-t, t=O, 1, 00., TO. For 0'=0,1,00.,5, let eO' denote the edge [O,expa7riO')] of the triangular grid TG, and let e~ denote the dual edge in TG*. Let zg denote the event Zg:={ST=O}n{e~C'y}. Fix some large R. Suppose that fJ3RcD and vo~fJ3R' Let extR/' denote the union of the components of /,\fJ3 R containing Xa and Ya. (If /,nfJ3 R =0, then extR/'=/") If there is an interface of h containing
e~,
denote it by (3=(30', and let (3=(3'R be the connected
component of bnfJ3 R that contains e~. Otherwise, set (3=b=0. Let intRS denote the part of S up to the first exit of fJ3 R, and let extRS denote the part of S up to the first entry to fJ3 R. Set if!R:=(D, 8+, ha, va, extR/', extRS) and 8 R =8 R (0'):=((3'R, intRS). Our goal is to show that conditioned on zg, the distribution of (3 does not depend strongly on if!4R' (A precise version of this statement is given in Corollary 3.16 below.) To this end we will use something like
P[8 = rJ I if! R
ZO'] = P[8 R = rJ I if!3R]P[Zg 18 R = rJ, if!3R] 3R, a P[zg I if!3R] .
(3.33)
This equality is obtained by applying Bayes' formula to the measure P[·1if!3R]. The following lemma takes care of the first factor in the numerator on the right-hand side. LEMMA
3.13. Assume (h), (D) and (S). There exists a constant c=c(A»O and a
function PR ( .) such that if R> 50, R' E [~R, 3R], D -:J fJ3 4R and va ~ fJ3 4R , then for all rJ=(~, S) such that ~=I=0 one has 1 -PR( rJ) :s; P[8 R = rJ I /,nfJ3 4R =1= 0, if! R'] :s; CPR (rJ). C
The function PR may depend on Rand rJ, but not on anything else (in particular, not on D, Va, if!R' or ha). Proof. The corresponding statement with 8 R replaced by (3, the first coordinate of 8 R , is an immediate consequence of Proposition 3.7. We assume that vO~fJ34R' The configuration if!R' determines the first vertex, say q, inside fJ3 R' visited by the random walk S. The continuation of the walk is just a simple random walk starting at q. Suppose that we had another such walk starting at a vertex q' EfJ3R', It is easy to see that with probability bounded away from zero the walk starting at q visits q' before 0. If that happens, we couple the continuation of the walk to be the same as the walk which starts at q' (otherwise, we let them be independent). On the event that the walk started at q hits q' before 0, the corresponding intRS for both walks will be the same. This proves the corresponding statement about the second coordinate of 8 R. Since the two coordinates are independent given if! R', the lemma follows. D 1082
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
Proving an analogous result for the second factor in the numerator of the right-hand side of (3.33) will be considerably more difficult. To this end, we now define a measure of the quality Q=QR of the configurations IPR and GR. If /,n~R#0, let x R (respectively, yR) denote the endpoint in 8~R ofthe component of extR/' containing Xa (respectively, ya). When VO~~R' let qR denote the vertex in ~R first visited by S. If /,n~R=0 or extRS visits 8D(extR/') , then set Q(IPR)=O. Otherwise, define
Q( IP ):= dist(x, extRS)l\dist(y, extRS)l\dist(q, extR/,)l\lx-yl R
R
1\
_1_ 100'
where x=x R , y=yR and q=qR. This is a measure of the separation between the strands comprising IPR. Similarly, define Q(G R ), as follows. Suppose that
vO~~RCD
and let
qR be the first vertex outside of ~R visited by S. Fix an orientation of e~. If j3ct~R' let R and yR be the two endpoints of the component of f3=f3R containing e~, chosen so that the orientation of the arc of f3 from R to yR agrees with that of e~. If j3C~R or if
x
x
S visits any vertex in 8D(f3R) \ {O}, then set Q(G R) =0. Otherwise, set Q( G ):= dist(x, intRS) I\dist(y, intRS) I\dist(q, (3) 1\ Ix-yl
R
R
1\
_1_ 100'
LEMMA 3.14. (Compatibility) Assume (h), (D) and (S). For every E>O there is a constant C=C(E, A) >0 such that
~ 1{Q(8 R»c:} l{Q(iI>R' »c:} ~ P[zg
1
G R , IPR' ]log R
~c
holds whenever R>c, 5R>R'>~R and VO~~6RCD. Proof. We start by proving the lower bound on P[zg G R , IPR'], which is the harder estimate. Assume that Q(G R )I\Q(IPR' »E, R>c>10 10 IE, 5R>R'> ~R and VO~~6RCD. Let Ds denote the connected component of ~ R \f3R that intersects intRS and let Ds denote the connected component of D\ (extR'/'U~ R') that contains Vo and therefore extR'S, Note that the sign of h on vertices in Ds adjacent to f3 is constant, as is the sign 1
of h on vertices in Ds adjacent to extR'/', Let V denote the event that these signs are the same, namely, the sign of h on vertices in V(extR,/,)nDS is the same as on vertices in V(f3R)nDS' Using symmetry, Proposition 3.7 immediately implies that P[DIIPR', G R ] is
bounded away from zero. (Although f3 is determined by G R , its orientation as a subarc of an oriented zero-height interface of h is not determined by GR.) We now construct some barriers, as illustrated in Figure 3.6. Let a be the initial point of the arc
8~R,nDs,
when the arc is oriented counterclockwise around 1083
~R"
and
74
o.
SCHRAMM AND S. SHEFFIELD
SE R ,
Figure 3.6. The construction of the barriers.
let b be the other endpoint of this arc. Likewise, let
a be
the initial point of the arc
8Pi R nD s , when the arc is oriented counterclockwise around Pi R , and let endpoint of this arc. Note that {o',b}={xR,yR} and {a, b}={x R', yR'}.
b be the other
We now describe a path lla connecting a to a and a path llb connecting b to b and a path llq connecting fjR to qR' such that these paths do not come too close to each other. For example, if the arguments arg a, arg b, arg qR', arg a, arg band arg fjR are chosen so that argo'<argfjR<argb<argo'+27r, arga<argqR<argb<arga+27r and then we may take lla to be defined in polar coordinates by ()=s+tr, with sand t chosen so that a and a are on the path, and similarly for llb and llq. It is easy to check that our assumptions guarantee that the distance between any two of these [arga-argo'[~7r,
paths is at least
Cl
eR for some constant
Cl
> 0. Set e' =
/0 Cl e.
Let D+ be the connected component of D\(extR,"(UfJRUllaUllb) that contains
v+(extR'''()' and let D_ be the other connected component. Let ll~ be the connected component of {ZED+:dist(z,lla)=e'R} that meets the circle 8B(O, ~(R+R')). Note that ll~ is a simple path which intersects the circle 8B(O, ~(R+R')) at one point. Let ao denote that point. We want to construct a pertubation of ll~, which will be some (ell, 12R)- barrier, with e" not much smaller than e'. Let al be the closest point to ao along ll~ such that the distance from al to 8D(extR'''() is /oe'R, and let 0,1 be the closest point to ao along ll~ such that the distance from 0,1 to 8D(fJR) is 110e'R. Let ll~ be the path which is the union of the arc of ll~ connecting a1 and 0,1 together with a shortest 1084
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
line segment connecting a1 to 8D(extR',) and a shortest line segment connecting
75
ih to
8D(f3R). We claim that a~ is an (c", 12R)-barrier for the configuration (D, extR"Uf3R) with c"= 10100c'. Indeed, conditions (1), (2), (4) and (3) (b) in the definition of the barrier clearly hold. To verify condition (3) (a), let Z1 be the endpoint of a~ on 8D(extR',), and let z~ and z~ be the two endpoints of XZ 1 on 8D(extR"). Consider the simple arc X connecting z~ to z~ in 8D(extR"). By the Jordan curve theorem, XUXZl separates the plane into two connected components. Since a~ crosses XZ 1 , it follows that the part of a~ inside ~R' is outside of A Z1 ' and thus the endpoints x R', yR' and also f3R are all outside A Z1 . It follows that 8Azl =XZl UX, as required. A similar argument applies near the endpoint of a~ on 8D(f3R). Thus, a~ is indeed an (c", R')-barrier. Note also that the above easily implies that 8Azl CD+. This will be useful below when we apply the barriers theorem (Theorem 3.11). We similarly construct a path a", in D_ close to aa. Likewise, we construct barriers a~ and a~ near the path abo The construction is the same, except that we replace aa byab. On the event D, we may apply Theorem 3.11 with Y+ = { a~, a~}, y_ = {a"" a~}, Y =Y+UY_ and ;Y=extR"Uf3R. (Here we use the assumption that R>c.) Note that conditioning on 8 R ,
y:= {(,\;y)n(U Y) = 0}. Let Aa be the connected component of
D(extR',Uf3R)\(a~Ua",)
that contains aa, and
let Ab be the connected component of D(extR"Uf3R)\(a~Ua~) that contains abo Again, using the Jordan curve theorem, it is easy to verify that A a nAb=0. On the event y, there is no other choice for the strand of, extending extR" at a, but to be confined to Aa until it hooks up with f3 at fl, since every other exit from Aa is blocked. Consequently, on y, we have ,~f3. A similar argument applies to Ab, and we get
We now turn to the random walk S. For Zo to hold, we must make sure that
{St:t
o.
76
SCHRAMM AND S. SHEFFIELD
are assuming that Q(IPR'»E. (Recall that {a,b}={xR',yR'}.) Thus, extR,SnAa=0, and we may also conclude that extR'S does not visit any vertex adjacent to Aa when R is large. Similar arguments apply to Ab and to intRS. Thus, intRSUextR'S does not visit any vertex adjacent to "( on the event yn{Q(IPR,»E,Q(8 R »E}. Now let S* be the walk S from the first time it visits qR' until the first time it visits qR. Then, conditioned on 8 Rand IP R', S* is just a simple random walk started at qR' conditioned to hit qR before hitting o. Let Aq denote the E'R-neighborhood of O'.q. Clearly, dist(Aq, AaUAb);?E'R. The probability that S* gets within distance ~E'R of qR before exiting Aq is at least some (perhaps small) positive constant depending only on
E' (and hence on E). Conditional on this event, the probability that S* visits qR before exiting Aq is within a constant multiple of l/log(E'R), by (2.6). Now let S** be the walk S from the first visit of qR to the last visit of qR before time
TO.
Note that S** and
intRS are independent given qR. Thus, given IP R', 8 R, V and S*, we may sample S** by starting a random walk from qR, stopping when it hits 0, and then removing the part
of that walk after the last visit to qR. When the latter walk first gets to distance ~E' R from qR, it has probability bounded away from zero (by a constant depending only on E') to hit 0 before qR. (This follows, for example, from Lemma 2.2 applied to the function giving for every vertex the probability to hit 0 before qR for a random walk started at that vertex.) Thus, conditioned on (S*, 8 R , IPR'), with probability bounded away from zero, S**cAq. Since
we conclude that
Above, we have argued that P[VIIPR', 8 R ] is bounded away from zero, and so we conclude that the lower bound estimate in the proposition holds. It remains to prove the upper bound. Conditional on ,,(, intRS and extR'S, the
probability that S* (as defined in the proof of the lower bound) hits qR before hitting "( is clearly O(l)/log R, since the conditional law of S* is that of a random walk started at qR' and conditioned to hit qR before 0, and the probability that an ordinary random walk started at qR' hits qR before 0 is bounded away from zero. The upper bound now follows, and the proof is complete. D To make the previous lemma useful, we will need to argue that configurations with quality bigger than E are not too rare, in an appropriate sense. This is achieved by the following lemma. 1086
CONTOUR U NES OF THE TWO-DlJ\IENSIONAL DISCRETE GAUSSI,\ N FREE FIELD
ext ", S
77
- -+-7
Figure 3.7. An example for Q j = O:#, Q (
'B Tj
is shaded.
LEMMA 3.15. (Separation) Assume (h ), (D) and (S) . Let p< 1. There exists some constant C=C(IJ, A»O such that if R>l / c and Vo~P,6I/ C D, then (3.34) provided that P [Z818R, «:I4R]> O. T he proof of t his lemma is modeled after Lawler's separation lemma for Brownian motions from [L2, Lemma 4.2]. Proof. To keep the notation simple, we start by proving a simpler version of the
lemma , where we also assume that Q(8u);O:: we prove that
,60, say (and t herefore Q(8n)= ,60)' and
P [Q(
...
(3.35)
as follows. Set 1'o:=4R. Suppose that
is defined . Set if P [Z81 «:1,." 8n ]> 0, otherwise,
and 1'j+ 1:= (1 - lOQj )1'j . Note that Q j = 0 impl ies that P [Z8 I cI)" j' 8 17 ] = 0. (An example showing that Qj =O#Q(<(:I,., ) is possible is given in F igure 3.7. Such a situation can only occur when IX"j - y'·jl= O(l).) Also note that P [Z8 1(1)"j, 8 n j>O if and only if there are paths "f. and S. satisfying the following: (1 ) "f. is a simple TC* -path in 15 containing j3R and extrj"f, (2) S. is a. TC-pa.t h in D containing extrj S and t he reversal of int RS
and (3) S. does not visit any vertex in 8D(,,(.), except for O. Vo,le claim t hat for every J E N, (3.36) 1087
78
O. SCHRAMM AND S. SHEFFIELD
...
... ...
......
... ... ...
...
... ...
,
\
\
/
I
I
I
-- --
-- --
/
/
/ /
/ /
I
I
Figure 3.8. The construction of the barriers giving (3.36).
for some constant co=co(A»O. Clearly it suffices to prove this in the case Qj>O. The gap between 8Pi r j and 8Pi rH1 is larger than but comparable to 5rjQj. Note also that because Pi r j is a union of TG*-hexagons, rjQj?;;1/v'3. If rjQj>20, say, then, we can easily use barriers as in the proof of Lemma 3.14 (see Figure 3.8) to direct and separate the two strands of extrH11'\extr/y and the walk extrH1S\extrjS so as to obtain (3.36). Now assume that rjQj ~20. In this case the discrete structure of the lattice is "visible". Let q' be the point on 8Pi r j crossed by extrj S in its last step, and let Ct be a longest arc among the three connected components of 8Pi rj \ {q', x rj , yrj}. Suppose first that q' is not an endpoint of Ct. Let Ct be oriented counterclockwise around Pi rj , and let a and b be the initial and terminal points of
Ct,
respectively. Let T/a (respectively, T/b)
denote the connected component of extrHll'\extrjl' that has a (respectively, b) as an endpoint. Assume that Iq'-al <500 and Iq'-bl <500. Consider the event X that T/a goes as far to the right as possible subject to the conditions that it remains inside B(q', 550) and avoids ext rj I' and that T/b goes as far to the left as possible subject to the conditions that it remains inside B(q', 550) and avoids extrjl'. See Figure 3.9. It is easy to see that P[zgl
Iq'-bl <500 do not hold.
1088
CONTou n LI NES OF T I'II!; TWO-DI~IENSIONAL DISCRETE GAUSSIAN FREE FIELD
79
Figure 3.9. The interfOCI)!; spreading out.
If q' is an endpoint of 0' a similar argument may be used. Suppose, for example, that q' is the initial point of 0 , that yrj is t he other endpoint of a and that
Iq' _ yr j 1<300.
Then
we may consider the possibility that the connected component of extrH1 ,\extr/ 1' that has y" j as an endpoint goes as far to the left as possible subject to the requirements that it stays inside B(yr j , 5000) and avoids hexagons containing vertices visited by ext rj S , and tha t t he connected component of ext rHI 1'\ext" j ' that has xrj as an endpoint goes as far to t he left as possible subject to t he requirements that it stays inside B(x" j, 4000) and avoids t he previous strand extend ing ext rj l' at yrj and finally, the random walk avoids
8D(exi rj+l') and stays in B(q' , 300) until it hits Iq' - y" j I ~300. This proves (3.36).
(]3 "Hl '
A similar argument applies if
We now prove that fo r every j E N,
(3 .37) for some CI=CI{A»O. Let x=x" j+ ' , y=y" Hl and q=qrj +l . Let
st
be the part of the
walk S from the first visit to qrj up to the first visit to q. Note t hat
if Q j> O, and similarly for y . T hus, the event Qj+l <2Qj is the union of the fo llowing five events Mo ' ~
{QH' ~O},
M, , ~ {d;st(St , (x , y}) < (2Qjrj+,)A Ix - yl),
M2 := {Tj +1Q j+1 = dist (q, extrH l , ), Qj+ 1 < 2Qj}, 1089
80
o.
SCHRAMM AND S. SHEFFIELD
M
3
:= {O < Ix-YI =rj+lQj+1,Qj+l < 2Qj},
M
4
:= {Qj+l
=
160' Qj+l < 2Qj}.
(In the definition of M 1 , dist(St, {x, y}) means the least distance from a vertex visited by st to x or y, of course.) Clearly, (3.38)
160
implies that the holds for k=O. The same is also true for k=4, because Qj+l = random walk started at q has conditional probability bounded away from zero to hit extrHl'"Y before 81J3 rH2 . A similar argument gives (3.38) when k=2. Now condition on M 1 , and let v be the vertex first visited by st that is at distance less than (2Qjrj+I)i\lx-YI from {x,y}. Conditioned additionally on 8 R , extrHl'"Y and the walk st until it hits v, there is clearly probability bounded away from zero that
st hits a vertex adjacent to extrH1'"Y before IJ3 rH2 , and in this case we have Qj+2=0. Consequently, (3.38) also holds for k=l. Now condition on M 3 ,
{Qj+1<2Qj}=U!=oM k , and (3.38) holds for k=0,1, ... ,4, it follows that (3.37) holds as well. Set sn:=2RI1~~~(1-2-k/10)-I. It follows from Lemma 3.14 and our assumption that Q(8R)~
160 that for any JEN,
for some C2=C2(A»0. An appeal to (3.36) therefore implies that
Continuing inductively, we get for every nEN,
Since
2-k)-1
< 2R 1- L 10 00
SUPSn n
(
k=O
1090
=
5
"2 R ,
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
81
we get for every j EN, [
a
I
-log2 Qj
j
P Zo rj,8R ~
Co
logR
Q-log2 Co C2
1{rj)5Rj2}=C2
j
logR
1{rj)5Rj2}·
(3.39)
From (3.37) we get that for every j the conditional probability that there is some
k>j such that rk>2R, Qk<2Qj and Qk+1>O, given rj and 8 R , is at most 1-Cl. Let mn denote the number of kEN such that rk>3R and QkE(2-n, 21 - n j. Fix some nEN, and suppose that P[m n >014R,8 R j>0. On the event mn>O, let k n be the first k such that QkE(2-n,21-nj. By induction and (3.37), for every mEN,
An appeal to the upper bound in Lemma 3.14 gives a
P[Zo ,mn >2m Imn >O,rk n ,8Rj ~
c3(1- Cl)m 1og R
for some C3=C3(A). On the other hand, (3.39) gives
Comparing the last two inequalities, we get
In particular, there is an no=no(A,p)EN and a C4=C4(A) such that
Let
nl
be the least integer larger than 3 such that I1~=nl (1-10.2 1 -
n )c4 n
>i, and let
n2=nl Vno· On the event zOnnn>n2 {mn~c4n} we must have some JEN with rj>~R and Qj~2-n2 (because rj+1/rj=1-10Qj and n2~nl). Consequently,
!
P [there exists j : Q j > C5, r j > ~ R I Zo, 4R, 8 R] ~ 1- (1- p)
(3.40)
holds for some C5=C5(P, A»O. Note that this almost achieves our goal of proving (3.35). The difference between (3.40) and (3.35) is that in the latter the radius r at which Q( r) is bounded from below is variable. Let A be the event that there is a JEN with Qj>C5 and rj>~R, and on A, let jo denote the first such j. We have from (3.39) that
(3.41)
1091
82
O. SCHRAMM AND S. SHEFFIELD
Fix some s>O small. We now argue that
a 3R 3R 0S~ P[ZO IA,<prJO ,eR,lx -y l<sRJ~-logR
(3.42)
for some positive constants C7 and C6 depending only on A. The argument is similar to the one given in the proof of the case k=3 in (3.38). Let z be the midpoint of the segment [x 3R , y3RJ. We construct a barrier as a perturbation of the connected component of oB(z,2sR)\oD(ext3R'Y) that intersects lE 3R. If that barrier is hit by the extension of ext3R'Y (which happens with probability bounded away from 1), then we condition on the extension up to that barrier, and construct another barrier at radius 4sR, instead. We continue in this manner, constructing barriers at radii 2n sR up to the least n such that 2n s> 10100' say. Because the probability of avoiding the nth barrier given that the (n-1)th barrier has been reached is bounded away from 1, we find that P['Y=:J;JRIA,<prjo,eR,lx3R_y3RI<sR] is bounded by a constant times some positive power of s. The estimate (3.42) follows by considering the behavior of S. Suppose now that the random walk S after its first hit to lE r JO· but before its first hit to lE3R gets within distance sR of ext3R'Y. Then, by Lemma 2.1, conditional on S up to the first time this has happened and on A,
P[zg I A,
C7SC6 +CSS(l logR
Comparison with (3.41) now gives
a P[Q(
P[Q(
and take Tj+1=(1-lOQj)Tj and Tj+1=(1+10Qj)Tj. The proof proceeds essentially as above. The straightforward details are left to the reader. D 1092
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
COROLLARY
83
3.16. There exists a constant c=c(A) >0 such that the following esti-
mate holds. Let (D',o~,o~,h'a,h',,,!',vb) satisfy the same assumptions as we have for (D,o+,o_,ha,h,,,!,vo). Let R>c, and assume that fJ3 6R cD'nD and vo,VbjtfJ3 6R . Let 8~, ~R and zg' be the objects corresponding to 8 R, 4R and zg for the system in D'
(with the same u, that is, u'=u). Then
P[8 R = {} I zg, 4RJ ~ cP[8~ = {} I zg', ~RJ
(3.43)
holds for all {} and for all 4R and ~R satisfying P[zg I4RJ >0 and
P[zg'l ~RJ >0,
respectively. Consequently, under the same assumptions, there exists a coupling of the conditional laws of 8 R and 8~ such that
Proof. It is enough to prove the first claim, since the latter claim immediately follows. Let c'>O be the constant denoted as c in the separation lemma (Lemma 3.15) with p=~.
Let Q denote the event Q(3R)i\Q(82R)~C'. Let X be the collection of all () such that 8 2R =() is possible and Q(())~c', and let X{} be the collection of all ()EX that are compatible with 8 R={); that is, such that {8 2R =() and 8 R={}} is possible. In the following, f~g will mean that fig is contained in [l/c,cJ for some constant c=c(A»O. By Lemma 3.15 and the choice of p,
P[8 R = {} I zg, 4RJ ~ P[8 R = {} I Q, zg, 4RJ =
L
P[82R = () I Q, zg, 4RJ.
(3.44)
OEXiJ
Now, if Q(3R»C' and ()EX, then
P[8
2R
= ()
I Q zu
' 0 ' 3R
= (), Q, zg I 3RJ P[zg, Q I 3RJ
J = P[8 2R
P[82R = (), zg I3RJ P[zg, Q I3RJ P[82R = () I 3RJP[Zg I 8 2R = P[zg, Q I 3RJ
(),
3RJ
We apply Lemma 3.13 to the first factor in the numerator and Lemma 3.14 to the second factor, and get
P[8
2R
I
= () Q
zu J ~ P2R(())/log R ' 0 ' 3R P[zg, Q I 3RJ·
The sum of the left-hand side over all ()EX is 1. Consequently,
1093
84
O. SCHRAMM AND S. SHEFFIELD
By taking expectation conditioned on Q, Zo and
pre
R
= {)
I Z
0'
4R
1~ ~
LOEXt/
'"
L..OEX
P2R(0) (0) . P2R
This implies (3.43), and completes the proof.
D
Our intermediate goal to show that the dependence between the local behavior near
Sr and the global behavior far away is now accomplished. Roughly, the next objective will be to show that it is unlikely that, contains an arc with a very large diameter whose endpoints are both relatively close to Sr. For R>r>O, let J =J(r, R) denote the event that there are more than two disjoint arcs of , connecting 123 r and 8123 R or that S exits 123 R between the time it first hits 123 r Set Zo:=U;=o Zoo Our next objective is to show that conditioned on Zo or Zo, J(r, R) is unlikely if R»r>O. More precisely, the claim is as follows.
and
TO.
LEMMA
3.17. Assume (h), (D) and (S). For every P>O there is some a=a(p,A»10
such that if r>1, R>ar, VOrjc123 4RCD and P[ZOI
and also P[J(r, R) I
TO.
Next, we identify a pair of arcs
0:1
and
0:2
that are defined from
of which has one endpoint on ext3rS and the other on ext3r" A barriers argument is then used to show that with high conditional probability ,\ext3r, does not hit In this case, we see that
0:1 U0:2
0:1U0:2.
has an alternative definition in terms of, and S, which
is in some sense more symmetric. Next we define another pair of arcs (Xl U(X2, which have a similar definition as
0:1 U0:2,
except that they are defined from
e R/3'
Again, the same
barriers argument can be used to show that with high conditional probability these arcs are not visited by ,\f3R/3' In this case, these arcs have a more symmetric definition, 1094
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
85
which leads us to conclude that with high conditional probability 0<1 U0<2 =0;1 U0;2. This is then used to establish that the endpoints of these arcs on 'Y belong to ext3r'Y as well as i3R/3. Next, we prove that these two endpoints belong to different connected components of 'Y\ e~. This then implies that each of the two strands of ext3r'Y merges with i3R/3, which implies that there are no more than two disjoint crossings between aSE Rand SE 3r in 'Y. Proof. The second claimed inequality with p replaced by 6p follows from the first
inequality and taking conditional expectation, since Zo = U~=o zg. Thus, we only need to prove the first inequality. By Lemma 3.15, there is a constant co=eo(A,p»O such that
(3.45) Let r' be in the range [vrR, vrR+ 1], chosen so that the circle aB(O, r') does not contain any TG-vertices nor any TG* -vertices. Let 8* denote the part of the walk 8 from its first visit to q3r until its last visit to it prior to TO. Conditional on
By the lower bound in that lemma, on the event Q(
Consequently, a may be chosen sufficiently large so that
Hence, (3.45) implies that
(3.46) Let 8' be the path traced by ext 3r 8 from the last time in which ext 3r 8 was outside SE R/ 3 until its terminal point q3r ESE 3r . Let 8 be the path traced by intR/38 from the last time in which intR/38 was inside SE 3r until its terminal point qR/3. Observe that 8' is
86
o.
SCHRAMM AND S. SHEFFIELD
Observe that there is a connected component a of aB(O, r') \5' such that 5'Ua separates aD from ~3r. We fix such an a, and if there is more than one possible choice, we choose one in a way which depends only on 5'. On the event Q(<J>3r»0 each strand of exhr'Y connects aD with ~3Tl and hence ext3r'Y intersects a. Thus, there are precisely two connected components of anD(ext3r'Y) which have one endpoint in 5' and the other in aD(ext3r'Y). Let al and a2 be these two arcs. We now argue that (3.47) if a is sufficiently large. The basic idea of the proof of (3.47) is to construct a sequence of barriers separating from al in D( ext3r'Y) such that if 'Y hits a barrier in the sequence, the conditional probability that it will hit the next barrier is bounded away from l. ~3r
Let D' be the connected component of D(ext3r'Y)\~3r that contains al. Note that D' is a simply connected domain. See Figure 3.10. Let Zal denote the endpoint of al on aD' and let 6 and 6 be the two connected components of aD'\({ZaJU$3r). (Both have zal as an endpoint and the other endpoint in a~3r.) Note that any path in D'\al connecting 6 and 6 separates al from a~3r in D(ext3r'Y). For each eE(3r+3, r'-3) let A(e) denote the connected component of D'\B(O, e) that contains aI, and let a(e) denote the connected component of aA(e)naB(O, e) that separates al from ~3r in D'. Observe that a(e) has one endpoint on 6 and the other on 6. If 3r+3<e<e'
7](n) := min{ dist(w, 6) : w E A~, dist(w, 6) = dist(w, 6)}. First, assume that 7](n»8en. In that case, a barrier Y n is defined as follows. By continuity, there is a subarc Y' of a(en)CA~ with endpoints Zl and Z2 such that dist(zj,~j)= 1108en' j=1,2, and dist(Y',aD')=1108en. Let zj be a point in ~j at distance l08en from Zj, j = 1, 2. Then we take Y n as the union of Y' with the two line segments
[Zl' zil and [Z2' z~l. Recall the definition of dist(·, .; .) from (3.30), and note that dist(zj, 6-j; D'» 1108en' j=1,2, for otherwise, by continuity again, there would be a 1096
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
87
oB(O,r')
Figure 3.10. The domain D'.
l
point wEI)' satisfying dist(w, zj; D')~ 0 8en (and therefore wEA~) that is at equal distance from 6 and from 6, which would contradict our assumption 7]( n) > 8 en. It easily follows that in this case Y n is a (210 8, 2en )-barrier. We now assume that 7](n)~8en. Let wIEA~ be a point satisfying
For j=1,2, let PjE~j be a point satisfying IWI-Pjl=dist(WI,aD'). If
then we may take as our barrier the union [Pl,Wl]U[Wl,P2]. This will be a (~8,28en) barrier. Otherwise, fix a point W2 satisfying
and consider the above construction with W2 in place of WI. It may happen that the construction succeeds now, and we construct a (~8, 282en)-barrier. Otherwise, we find a point W3ED' satisfying dist(w3,wl;D')~(8+82+83){!n such that
We continue this procedure until some (~8,28m{!n)-barrier is obtained. The procedure must terminate successfully at some finite m, for otherwise the points Wm would converge to some point in
6n6 within distance 28{!n from 1097
A~, which is clearly impossible. Note
88
o.
SCHRAMM AND S. SHEFFIELD
that the barrier Tn thus constructed is contained in A~o. Thus, when 1 ~ n' < n ~ N,
n,n'EN, we have dist(T n , Tn,;D'»~en and Tn separates 001 from Tn' in D'. Suppose nE{l, ... , N -I}. Note that (contrary to what appears in Figure 3.10, which does not show the scale of the lattice) the endpoints of Tn are not on ,,/, since 6 U6 are disjoint from ,,/, by construction. On the event ,,/nTn #0, let "/1 and "/2 be the two arcs of,,/ extending from the endpoints of 0+ to the first encounter with Tn. Now we apply Theorem 3.11 with Y ={T n+d. Our careful construction above ensures that T n+l is an (c:, diam T n+l)-barrier for some universal constant c:>O. Note that 6 and 6 contain vertices on which h takes the same sign. We conclude from the theorem that
for some Cl=Cl(A»O. The above implies that
which gives (3.48) Conditioned on ,,/, 3r and 8 2n the probability of of the upper bound in Lemma 3.14. Thus,
Zo is at most O(l)/log r, by the proof
On the other hand, the lower bound tells us that on the event {Q(3r)I\Q(2r);)co}, we have O,\,p(1)P[ZOI3n82r];)1/logr. Thus, (3.47) follows. Clearly, (3.47) also holds for 002. On let a]' and 00 2 be the two connected components of oonD("() that have one endpoint in 8' and the other in oD("(). Note that when
Zo
Zo holds and ,,/n(001UOO2)=0, we have oo]'Uoo2=OOlUOO2. Thus, (3.47) for 001 and for 002
together with (3.45) now gives (3.49) We now follow an analog of the above argument with the roles of inside and outside switched. Observe that there is a connected component a of oB(O, r') \S such that SUa separates aD from Sl3 3r . We fix such an a, and if there is more than one possible choice, we choose it in the same way in which a was chosen from 8'; that is, we make sure that
oo=a if 8'=S (as unoriented paths). The point is that although a a is 8 R / 3-measurable, we have oo=a if 8'=S. 1098
is 3r-measurable and
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
89
On the event Q(8 R / 3»0, let al and a2 be the two connected components of anD((3R/3) that have one endpoint on S and the other on aD((3R/3). On the event
Zo, let ai and a 2 be the two connected components of anD(--y) that have one endpoint
on S and the other on aD(--y). Essentially the same proof which gave (3.49) now gives
(3.50) But observe that when
Zo and 8'=S hold, we clearly have aiUa2=c¥iUc¥2. Also recall
that 8'=S when 8*clErl/3. Thus, from (3.46), (3.49) and (3.50), we get
Zo,
al Ua2 =c¥i UC¥2 =C¥1 UC¥2 and 8* cIEri /3 hold. It remains to show Assume that that in this case the path '/ has no more than two disjoint arcs connecting IEr and alE R . Recall that al has an endpoint on aD((3R/3). This endpoint is on a TG-triangle containing a TG*-vertex VIE(3R/3. Similarly, there is a TG*-vertex V2E(3R/3 for which
the TG-triangle containing it has an endpoint of a2. From al Ua2 =C¥1 UC¥2, we conclude that Vl,V2Eexhr'/ as well. Shortly, we will prove that VI and V2 are in separate connected components of This implies that each connected component of (3R/3\e~ intersects ext3r'/. Since (3R/3Uext3r'/C,/, and,/ is a simple path, it easily follows that ,/=(3R/3Uext3r'/, which implies that there are at most two disjoint crossings in '/ between IEr and alE R. (3R/3\e~.
It remains to prove that VI and V2 are in different connected components of (3R/3\ e~. This will be established using planar topology arguments. Let a consist of al Ua2, a simple path SacS connecting them, and short line segments (contained in the TGtriangles containing VI and V2) from the endpoints of al and a2 on aD((3R/3) to VI and V2. Then a is a simple path and only the endpoints of a are on '/. Let ~ be the connected component of (3R/3\ {VI, V2} with endpoints VI and V2. Then ~Ua is a simple closed path and it suffices to show that e~ C~. If ~Ua separates 0 from aD, then ~ must contain e~, because each connected component of '/\ {e~} connects aD to TG* -vertices adjacent to 0 and is disjoint from a\ {VI, V2}. Suppose that ~Ua does not separate 0 from aD. Recall that aUS separates aD from 1E3r and therefore from O. Since S itself does not separate aD from 0, it follows that the winding number of aUSo around 0 is ±1 (depending on orientation). Since ~Ua does not separate 0 from aD, its winding number around 0 is zero. If we remove from the union of the two paths aUSo and ~Ua all the non-trivial arcs where they agree, we get a closed curve X, which consists of ~, a segment of a and the two short connecting segments near VI and V2, and X has odd winding number around O. Consequently, it 1099
90
o.
SCHRAMM AND S. SHEFFIELD
separates 0 from aD. But observe that S is disjoint from X. Moreover, since we are assuming S* C~r' /3, it follows that int R/ 3S is also disjoint from it. But this contradicts the fact that X separates aD from 0, since intR/3S can be extended to a path disjoint from X and connecting 0 and aD. Thus, the proof is now complete. D
3.6. Coupling and limit In this subsection, we retain our previous assumptions (h), (D) and (S) about the system D, a+, a_, ha, Vo, h, 'Y. Moreover, we also consider another such system D', a~, a~, h~, vb, h', 'Y', which is supposed to satisfy the same assumptions. In particular, Ilh~ 1100 ~A.
Generally, we will use' to denote objects related to the system in D'. For example, .:J' (r, R) will denote the event corresponding to .:J(r, R). Definition 3.18. Fix R>r>O, and suppose that ~RcDnD'. Consider the intersec-
tion extr'Yn~R as a collection of oriented paths, oriented so as to have vertices in V+h) on the right. We say that
3.19. For every 8>0, r;?l0 and R*>r+3 there is an R=R(8,r,R*,A»R*
such that the following holds. Suppose that vb,VOtt.~RcDnD', P[-'.:J(r,R*),Zol
Here, Ilv-v'll denotes the total variation norm L:'I9lv['Y*=OJ-v'['Y~=OJI. Proof. Assume that the orientation of extr'Yn~ R agrees with that of extr'Y' n~ R.
This involves no loss of generality, since we may replace a~ with a~, replace h' by -h', etc. Since we are assuming that P[-'.:J(r,R*),Zol
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
91
neighbors of v, the conditional law of h( v) is a Gaussian with some constant positive variance. It follows from (3.2) that with high probability (conditioned on the success of the previous steps) the mean of this Gaussian random variable is unlikely to be large. Thus, the probability for either sign is bounded away from zero, which means that each step is successful with probability bounded away from zero. By (3.3) it is unlikely that h( v) will be very close to zero. Proposition 3.3 therefore implies that if R> R* is very
large, the probability for a successful one-step extension for 'Y' is almost the same as for 'Y. Thus, we conclude that for sufficiently large R> R*,
holds for all
-aEro
It is moreover clear that
because under -,J(r, R*) the random walk S cannot get close to any place where extr'Y differs from extr'Y' between the first visit to qr and time
TO.
Thus,
D
The lemma follows (though perhaps 8 needs to be readjusted). The next lemma shows that given R the events for different a. LEMMA
Zo have comparable probabilities
3.20. As usual, assume (h), (D) and (8). There is a constant c=c(A)~l
such that for all R sufficiently large and every a, a' E {O, 1, ... , 5} we have
Proof. The statement is clear when R= 100, because in that case if it is at all possible
Zo
zo'
to extend R in such a way that holds, then the probability that holds is bounded away from zero (by a function of A). (We may choose the continuations of 'Y and S as we please, and as long as the continuations involve a bounded number of steps, the probability for these continuations are bounded away from zero, as in the proof of Lemma 3.19.) When R>100, we may just condition on the corresponding extension of R up to radius 100.
D
We now come to one of the main results in this section-the existence of a limiting interface. 1101
92
O. SCHRAMM AND S. SHEFFIELD
THEOREM 3.21. (Limit existence) There is a (unique) probability measure f1CXJ on the space of two-sided infinite simple TG* -paths i which is the limit of the law of r (unoriented) conditioned on Zo and R, in the following sense. Assume (h), (D) and (S). For every finite set of TG* -edges Eo and every 15>0 there is an Ro=Ro(J, Eo, A) such that if R>Ro, vorjc'iJ3RCD and P[ZO[RJ>O, then
[P[Eo C r [Zo, RJ- f1CXJ[Eo C iJ[ < J. Proof. Clearly, it suffices to show that, for every r>O, if R is sufficiently large, vo, vbrjc'iJ3 R cDnD ' , P[Zo [RJ >0 and P[Zb [kJ >0, then we may couple the conditioned laws of r given R and Zo, and r' given k and Zb such that P[r\extrr=r'\extrr' [R' Zo, k, ZbJ > 1-15.
(3.51 )
(Here, the equivalence is an equivalence of unoriented paths.) Let an be the constant
a given by Lemma 3.17 when one takes p=Jn:=~2-nJ. We define a sequence of radii ro, rl, ... inductively, as follows. Let Cl be the constant c given by Corollary 3.16, and set ro:=rVlOVcl. Given r n , let Tn be the R promised by Lemma 3.19 when we take rn for r, I n for 15 and anrn for R*. Finally, set rn+l =4anTn. Let C2 be the constant promised by Lemma 3.20. We assume, with no loss of generality, that J<1j4cl' Let NEN be sufficiently large so that (1-1j72clC~)N-l<~J. We will prove (3.51) on the assumption that R>6rN. The construction of the coupling is as follows. First, we choose rN_l and ~N_l independently according to their conditional distribution given R, Zo, k and Zb. We proceed by reverse induction. Suppose that nE[1,N-1JnN and that rn and ~n have been determined. If rn and ~n match inside 'iJ3 fn , then we couple rand r' in such a way as to maximize the probability that r\ext rn r=r'\ext rn r', subject to maintaining their correct conditional distributions given the choices previously made. If they do not match, then we couple rn_l and ~n_l in such a way as to maximize the probability that they match in 'iJ3 fn _ 1 , subject to their correct conditional distributions. If ro and ~o have been determined, but rand r' have not, then we couple rand r' arbitrarily, subject to their correct conditional distributions. We claim that the coupling just described achieves the bound (3.51). Let Mn denote the event that rn and ~n match inside 'iJ3 fn , where nE{1, ... , N -1}, and let M denote the union of these events Mn. It follows from the choice of Tn that if Mn holds and n is the largest n ' E {1, ... , N -1} with that property, then there is a coupling of the appropriately conditioned laws of rand r' such that
Pb\ ext rn r # r'\ ext rn r' [ rn , ~n' Zo, zbJ ~ I n +P[..1(rn, anrn) [rn' ZbJ+P[J'(rn' anrn) [~n' ZbJ, 1102
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
and hence this also holds for our coupling. By taking conditional expectation and summing over
n, we get
P[M, 'Y \ extr'Y i=- 'Y'\ extr'Y' I R, ZO, ~, ZbJ N-l
:::;
~6+ L (P[.J(rn, anrn) I R, ZoJ+P[.J'(rn, anrn) I ~, ZbJ):::; ~6. n=l
(where the last inequality follows by the choice of an). Now fix some nE{l, ... , N -2}, and suppose that none of the events Mn"
n'>n,
occurs. Fix some arbitrary aE{O, 1, ... , 5}. Conditional on rn+l and Zo, by the choice of C2, there is probability at least 1/6c2 that Zo holds, and the same is true for the system in D'. If we additionally condition on Zo and Zo', then, by the choice of Cl, there is a coupling of the appropriate conditioned laws of e rn +t/4 and e~n+t/4 such that
But note that if ern+t/4=e~n+t/4 and -'(.J(rn, ~rn+l)U.J'(rn, ~rn+1)) both hold (as well as ZonZo '), then Mn holds as well. We may then consider a coupling of the two systems which first decides the two events Zo and Zo' independently, and if both hold (which happens with probability at least 1/(6c2?), then with conditional probability at least l/Cl we also have ern+t/4=e~n+t/4' Thus, under this coupling,
(6C2)2p[Mn I rn+l' ~n+l' Zo, ZbJ ~ c1l -p [.J(rn' ~rn+l) U.J' (rn' ~rn+l) I rn+ll ~n+l' zg, zg'J ~ c1l -26n (by the choices of rn+l and an) (by our assumption 4C16 < 1). This must hold for our coupling as well. Consequently, induction gives
which is less than ~6 by the choice of N. Therefore (3.51) follows, and the proof is D
complete.
3.7. Boundary values of the interface Consider the random path 'Y whose law is the measure /-Loo provided by Theorem 3.21. We orient 'Y so that the edges
e~,
a=O, 1, ... ,5, that are in 'Yare oriented clockwise around the 1103
94
o.
SCHRAMM AND S. SHEFFIELD
hexagon U~=o e~ centered at O. Let U+ denote the set ofTG-vertices adjacent to 'Y on its right-hand side, and let U_ denote the set of vertices adjacent to 'Y on its left-hand side. Using the heights interface continuity (Proposition 3.3), it is clear that given 'Y we may define the DGFF h on all of TG conditioned to be positive on U+ and negative on U_, as a limit of an appropriately conditioned DGFF on bounded domains. Moreover, many properties of the DGFF on bounded domains easily transfer to
h.
In particular, (3.2)
applies, to give E[lh(0)11'Y]=O(1). Set
A:=E[h(O)].
(3.52)
Clearly, O
in probability, under the assumption that (8) where Ao=Ao(A»O is the constant given by Lemma 3.9. The importance ofthe assumption (8) is that, by Remark 3.12, it enables the application of Theorem 3.11 to barriers with endpoints on 8D, provided that the barriers in Y+ do not have endpoints in 8_ and those in Y_ do not have endpoints in 8+. This allows us to prove the following result. THEOREM 3.22. Assume (h), (D), (S) and (8). There are positive constants (2= (2(A) and c=c(A) such that the following holds true. Let z be any point on 8+. Then for every r>1, r
P[dist(z, /'; D) < r] ~ c ( d· ( 8. ) 1St z, _, D
)(2
Proof. Let /31 and /32 be the two components of 8+\ {z}. Let R=dist(z, 8_; D). For
each eE (0, R), let A(e) denote the connected component of B(z, e)nD that has z in its boundary, and let a(e) denote the connected component of 8A(e)\8D that separates z from 8_ in D. Using this construction, the proof proceeds as in the proof (3.48), except that the barriers start from the outside and get closer to z, and we appeal to Remark 3.12 instead of the barriers theorem. We leave it to the reader to verify that the proof carries over with no other significant modifications. D Our next lemma shows that it is unlikely that ST is adjacent to /' and is near 8D. 1104
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
LEMMA
95
3.23. Assume (h), (D), (S) and (a). For all p>O there is some 8=8(p, A) >0
such that
prO < dist(Sn aD) < 8 dist(vo, aD)] <po Proof. Let
TD
be the first t such that StEaD, and let
We will prove the stronger statement P[dist(M, aD) < 8 dist(vo, aD)] <po
(3.53)
(By convention dist(0, aD)=oo.) Fix r>O and set R=dist(vo, aD). Conditioned on dist(M,aD)
since the random walk started at any vEM such that dist(v,aD)
(3.54)
for 8=8(p, A) >0. Let A+=A+(8) denote the event dist(STD' a_; D) >8 1 / 2 R, and similarly define A_ with a_ replaced by a+. By conditioning on STD' Theorem 3.22 shows that P[A+, dist(STD' 1'; D) < 8R] < ~p
for an appropriate choice of 8. A symmetric argument applies on A_. Consequently, it is enough to prove that P[-.(A+UA_)]<~p for an appropriate choice of 8. Fix ro :=P/2 R, let Lro denote the set of points that lie on some path in 15 of diameter at most ro connecting a+ and a_, and let D* be the connected component of Vo in D \ Lro' (See Figure 3.11.) We now prove that aD* \ aD is contained in the union of two balls of radius 2ro.
(3.55)
Every path connecting a+ and a_ in 15 must separate Vo from Xa or from Ya in 15 (because, by Jordan's theorem, it separates Xa from Ya in 15). Let r 1 (respectively, r 2 ) denote the collection of paths in 15 of diameter at most ro that connect a+ and a_ and separate Xa (respectively, Ya) from va. Then aD* \aD is contained in the union of the set of points belonging to a path in r 1 and the set of points belonging to a path in r 2 . Suppose that a and a' are two paths in r 1 , both of which intersect aD*. Let (3 be a path connecting Xa with aUa' in 15, which is disjoint from aUa', except for its endpoint. If (3nayf0, 1105
96
o.
SCIHIAM~r
AND S. SHEFFIELD
D.
Xa Figure 3.11. The set
I,~o
(shaded) and D •.
t hen one can connect Xo to Vo in !3Uo:UD., and therefore ana':r60 (since a' separates from X{J in 15). Similar reasoning applies if {3na't-0. It follows that any two paths in f l that intersect aD. must intersect each other, and hence the collection of a ll such paths is covered by the ball of radius 21"0 centered at any point on any such path . Since a similar argument applies to f 2, (3.55) follows. By (3.55) and Lemma 2.1 ,
'00
P [there exists t ~ TD: Sf E LroJ
< 3P
for all sufficiently small 8> 0. Thus P [-,(A+uA _)]= P (STD E Lrol < com plete.
3P, and the proof is 0
Next, we show that So is unlikely to be close to Vo by proving the same for f.
LE1\]1\IA 3.24. Assume (h) and (D). There are constants c> O and (3) 0, both depending only on
A,
such that
/07'
eve111 15>0,
P [dist(vo, ,) < 15 dist(vo, tiD)] < cJc. We expect that the left-hand side is bounded by
"esul t [RS[
[0<'
Jl /2+Q( I ),
using the corresponding
SLE(4) ,
Proof. Let T " denote the circle of radius 2- ,, - 1 dist(vo,tlD) about va . As in the
proof of (3.48) , Theorem 3.11 implies that , given that, intersects T " (where nEN), t he conditional probability that I doC>:! not intersect T "+I is bounded away from zero by a funct ion of A, provided t hat 2-,,- 1 dist(vo,8D} > 1O, say. The lemma follows by induction.
0 1106
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PROPOSITION
97
3.25. Assume (h), (D), (S) and (8). For every E>O there exists an
R=R(E, A) such that
(3.56) holds, provided that dist(vo,8D»R, where A is the constant given by (3.52).
The proof is based on the simple idea that given a single instance of 'Y we consider two independent copies of (h, S). Proof. Set X:= E[(lh(Sr )1-A)l{sT~aD} I'YJ.
To get a handle on E[X2], let (h', S') be independent of (h, S) given'Y and have the same conditional law as that of (h, S) given 'Y. Thus, (h, S, 'Y) has the same law as (h', S', 'Y). Let T':=min{t:S~E8Db)}, y:=E[lh(Sr)I-AISn 'YJ and y':=E[lh'(S~, )I-AIS~" 'YJ. Then X2=E[yy'l{sT,s~,~aD} I'YJ and hence
(3.57) Fix some r3»r2»rl»O, and assume that dist(O,{vo,8D}»r3. Suppose that we condition on Zo; that is, on Sr=O. Then, with high conditional probability IS~,1>2r2' and moreover dist(O,{Sb,S;:, ... ,S~,}»2r2. By the heights interface continuity (Proposition 3.3), given extrl'Y and S~, and 'Y\extrl'Ycs.Br2' the actual choice of 'Y\extr1'Y can change the value of y' by very little if IS~, I>2r2. Thus, we conclude that y'ls~,~aD is nearly independent of y given Zo, -':1 (rl , r2) and extrl 'Y. Since y and y' are bounded, in the limit as r3 -+00, we have E[yy'l{s~,~aD} I Zo, extrl 'Y, -,:1J
= 0(1)+ E[y'l{s~,~aD} I Zo, extrl 'Y, -':1JE[y I Zo, extrl 'Y, -,:1],
(3.58)
where :1=:1(rl,r2). By Lemma 3.17, P[:1IZoJ-+O as rdrl-+oo. Thus
in probability as r2/rl -+00. A similar remark applies to the other terms in (3.58). Since y and y' are bounded, taking conditional expectation given Zo in (3.58) gives
By the limit existence theorem (Theorem 3.21), given Zo and extrl 'Y, near 0 the path 'Y when rl is large. Consequently, Proposition 3.3 implies
'Y* is close in distribution to
that E[lh(O) II ZO, extrl 'YJ- A=o(l) as rl -+00, which gives that
1107
98
o.
SCHRAMM AND S. SHEFFIELD
Now (3.59) implies that (3.60)
as dist(O,aDU{vo})-+oo. Lemmas 3.23 and 3.24 tell us that, for ro
P[5r
~
aD, dist(5T) aDU{ vo}) < ro]-+ 0
as dist(vo, aD)-+oo. Since there is nothing special about the vertex at 0, except for our assumption that dist(O, aDU{ vo}) is large, we conclude from (3.60) that
in probability, as dist(vo, aD)-+oo. Now, equation (3.57) implies that E[X2]-+0, since yy/1{sT,s~/rf.aD}
is bounded. This gives (3.56) and completes the proof.
D
Let F be the function that is equal to ha on aD, A on V+ ('y), - A on V_ ('y) and is discrete-harmonic on all other TG-vertices in D. Since E[h(vo) h]=E[h(5r sition 3.25 gives
E[h(vo) h]-F(vo)-+O
)
hJ,
Propo(3.61)
in probability as dist(vo,aD)-+oo. We now need to generalize the proposition and (3.61) to apply when "( is replaced by an appropriate initial segment of "(. Let T be some stopping time for "( started at Xa and let "(T denote "( stopped at T. (Note that the relevant filtration here, the one generated by intitial segments of ,,(, only reveals the signs of h on vertices adjacent to these initial segments, but not the actual values of h.) Let ZT denote the vertex in aD('yT) first visited by 5, and let 5 T denote the initial segment of 5 up to its first visit to ZT. LEMMA
3.26. Assume (h), (D), (S) and (a). For every Po>O there is some s=
s(Po, A) >0 such that P[dist(zT, "(\ "(T) < s dist( vo, aD), ZT ~ aD] < Po. Note that we could rather easily prove the estimate with dist(zT,,,(\,,(T;D('yT)) instead of dist(zT, "(\ "(T), by the argument giving (3.48), but this is not sufficient for our purposes. The idea of the proof of the lemma is to first show that 5 r =ZT is usually not too unlikely given "(T and 5. Then Lemma 3.17 may be used in conjunction with the argument giving (3.48) to deduce the required result. Note that the event 5 r =ZT is the event that "(\ "(T is not adjacent to any vertex visited by 5 prior to ZT. 1108
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
99
Proof. We first show that for every c:>0 there is a p>O and an R>O, both depending only on c: and X, such that
P[P[ZT=ST IS,'"'?]
(3,62)
holds provided that dist(vo, aD»R. We choose 8=8(c:, X) >0 very small. Set r=dist(vo, aD), and assume that r> 1008- 2. Set further DT:=D(ryT). Let bT be the point on aDT near the tip of IT that is at equal distance from V+(ryT) and V_(ryT) along aDT . Let and a!. denote the two connected
ar
components of aDT \ {Ya, bT } that have Ya and bT as their endpoints, with one containing vertices in a+.
ar being the
Let A I be the event dist (ZT' ar; DT ) Vdist (ZT' a!.; DT ) ~ 28r, let A2 be the event diam(ST)<8- l r, let A3 be the event that the diameter of the segment of ST after the first time at which it is distance at most 82r from aDT is less than !8r, and let A4 be the event dist(vo,IT)~P/2r. Lemma 3.24 shows that if 8=8(X, c:) is sufficiently small, then P[....,A4 ] < ~c:. On the other hand, Lemma 2.1 implies that, by choosing 8 sufficiently small, one can ensure that P[....,AjItT]<~c: for j=2,3. We now prove the same for j=1. Assume that A4 holds. Let L be the set of points in DT that lie on a path of diameter at most 48r in DT connecting
ar and a!., and let D* be the connected component of DT\L that contains
Vo (we know that votf-L, since 8 is small and A4 holds). By (3.55) applied to DT in place of D, aD*nDT may be covered by two balls of radius 88r. Thus, Lemma 2.1 shows that if 8=8(c:) is chosen sufficiently small, then P[S hits L before aDT'~] < ~c:. This implies that P[....,AI,A4]<~c:. Thus P[....,A]
P[ST = ZT I S, IT]
100
o.
SCHRAMM AND S. SHEFFIELD
Suppose now that BT does visit vertices adjacent to 10. We can then construct a path 1 whose image is 10 as well as all the boundaries of hexagons visited by BT that are not separated from a'!. by 10. Since we are assuming that A2 holds, diam 1:::;;20- 1 r. As A3 holds, dist(l\ 10,aDT»~02r and therefore also diam 10>~02r. Consequently, 1 is a (~0103, 20- 1 r)-barrier. Now Theorem 3.11 and Remark 3.12 may be used again to give a similar lower bound on P[Br=ZTIB,')'T]. As a similar argument applies when
BrEa'!., the proof of (3.62) is now complete. We now choose E:= ~po and take a p>O and R>O depending only on E: and A and satisfying (3.62). Let a be such that the estimate given in Lemma 3.17 holds with the p there replaced by ~Pop. Let o=o(Po,A»O be sufficiently small so that o-l>RVa. We assume that r> 100- 5. For ZED, let Jz denote the event that there are more than two disjoint arcs in ')' joining the two circles aB(z,04r) and aB(z,05 r ), and let f[ denote the event that there are more than two such arcs in
')'T.
By the choice of a and 0, the
probability that dist(BTl aDU{vo}»40 4r and JS holds is at most ~pop. Consequently, the same bound applies for the probability that dist(zT, aDU{ vo}) >404r, zT=Br and Jz~ holds. Thus, as the events dist(zT, aDU{ vo}) >404r and Jz~ are h T , B)-measurable, T
~poP ~ P[ZT = BTl dist(zT, aDU{ vo})
> 40 4r, Jz~]
h T , B]] Br I ')'T, B] 1{dist(zT,8DU{vo}»48 r} l:Tl;J
= E[P[ZT = BTl dist(zT, aDU{vo}) > 40 4r, J'f; = E[P[ZT =
4
By (3.62) and our choice of E:, we therefore have P[dist(zT, aDU{ vo}) > 40 4r, J'f;] :::;; ~po. Let
1{
(3.63)
denote the event h\')'T)naB(zT' 05 r )#0. Now condition on ')'T and B such
that dist(zT, aD) >Or and -'Jz~ holds. Suppose also that dist(zT, bT ) >Pr. Then there are precisely two connected components of B(zT,04r)nDT that intersect aB(zT,05 r ). Let WI, w2EaB(ZT' 05 r )nDT be points in each of these two connected components. By constructing barriers as in the proof of (3.48), it is easy to see that if o=o(po, A) is sufficiently small, then
for j=1,2. Now note that if dist(wj,,),\,),T;Dh T ))>o4r for j=1,2, then Consequently,
1110
-,1{
holds.
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
101
We combine this with (3.63), and get
P[Ji, dist( vo, ZT) > 48 4 r, dist(zT, bT ) > 83 r, dist(ZT, aD) > 8r] ::( ~po.
(3.64)
Since r=dist(vo, aD), provided that we take 8=8(po, A) >0 sufficiently small, Lemma 3.24 gives P[dist( vo, ZT) ::(8r] < ~Po, Lemma 2.1 gives P[dist( vo, ZT) >8r, dist(zT, bT ) ::(82r] < ~Po and (3.53) gives P[dist(zT, aD) ::(8r, zT~WD] < ~Po, These last three estimates may be combined with (3.64), to yield P[Ji, ZT ~aD]::( ~po, which completes the proof. D We now prove the analog of (3.61) with ,T replacing ,. Let FT denote the function that is +A on V+(tT), -A on V_(tT), equal to hG on TG-vertices in aD, and is discreteharmonic at all other vertices in D. PROPOSITION
3.27. Assume (h), (D), (S) and (a). Then
in probability as dist (Vo, aD) --+ 00 while A is held fixed. Proof. Fix 10>0 and set r:=dist(vo, aD). We have, by the heights interface continuity (Proposition 3.3),
IE[h(ZT) It,zT]-E[h(ZT) It T ,ZT]1 <10 if dist(zT, '\'T »Ro, where Ro=Ro(c, A). Therefore, Lemma 3.26 with po=cjOX(l) >0 gives
E[IE[h(ZT) It,zT]-E[h(ZT) It T ,ZT]IJ <210
(3.65)
when r>s-l R o, and s is as given by the lemma.
(Note that E[h(ZT) It, zT]=E[h(ZT) itT, ZT] when zTEaD.) In the following, we will use a parameter 8>0. The notation 0(1) will be shorthand
for any quantity g satisfying limo-to limr-too Igl =0 while A is fixed. Let Z be a maximal set of TG-vertices in DnB(vo, 28- 1 r) such that the distance between any two such vertices is at least 82r and the distance between any such vertex to aD is at least 83 r. Then IZI=O(8- 6 ). By (3.61) (with each uEZ in place of vo), we therefore have P[there exists u E Z: E[h(u)-F(u) I,f > 10] = 0(1).
(3.66)
Let to be the first tEN such that dist(St, aD(tT)) ::(8r. By Lemma 2.1, (3.67) By (3.54), P[dist(zT, ,; D) < 8 1 / 3 r, ZT E aD] = 0(1), 1111
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O. SCHRAMM AND S. SHEFFIELD
while Lemma 3.26 gives
Thus, P[dist(zT, 'Y\'YT; D)
< 81/ 3r] = 0(1).
This and (3.67) imply that P [dist(Sta, 'Y\ 'YT; D) < ~81/3r ] = 0(1).
(3.68)
Since dist(Sta, aD('YT)) <8r, this and Lemma 2.1 imply that with probability 1-0(1) the L1 norm of the difference between the discrete-harmonic measure from Sta on aD('YT) and the discrete-harmonic measure from Sta on aD('Y) is 0(1). Because E[h(Sta) b, Sta] is the average of E[h(z) I'Y]' where Z is selected according to the discrete-harmonic measure on aD('Y) from Sta and similarly for 'Y T , we conclude from the above and (3.65) that
in probability. Now, Lemma 2.1 implies that P[dist(Sta,vo»8- 1r]=0(1). On the event dist(Sta' vo)::::;8- 1r, fix some zoEZ within distance 82 r from Sta (if there is more than one such Zo, let Zo be chosen uniformly at random among these given (h,'Y,S)). By the discrete Harnack principle (Lemma 2.2) and (3.68), we have
E[h(Sta) 1 'Y, Sta]-E[h(zo) 1 'Y, zo] = 0(1) in probability, which in conjunction with (3.66) yields E[h(Sta) b, Sta]-F(zo)=0(1). The discrete Harnack principle now implies that F(zo)-F(Sta)=o(1) in probability, and (3.68) gives F(Sta)-FT(Sta)=0(1) in probability. Consequently,
in probability. Since E[h(·) bTl and FT are discrete-harmonic in D('YT) , the proposition follows.
D
We can now prove the height gap theorem. THEOREM 3.28. Assume (h), (D), (S) and (a). As above, let T denote a stopping time for 'Y, let 'YT :='Y[O, T], let DT be the complement in D of the closed triangles meeting 'Y[O, T). Let hT denote the restriction of h to VnaD T and let Vo be some vertex in D. Then E[h(vo) I'YT,hT]-FT(vo)-+O
in probability as dist( Vo, aD) -+00 while A is held fixed, where FT is as in Proposition 3.27. 1112
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103
Proof. Set
X:=E[h(vo) ["?,hT]-E[h(vo) I"~?]. By Proposition 3.27, it suffices to show that X -+0 in probability as dist(vo,8D)-+oo. For vEVn8DT , let av denote the conditional probability that a simple random walk started at
Vo
first hits 8DT at v, given
X
L
=
"'IT.
Then
av(h(v)-E[h(v)
h T ]).
vEvn8 D r
Consequently,
v,u
Now Corollary 3.5 implies that it suffices to show that for every r>O,
v,u
in probability. This follows by Lemmas 2.1 and 3.24.
D
4. Recognizing the driving term In this section we use a technique introduced in [LSW4] and used again in [SS] in order to show that the driving term for the Loewner evolution given by the DGFF interface with boundary values -a and b converges to the driving term of SLE(4; a/ A-I, b/A-l) if a, bE [-Ao, X]. The reader unfamiliar with this method is advised to first learn the technique from [SS, §4] or [LSW4, §3.3]. The account in [SS] is closer to the present setup and somewhat simpler, but some parts of the argument there are referred back to [LSW4]. The present argument is more involved than those of the above mentioned papers, because we prove convergence to an instance ofSLE(x; (?1, t?2) rather than just plain SLE(4). The main added difficulty comes from the fact that the drift term in SLE(x; t?l, t?2) becomes unbounded as W t comes close to the force points. These difficulties disappear if a=b=A, in which case the convergence is to ordinary SLE and the argument giving the convergence of the driving term to scaled Brownian motion is easily established with minor adaptations of the established method. We therefore forego dwelling on this simpler case, and move on to the more general setting, assuming that the reader is already familiar with the fundamentals of the method. 1113
104
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SCHRAMM AND S. SHEFFIELD
4.1. About the definition of SLE(x; (11, tl2) Throughout this section, given a Loewner evolution defined by a continuous Wt, we let
Xt and Yt be defined as in §1.4 by
and we make use of the definition of SLE(x; th, e2) by means of the SDE (1.5). As we mentioned in §1.4, some subtlety is involved in extending the definition of SLE(x; el, e2) beyond times when W t hits the force points, and in starting the process from the natural initial values Xo = Wo =yo =0. This is closely related to the issues involved in defining the Bessel process, which we presently recall. The Bessel process Zt of dimension 8>0 and initial value x#O satisfies the SDE
8-1 dZt = 2Zt dt+dBt ,
x,
(4.1)
8-1 Zt =x+ Jo 2Zs ds+Bt-Bo,
(4.2)
Zo
=
which we also write in integral form as
t
up until the first time t for which Zt=O. When defining Zt for all times, this SDE is awkward to work with directly since the drift blows up whenever Zt gets close to zero (and some of the standard existence and uniqueness theorems for SDE solutions, as given, e.g., in [RY], do not apply in this situation). However, for every 8>0, the square of the Bessel process turns out to satisfy an SDE whose drift remains bounded and for which existence and uniqueness of solutions follow easily from standard theorems. For this reason, many authors construct the Bessel process by first defining the square
Z;
of the Bessel process via an SDE that it satisfies and then taking its square root [RY]. (Recall also that when 8";; 1 the Bessel process itself does not satisfy (4.2) at all without a principal value correction. Even when 1<8<2, which, as we will see below, is the case that corresponds to SLE(x; e) that hit the boundary and can be continued after hitting the boundary, the solution to (4.2) is not unique unless we restrict attention to non-negative solutions.) The formal definition for SLE(x; e) with one force point (i.e., el =e and e2=0) was given in [LSW3]. It was observed there that in this case, (1.5) implies that the process Wt-Xt satisfies the same SDE as the Bessel process of dimension 8=1+2(e+2)/x up until the first time t for which Wt=Xt. Thus, to define SLE(x; e), the paper [LSW3] starts with a constant multiple of a Bessel process Zt of the appropriate dimension and defines the evolution of the force point Xt by Xt=xo+ J~(2/Zs) ds and the driving term by Wt=Xt+Zt. 1114
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105
Defining SLE( x; [h, (22) is a slightly more delicate matter since neither W t - Xt nor Yt - W t is exactly a Bessel process (although each one is quite close to a Bessel process when the other force point is relatively far away). Although this is not a very difficult issue, it seems that there does not yet exist, in the literature, an adequate definition of SLE( x; (21, (22) that is valid beyond the time that the driving term hits a force point. Since we prove the convergence to SLE(x; (21, (22), we have to define it. The approach we adopt is basically similar to the way in which the Bessel process (and hence SLE(x; (2)) is usually defined: we pass to a coordinate system in which the corresponding SDE becomes tractable. We will describe the coordinate change we use in §4.2. Within this new coordinate system, we then prove the convergence of the Loewner driving parameters of our discrete processes to those of the corresponding SLE(x; (21, (22) in §4.3 and §4.4. Then §4.5 describes the reverse coordinate transformation and use it to give a formal definition of SLE(x; (21, (22), Definition 4.14. We remark that there are many equivalent ways to define SLE(x; (21, (22) (for example, one can probably show directly that (1.5) has a unique strong solution for which Xt~Wt~Yt for all t), but ours seems most efficient given that the coordinate change also simplifies the proofs in §4.3 and §4.4.
4.2. A coordinate change
In this subsection, we recall a different coordinate system for Loewner evolutions, which is virtually identical to the setup used in [LSW2, §3j. Suppose that ,: [0, (0) -+ iHi is a continuous simple path that starts at ,(0)=0, does not hit lR\ {O}, satisfies limt-+oo 1,(t)l=oo and is parameterized by half-plane capacity from 00. Let gt:IHI\,[O,tj-+IHI be the conformal map satisfying the hydrodynamic normalization at 00, let Wt=gt(r(t)) be the corresponding Loewner driving term. Loewner's theorem says that gt satisfies Loewner's chordal equation (1.3). Now we introduce a I-parameter family of maps C;: IHI\,[O, tj-+IHI satisfying the normalization for t>O,
C;(oo) = 00,
C;((O, (0)) = (1, (0)
and
C;(( -00,0)) = (-00, -1).
That is, (4.3) where Xt and Yt (as defined earlier) are the two images under gt of
1115
°
and Xt
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O. SCHRAMM AND S. SHEFFIELD
By differentiating (4.3) and using (1.3) and (1.4) it is immediate to verify that G; satisfies
dG;(z) dt
1-G;(z? 8 (Yt-Xt)2 (GHz)- Wt)(1- (Wt)2)'
We now define a new time parameter
It is easy to verify that s is continuous and monotone increasing and s( (0,00)) = (-00,00).
Set Gs=G; and Ws=Wt when s=s(t). Differentiation gives (4.4) Consequently, this change of time variable allows us to write the ODE satisfied by G as
dGs(z) ds
1-Gs (z? G s (z) - W s
(4.5) '
where all the terms come from the new coordinate system. Later, in §4.5, we explain how to go back to the standard chordal coordinate system.
4.3. The Loewner evolution of the DGFF interface In addition to our previous assumptions (h) and (D) about the domain D and the boundary conditions, we now add the assumption that (ab) there are constants a and b such that ha=b on 8+, ha=-a on 8_ and
min{a,b} > -Ao, where Ao>O is given by Lemma 3.9 with A:=max{lal, Ibl}. In this case, clearly (8) holds. In the following, a and b will be considered as constants, and the dependence of various constants on a and b will sometimes be suppressed (for example, when using the 0(·) notation). Let 4>: D--+lHI be a conformal map that corresponds 8+ with the positive real ray. Let "( be the zero-height interface of h joining the endpoints of 8+, and let "(
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
107
Ws=Gs("y(s)) and have the differential equation (4.5). Our goal now is to determine the limit of the law of W as rad-l(i) (D) -+00. Set for xE [-1,1],
q1(X) :=2(I-x 2)
and
>'+a >'+b q2(X) :=-"""""i\(x-l)-~(x+l).
(4.6)
We extend the definitions of q1 and q2 to all of lR by taking each qj to be constant in each of the two intervals
(-00,
-1] and [1, (0). Consider the SDE
(4.7) where B is a standard I-dimensional Brownian motion. A weak solution is known to exist (see [KS, §5.4.D]). We also recall that the weak solution is strong and pathwise unique (see [RY, §IX, Theorems 1.7 and 3.5]).
4.1. There is a time-stationary solution Y: (-00, (0)-+[-1, 1] of (4.7). Moreover, for every finite 8> 1 and c: > 0, there is an Ro = Ro (8, c:) such that if R: = rad-l(i)(D»Ro and the assumptions (h), (D) and (ab) hold, then there is a coupling of Y s with h such that THEOREM
P[sup{lYs- Wsl: s E [-8, 8]}
> c:] < c:.
The following proposition is key in the proof of the theorem. In essence, it states that
W s satisfies a discrete version of (4.7). Let Fs be the (T - field generated by (W r : r ~ s). (Note that, although the filtration defining W s is discrete, there is no problem in considering Fs for arbitrary s, though the behavior of W r for r in some neighborhood of s might be determined by Fs.) PROPOSITION
4.2. Assume (h), (D) and (ab). Fix some 8>1 large and some 8, T]>O
small. There is a constant C>O, depending only on a, band 8, and there is a function Ro=Ro(8, 8, T]), depending only on a, b, 8, 8 and T], such that the following holds. If R:= rad-l(i) (D) > Ro and
So
and
Sl
are two stopping times for W s such that almost surely
-8~SO~Sl~8, ,6,s:=sl-so~82 and SUPSE[SO,SllIWs-Wsol~8, then the following two
estimates hold with probability at least 1- T]:
where ,6,W:=W S1
-
IE[,6,W -Q2(W so ),6,s IFso] I ~ C8 3 ,
(4.8)
IE[(,6,W)2 -Q1 (W so),6,s IFso] I ~ C8 3 ,
(4.9)
Wso'
To prepare for the proof of the proposition, we need the following easy lemma. The first two statements in this lemma should be rather obvious to anyone with a solid background on conformal mappings. 1117
108
o. LEMMA
SCHRAMM AND S. SHEFFIELD
4.3. Set Cl=Cl(S)=100e s . There are finite constants C2=C2(S»0 and
Ro=Ro(S»O, depending only on S, such that if R:=rad
Consequently, statement (2) holds once Ro>4/c~ rad(B). This takes care of (2), because l/rad(B) is bounded by a function of Cl. lt is easy to check that (3) follows from (4). lt remains to prove the latter. Let Xt and Yt be as in §4.2. Note that Xt<Wt
Ot(Yt-Xt) =
2(Yt-Xt) ;?: _8_. (Yt-Wt)(Wt-Xt) Yt-Xt
Therefore, Ot((Yt-Xt?);?:16, which gives
e2s (t) ;?: 16t. Observe, by (1.3), that
Ot((Imgt(z))2);?: -4. Thus, (Imgt(z)?;?:(Imz)2-4t;?:cr-4t. Another appeal to (1.3) now gives
Igt(z)-zl ~
2t
JCF4t.
1118
cr- 4t
(4.10)
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
109
By (4.3) and the definition of s(t), the above gives
Now, the first summand on the right-hand side is at most 1, because Yt>O>Xt. The second summand is also at most 1 in the range SE[-S,S], by (4.10) and the choice of Cl. This completes the proof of the lemma. D
Proof of Proposition 4.2. With the notation of Lemma 4.3, let Zj:=2Cli+~Clj for j =0,1, and let Vj be a TG-vertex satisfying condition (2) of the lemma with Zj in place of z. Then zj :=¢(Vj) satisfies in turn the assumptions required for z in the lemma. For k=O, 1, note that there is a stopping time Tk for,,( such that ¢o"((Tk)="((Sk)' Fix jE{O, 1} and set Clearly, (4.11) Recall the definition of the function FT from Proposition 3.27. Let Ak be the event
IXk -
FTk (Vj) I ~85. By that proposition and the fact that zj satisfies condition (1) of Lemma 4.3, if R is chosen sufficiently large then P[Akl<~1]85. Since FTk (vj)=Oa,b(l), and likewise X k =Oa,b(l) by (3.2), we get
Let
A be the event that P[AIIFsol >85 •
Then P[Al <
h
(since we are assuming prAll <
h85) and we have
Thus we have, from (4.11), (4.12) Let Hk be the bounded function that is harmonic (not discrete-harmonic) in D\ "([0, Tk], has boundary values b on 0+, -a on 0_, +,\ on the right-hand side of "([0, T k ], and -,\ on the left-hand side of "([0, Tkl. We claim that the difference Hk(vj)-FTk(Vj) is small if dist( Vj, oDU"([O, Tk]) is large. Indeed, this easily follows by coupling the simple random walk on TG to stay with high probability relatively close to a Brownian motion and using (3.54) and Lemmas 3.23 and 2.1 to show that with high probability the boundary value sampled by the hitting point of the Brownian motion is the same as that sampled 1119
110
O. SCHRAMM AND S. SHEFFIELD
by the hitting vertex of the simple random walk. Now, Lemma 3.24 guarantees that if R is sufficiently large, then with high probability dist( Vj, 8DU')'[0, T k ]) is large as well.
Consequently, if R is chosen sufficiently large, we have P[[Hk (Vj) - FTk (Vj) [?:05l < :h05. Let Bk be the event [Hk (vj)-FTk (Vj)[?:05, let B be the event P[B1 [Fso l?:05 and let A:=AouAUBouB. Note that P[Al <1]. The above proof of (4.12) from (4.11) now gives
(4.13) Now, the point is that H k ( V j) can easily be expressed analytically in terms of Zk=Zk,j:=GSk(zj)=GSk(CP(Vj)) and W Sk ' Indeed, conformal invariance implies that the harmonic measure of 8+ in D\,),[O, Tkl from Vj is the same as the harmonic measure of [1,(0) from Zk, which is 1-arg(Zk-1)j7f, because Gsocp corresponds 8+ with [1,(0).
Likewise, the harmonic measure of the right-hand side of ,),[0, Tkl is arg(Zk- 1) -arg(Zk - W Sk ) 7f
Similar expressions hold for the harmonic measure of the left-hand side of ,),[0, Tkl and of 8_. These give
(4.14) Recall that Zk=Gsk(zj). By (4.5), we have in the interval sE[SO,Sl], Gs(zj)=Zo+
i
s
So
1-G (Zl? ,r J dr. Gr(Zj)-W r
(4.15)
Note that conditions (3) and (4) of Lemma 4.3 imply that the integrand is 0 8 (1). Therefore G s (zj)-ZO=08(..6.S)=08(02) for SE[SO,Sll. Moreover, we have [Ws-Wso[::;;o in that range. Thus, it follows from (4.15) and condition (3) of the lemma that Zl -Zo =..6.s (
1-Z6 Zo- Wso
+0 8 (0)
)
=..6.s
1- Z 6 Zo- Wso
3 +08 (0 ).
(4.16)
We will now write an expression for H1(vj)-Ho(vj) and then use (4.13) to complete the proof. Let us first look at the term arg(Zk-WSk) on the right-hand side of (4.14), and see how it changes from k=O to k=l. For this purpose, we expand log(Z - W) in Taylor series up to first order in Z - Zo and up to second order in W - W So (since Zl-ZO=08(02) while W S1 -Wso=O(o)), as follows:
arg(Zl - WS1)-arg(Zo- W so ) = Im(log(Zl - WS1)-log(Zo- W so )) =Im( Zl-ZO _Ws 1 -WS O _(Ws 1 -Ws O ):)+08(03). Zo-Wso Zo-Wso 2(Zo-Wso) 1120
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
111
Similar (but simpler) expansions apply to the other arguments in (4.14). We use these expansions as well as (4.14) and (4.16), to write 7r ( HI (V j) -
H
°(v
j )) =
~s(1-Z6)
(oA - b) 1m _------=:-'-:c-------"-'-----_ (Zo - W so)(Zo -1) ~s(1-Z6) -~W(Zo- W so ) - ~(~W)2
-2oAlm--------------~------~-----
(Zo- W so )2
~s(1-Z6)
+(oA-a) 1m
+Os(83 ).
(Zo - W so )(Zo+l) With the abbreviations x:=Re(Zo - W so) and y:=lm Zo, the above simplifies to
We know from (4.13) that on -,.,4 the conditioned expectation given Fso of the left-hand side is Oa,b(8 5 ). Since (x 2+y2)/y=Os(1) (by statement (4) of Lemma 4.3), we have on -,.,4, E [X 2!y2
(~Sql(Wso)-(~W)2) +~Sq2(Wso)-~W IFs o] =
Oa,b,s(8 3 ).
(4.17)
Now, this is valid for zj, with j=O, 1. The choice of j only affects the left-hand side in the term x/(X 2+y2). By the choice of the points zj and by statement (4) of Lemma 4.3, the factor x/(x2+y2)=Re((Zo-Wso)-1) differs between the two zj by an amount that is bounded away from zero by a constant depending on S. Subtracting the above relation (4.17) for
zb
from that of
zi,
we therefore get (4.9) on -,.,4. When this
is used in conjunction with (4.17) again, one obtains (4.8) on -,.,4. This concludes the
D
proof of the proposition.
4.4. Approximate diffusions In this subsection we embark on the general study of random processes satisfying the conclusions of Proposition 4.2 and show that the proposition essentially characterizes the macroscopic behavior of the process. As one of the referees of this paper pointed out, one can try to do this more "traditionally" by proving tightness of the driving term and characterizing the subsequential fine mesh limit using the appropriate martingale problem. However, our approach is somewhat different (though not necessarily better). 1121
112
o.
SCHRAMM AND S. SHEFFIELD
Motivated by the proposition, we say that a continuous random W: [0, 8]-+ [-1, 1] is a (C,6,TJ)-approximate (q1,q2)-diffusion if it satisfies the conclusion of the proposition; namely, for every pair of stopping times 80 and 81 such that almost surely 0:( 80:( 81:( 8, .6.8:=81-80:(62 and sUPsE[sa,sllIWs -Wsa l:(6 we have with probability at least 1-TJ
that (4.8) and (4.9) hold with W in place of W.
4.4. Suppose that a, b~-)" and that Y s satisfies (4.7), where q1 and q2 are given by (4.6). Suppose that Yo E [-1, 1] almost surely. Then there is a C > 0 such that Y s is a (C, 6, 0) -approximate (q1, q2) -diffu8ion for every 6 E (0,1) and in every time interval LEMMA
[0,8]. Proof. First, note that q1(X)=0 and xQ2(X):(0 for Ixl~l. This clearly implies that {Ys:8~0}c[-1,
1] almost surely.
Now fix some 6>0 and two stopping times 80:(81
satisfying the assumptions in the definition of approximate diffusions. Let Fsa denote the a-field generated by (Ys:8:(SO). Then
The second summand is a martingale, and therefore
Since IYs - Y sa 1:(6 for 8E [80, 81] and Q2 is a Lipschitz function, we conclude that
Thus, Y s satisfies (4.8). We now use Ito's formula to calculate (YS1 - Ysa?:
The left summand is 0(63 ) and the middle summand is a martingale and therefore its expectation given Fsa is zero. Thus
because Q1 is Lipschitz. This shows that Y s satisfies (4.9), and completes the proof. 1122
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CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
PROPOSITION
113
4.5. Fix 8>2. Let ql, q2: [-1, 1]-+1R be defined as in (4.6), where
8uppose that WI: [0,8]-+[-1,1] is a (C,O,TJ)-approximate (ql, q2)-dijJusion and that W 2: [0,8]-+[-1,1] is a solution of (4.7) with the same ql and q2 and Wl(0)=W2(0) almost surely. Also assume that TJ<05/8 2 . Then there is
we assume that a,b>-A..
a coupling of WI and W 2 such that sUPsE[0,s_ljIWs1 -W.?1-+0 in probability as 0-+0, while C is fixed. Namely, for every c:>0 there is a 00>0, depending only on a, b, 8, C and c: such that sUPsE[o,s_llIWsI-W;I~c: with probability at least 1-c: if 0<00.
A useful tool in the proof of the proposition is the following lemma. LEMMA
4.6. Let W: [0,8]-+[-1,1] be a (C, 0, TJ)-approximate (ql, q2)-dijJusion, and
let TO and Tl be two stopping times for W satisfying 0 ~ TO ~ Tl ~ 8. Assume that Co < ~ . Let f: [-1, 1]-+ IR be a function whose second derivative is Lipschitz with Lipschitz constant 1 and which satisfies II f' II 00, II f" II 00 ~ 1. 8 et
Then there is a stopping time T{ satisfying TO~T{~Tl almost surely and P[T{#Tl]~TJ such that
Moreover, in the above the function f may be random, provided that it is Fro -measurable. Proof. We inductively define the stopping times Sj as follows.
Set So :=TO, and
Sj+l :=min{ S~Sj :S=Sj+02 or IWs - W Sj 1=0 or Sj=TI}. If there is a j EN such that W
sd, then let n be the minimal such j. (Note that the event that W does not satisfy (4.8) or (4.9) for (Sj, Sj+l) is Fsj-measurable.) Otherwise, let n be the minimal j such that Sj=Tl. Note that (sn' Sn+1) is a pair of stopping times and they do not satisfy both (4.8) and (4.9)
does not satisfy (4.8) or (4.9) for the stopping times (Sj, Sj+l) in place of (so,
unless Sn=Tl. Consequently,
(4.18) Since IWsHl - W Sj I~O using a Taylor series for f around W Sj we have
where fj.j W:= W SHl - WSj.
We may use (4.8) and (4.9) to estimate the conditioned
expectation of fj.j Wand (fj.j w)2 given FSj and get for j
114
O. SCHRAMM AND S. SHEFFIELD
By our assumptions about
f,
this may also be written as
We sum this over j from 0 to n-1, then take expectations conditioned on F TO ' to obtain
Now fix some JEN. On the event j+1
E[((~jW)2+Sj+l -sj)1{j
By (4.9), this gives
We take expectation conditioned on
.r
TO
and use the fact that ql is bounded, to obtain
We sum this over all j EN, to get
By our assumption that C8 < ~, this implies that
When combined with (4.19), this gives
By (4.18), this completes the proof with
Ti =Sn'
D
The next lemma bounds the expected time that Ws spends close to ±1. LEMMA
4.7. Let W be a (C, 8, ry)-approximate (ql, q2)-dijJusion W: [0,8]--+[-1,1]'
where ql and q2 are given by (4.6), C8 < ~ and b> - A. Suppose that 8~ 1. Given any 10>0 there is some xo<1, 8'>0 and ry'>O all depending only on 10, a, band 8 such that if 8<8' and ry
[1
S
1{W8>XO}
dS]
< E.
A similar statement holds for the set of times such that Ws is near -1, provided that
a>-A. 1124
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
115
Proof. Set JL(A):=E[Jos l{wsEA} ds]. Note that q2(1)<0. Fix some YoE[O, 1) such
that q2(x)~~q2(1) throughout [Yo, 1] and set Yn:=I-(I-Yo)2- n . Let f(x) be the twice continuously differentiable function that is zero on [-1, Yo] and satisfies
1" (x) = { min {x - Yo, Y1 - x},
on [Yo, Y1], on [Y1, 1].
0,
We apply Lemma 4.6 to f with TO=O and T1 =8. Clearly Lf(x)=O in [-1, yo]. On the interval [Yo, Y1], we have
1" (x) ~ ~ (Y1 - yo), l' (x) ~ 0 and q2 (x) < O.
Consequently,
On the interval [Y1, 1], we have f"(x)=O, 1'(X)=~(Y1 -Yo? and q2(x)~~q2(1)<0. Consequently, Lf(x)~-c(l-yo?, where c>O depends only on q2(1). Also note that
If(Ws)- f(Wo)1 ~ sup f(x) -inf f(x) x
x
=
f(l) -0 ~ (I-Yo)3.
Therefore, Lemma 4.6 gives
(I-Yo)2 JL([Yo, Y1)) -c(I-Yo)2 JL([Y1, 1]) ~ -(I-Yo)3 +O( C+ I)J8+0(8)TI. We assume that 15' and TI' are sufficiently small so that the right-hand side is larger than -2(I-Yo)3. Then we get
This implies that
JL([Y1, 1]) ~ 2(I-yo) + JL(~:'cl]) . A similar inequality applies to Yn and Yn+1. Induction therefore gives
lI.[y 1] ~ JL([Yo, 1]) + ~ 2(I-y.) < JL([Yo, 1]) +4(I-y ) ,... n, '" (l+c)n J (l+c)n 0 ,
f::o
provided that 15' and TI' are smaller than some functions of n, 8, C and Yo. Consequently, we first choose Yo such that in addition to the requirements stated in the beginning of the proof, 4(I-yo)~~c. Then we take n sufficiently large so that (l-c)-n8<~c. Then 15' and TI' are determined. This proves the first claim. The second one follows by symmetry. D The next lemma estimates the conditional expectation and conditional second moment of the time it takes Ws to move a distance of 15 beyond its location at a stopping time. 1125
116
O. SCHRAMM AND S. SHEFFIELD
LEMMA
4.8. Let W be a (C, 15, 7])-approximate (q1, q2)-diffusion W: [0,8]--+[-1,1]'
where q1 and q2 are given by (4.6) and 8>1. Let XoE(O, 1). There is a function 150 >0, depending only on Xo, C, a and b such that the following holds ifr5
A
denote the event {To<8- ~ }n{[WTo [<xo}. Then
(4.20) Moreover,
(4.21) Proof. Let f(x)=*(x-WTo )2, and let L be as in Lemma 4.6. Then
for xE [WTo -15, W TO +15]. Therefore, Lemma 4.6 gives
That is,
By choosing 150 sufficiently small, we make sure that O(C+1)r5<~q1(WTo) on A. Since [f(WTi)[~*r52, the above gives
We plug this and
into (4.22), simplify, and get
on A. Now (4.20) follows by dividing (4.23) by q1(WTo )+O(C+1)r5, taking expectation and recalling that P[T{ #T1] ~7]. 1126
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
117
Now define t n :=T1/\(To+16n8 2jq1(WTo )). If 80 is sufficiently small, then Lf(x» ~Lf(WTO)=~q1(WTO) throughout [WTo-8, W To +8l on the event A. Thus, we get by applying Lemma 4.6 to the stopping times tn and t n+1,
where
t~+1
is the stopping time provided by the lemma. Again, on A we may assume
that O(C+1)8<~q1(WTo)/\~. Thus,
which implies that P
[t~+1
=
tn +
q~~~:) I Ftn] ~ ~.
But if t~+1 #tn +16n8 2jq1(WTO )' then t~+1 =T1 or t~+1 #tn+1. If Bn denotes the event that tj=tj for all j=l, ... ,n, then induction gives
Lemma 4.6 gives P[t~+1#tn+1l~7] and therefore P[--,Bnl~n7]. (In fact, it is not hard to get the better estimate P [-,Bnl ~ 7].) Consequently, for n EN,
The above applies with n:=n/\ i-log2 7]l in place of n, and hence
We multiply both sides by 2(n+1)(168 2jq1(WTo ))2, and sum over n from n=O to the least m such that 16m82jq1(WTo)~S. The result on the left-hand side bounds
Consequently, the required bound (4.21) follows by taking expectations and using our assumed upper bound for
D
7].
The following lemma shows that when we discretize the approximate diffusion the resulting random walk has transition probabilities that can be well estimated from q1 and q2 away from the boundary. 1127
118
o.
SCHRAMM AND S. SHEFFIELD
LEMMA 4.9. Fix some xoE (0,1). Let W: [0,8]-+[-1,1] be a (C, 8, 7])-approximate (ql,q2)-dijJusion, where ql and q2 are given by (4.6), 8:;?:1 and 7]~85/8. Set Z:={k8:kEZ and 1k8I<xo}, so:=inf{s:;?:O:WsEZ or s=8} and inductively Sn+l :=inf{s:;?: Sn: WSn -=J Ws E Z or s=8}. Also set Xn:=WSn and ZO:=Z\{minZ,maxZ}. Let
and ±.=
rn·
~±8 q2(Xn ) 2
2ql(Xn )"
There is a 80>0, depending only on C, xo, a and b, such that if 8<80, then for all nEW,
Proof. We now use a different test function:
where a:=-Q2(Xn ){3/Ql(Xn ) and {3:=IQ1(Xn)/6Q2(Xn)I!\~ with {3=~ if Q2(Xn ) =0. The choice of a and (3 above is tailored to give Lf(Xn)=O and 14al+I{3I~1. The latter implies that 111"1100' 11f'1100~1 when 1 is restricted to the interval [-1,1]. We now apply Lemma 4.6 again with this 1 and stopping times Sn and Sn+1. Note that L1=0(8) in the interval [Xn -8,Xn +8]. Hence Lf(Ws )=0(8) for sE[Sn,Sn+l]. Lemma 4.6 gives
on the event XnEZo, where S~+l is the stopping time produced by the lemma. The above may be written
({38+a82)p~ +( -{38+(82)p~ = -E[1{Ws~+1IlZ}1(Ws~+J Fsn] 1
+0(C+1)8EW+s~+1 -Sn FsJ. 1
Set p~:=P[WS~+l ~{Xn -8, Xn+8} IFsJ. Then 1:;?:p~ +p;;, :;?:1-p~. Hence, the above gives
1128
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
which, by the definitions of 0: and
r~
119
may be rewritten
(4.24) By (4.20) and our assumption TJ~85/S, we have
(4.25) provided that 80 is sufficiently small. Since W s'n+l t/'-Z only when S~+l ~Sn+l or S~+l = Sn+l=S, on the event {XnEZO}n{Sn<S-!},
E[p~ JFsnl ~ P[S~+l ~ Sn+1J Fsnl+P [S~+l ;?: Sn
+! I Fsnl
~ P[S~+l ~ Sn+1J Fsnl+2E[sn+1-sn JFsnl·
(4.26)
Note that jJ-l=Oxo(1) and P[S~+l ~Sn+1]~TJ. Hence, we now obtain the result for p~ by taking expectation on the event {XnEZO}n{Sn<S-!} in (4.24) and using (4.25) and (4.26). A symmetric argument applies to
p:;; and r:;;, and the proof is complete. D
Next, we show that the time parameterization of W can be well approximated by a function of the discretized walk trajectory. LEMMA
4.10. Assume the setting and notation of Lemma 4.9 in addition to 8<80 .
For nEN let tn denote the time spent up to time Sn in segments [Sj, Sj+1] such that Xj=WsjEZO; that is,
Also let
n-l tn:= L l{XjEZ O }(Sj+l -Sj). j=O
82
n-l
(J"n:= L1{XoEZ o}-(-). j=o J ql Xj Let No:=min{nEN:sn;?:S-!}. Then for all nEN,
(4.27) Proof. Let Vj :=(Sj+1 -sj)l{xjEzo}l{j
82
Wj:= - ( - ) l{xEzo}l{j
Now, Mn:=L;~~(Vj-Uj) is clearly a martingale. Consequently, Doob's maximal inequality for £2 martingales [RY, 11.1.6] gives n-l E[max{JMjJ:j = 1, ... ,n}]2 ~ O(l)E[M~] =0(1) L E[(Vj-Uj)2]. j=O 1129
120
o.
SCHRAMM AND S. SHEFFIELD
Since Uj=E[vjIFsj ]' we have E[(Vj-Uj)2]::;;E[vJ]. By Lemma 4.8, E[vJ] is bounded by the right-hand side of (4.21). Now, the right-hand side of (4.20) bounds E[luj-wjlJ. The result follows by our assumption 'f/::;;8 5jS2, since for every m::;;nI\No, m- I I m-I lam-tml = I ~(Vj-Wj) ::;; IMml+ ~ IUj-wjl n-I ::;;max{IMjl :j= 1, ... ,n}+ L IUj-Wjl· D j=O Proof of Proposition 4.5. By Lemma 4.4, W 2 is a (C', 8, O)-approximate (ql, q2)-
diffusion for some fixed constant C'>O and every 8>0. We may assume, with no loss of generality, that C~C'. Let 10>0. Let X~E(I-E, 1) satisfy Lemma 4.7 with this given 10, and assume that 8 is sufficiently small so that that lemma is valid. Take xo=~(I+x~). Let Z and ZO be as in Lemma 4.9, let sj be the corresponding stopping times introduced there for Wk and let pL denote the random transition probabilities for Wk defined there. Also abbreviate Xj:= W~. Let F: denote the filtration of W k , k=I,2. Let Y/=1 if Xl+ 1 -Xl=8, Y/=-1 if Xl+1-Xl=-8 and Y/=O if IX1+1-X1I=l8. Then P[Y.k=±IIF\]=pt· and P[yJk =OIF\]=I-Pk ·-Pi:, . if XkEZo. J Sj,J Sj ,J,J J
For the coupling of WI and W 2 we use an independent identically distributed sequence Uj of uniform random variables in [0,1]. The coupling proceeds as follows. Up to their corresponding stopping times s~, k=l, 2, let them run independently. Inductively, we suppose that the coupling has been constructed up to their corresponding stopping times sj, k=I,2. For each k=I,2, we take
I, { Y/= -1, 0,
if Uj ::;;pL, if Uj ~ I-pi:,j' ifUjE(Pk,J.,I-pi:,].).
(In other words, we try to match up X]+I -X] with X]+1-X] as much as possible.) These choices respect the correct conditional distributions for these variables. Now we sample the restriction of WI to [S], S]+1] and the restriction of W2 to [sJ' sJ+1] independently of their corresponding conditional distribution given (F;j' ~I) and (F;j' ~2), respectively. This completes the description of the coupling. Let N:=min{n:s~vs;~S-n. Let Aj be the event {X],X]EZO} and set n
Qj:=IAjIY~l-Yll
and
Qn:=LQjl{j
Note that on .Aj we have IX]+I-X]+1I::;;IX]-X]1 unless S]+IVSJ+I=S. Moreover, IXJ-X61::;;8 and when Y/=O we have Sj+I=S. Consequently, I
2
~
IXn -Xnll{n
(4.28)
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
121
We now proceed to estimate Bj :=E[Q j 1{j
Let r~,j be the rj in Lemma 4.9 corresponding to the process Wk. Then
Ipi --p~ -I::S; Ipi --ri -1+lri --r~ -1+lr~ --p~ -I· ,]
,]
,]
,]
,]
,]
,]
,]
By that lemma,
Using the expression given for r~,j' we deduce that
Thus, we get
In conjunction with (4.28), this gives
Induction therefore implies
Taking note of (4.28), we infer that (4.29) Now let mo:=i4S8- 2 max{Q1(x):lxl::s;xo}l Observe that at least one of every two consecutive JEN satisfies X;EZo or sj=S. Consequently, mo
Set X*:=maXj
> 81 / 2 or N > m o] '" ~ 0 xo,C,B (8 1 / 2 ). 1131
(4.30)
122
O. SCHRAMM AND S. SHEFFIELD
Now for each s~S let J(s):=min{jEN:sJ~s}, Then sup
sE[O,S-l]
IW;-W.?I ~
sup
sE[O,S-l]
+
IW;-X](s)1
sup
sE[O,S-l]
IX](s)-XJ(s)l+
sup
sE[O,S-l]
IW;-XJ(s)l.
(4.31)
First, it is clear that sup
sE[O,S-l]
IW;-X](s)1 ~E+8.
(The left-hand side is usually at most 8 but can be as large as 1-xo+8 if, for example, We leave aside, for now, the estimation of the second summand in (4.31) and consider the last. Set XJ(s)-l =maxZ=xo.)
t *·.Since
sup
sE[O,S-l]
Is-sJ(s)' 2 I
2 XJ(s) = WSJ(s) ,we have 2
sup
sE[O,S-l]
IW;-XJ(s)l~
sup
sup
sE[O,S-l] tE[O,t*]
IW;-W;+tl.
(4.32)
Observe from (4.27) that for k=1, 2, . max { 10'] -
J~N/\mo
t] I} --+ 0
tJ
in probability as 8--+0, where is the tj of Lemma 4.10 corresponding to Wk. Now the choice of xo (via Lemma 4.7) implies that
Lemmas 4.7 and 4.8 imply that E[s~l ~EVOxo,c(82). Consequently, we have (4.33) in probability as
E,
8--+0. Now,
1132
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
123
The right-hand side is monotone non-decreasing in n. When n~N I\mo the first sum is at most Oxo(0"2)moX*. It is easy to see that for n=N I\mo the iterated sum on the right tends to 0 in probability: this follows from the proof of (4.33), because if we replace
Xo by xo+O", the terms appearing in this iterated sum are included in an. We now get, from (4.30), maxla}-aJI-+O J::;;N
in probability as E,O"-+O.
By (4.33), this also gives max Is} -S] 1-+ 0 J::;;N
in probability as
E, 0" -+ O.
In particular, (4.30) and (4.33) imply SUPj::;;N-l(SJ+l -sJ)-+O in probability, because aj+l-aj~Oxo(0"2). One consequence is that P[J(S-l)~N]-+O. Furthermore, we now have
t* ~
sup Is-s}(s)l+ sup Is}(s) -s}(s)I-+O sE[O,S-l] sE[O,S-l]
in probability. Now (4.32) implies that SUPSE[O,S-l] IW.? - X](s) 1-+0 in probability, because the right-hand side in (4.32) is smaller than (t*)1/3 with probability going to 1 as t*-+O, since W2 is a solution of (4.7). This takes care of the last summand on the right-hand side of (4.31). The middle summand on the right-hand side of (4.31) also tends to 0 in probability because, as we have seen, P[J(S-l)
be two independent solutions of (4.7) (with respect to two independent Brownian motions) which start at Xl and X2, respectively. We claim that s=:=min{s:~1=Ys2}<00 almost surely. The argument is quite standard. Suppose without loss of generality that
X2 > Xl. By Lemma 4.7, it is unlikely that Ys2 stays very close to 1 for a long time and unlikely that ysl stays very close to -1 for a long time. It is therefore easy to conclude from Lemmas 4.8 and 4.9 that there are constants so, Co >0 (which do not depend on Xl or X2) such that P[Ys~ <0] >co and P[Ys~ >0] >co. This implies that P[s= <so] >c6. By strong uniqueness of solutions of (4.7), it follows that the solutions are Markov and have stationary transition probabilities. Consequently, we get by induction P[s= >nso] < (1-c6)n for all nEN, which proves that s=
124
O. SCHRAMM AND S. SHEFFIELD
We now argue that P[s=<00]=1 and the uniqueness in law of solutions of (4.7) implies that for every Borel subset AC[-1, 1] the limit JL(A):= lim P[Y/ E A] r--+oo
exists. We may couple a solution started at some time so
17: [0, (0) --+ [-1, 1] of (4.7) such that the distribution of Yo is given by JL is time-stationary. To get a timestationary solution Y: (-00,00)--+ [-1,1]' we may take the weak limit oftime-translations ofY. Now let 8' be much larger than 8. By Proposition 4.2 with 8' instead of 8 and Proposition 4.5 translated to start at time -8' and an appropriate choice of the 8 appearing there, we may couple W s so that with probability close to 1 it stays close to a solution W; of (4.7) starting at W -8' throughout [-8',8']. We may at the same time couple W; so that with high probability it agrees with Y s inside the interval [-8,8]. Then with high probability Ws stays close to Y s in [-8,8], which concludes the proof of the theorem. D
Remark 4.11. At this point, it may be worthwhile to point out which properties of the functions ql and q2 played a part in the proof. The only properties that are essential for the above proof are that ql and q2 are both Lipschitz continuous in [-1, 1], that ql > 0 in (-1,1) and ql =0 on {-1, 1}, and that q2(1) <0
4.5. Back to chordal
In §4.2, we described the transition from the chordal Loewner system to the setup with the points ±1 fixed. We now describe the reverse transformation. We start with some continuous Y: (-00, (0)--+[-1,1]. Set (4.34)
1134
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
125
Also define
11
t*(s):=8
s
e2u (1-Y,:)du,
s*(t) :=sup{sE (-00,00) :t*(s)
~t}.
-(X)
Now set Yt:=ws'(t) for t>O, Yo:=O, and observe that Y is continuous provided that there is no non-trivial time interval in which YE{±l}. LEMMA
4.12. If Ys=Ws is defined from W t as in §4.2, then yt=Wt .
Proof. By the definition of s(t) in §4.2, we have eS=Yt-Xt. Consequently, (4.4) implies that Ot(t*(s(t)))=1. Since s(O)=-oo and t*( -(0)=0, it follows that t*(s(t))=t for all t~O and s*(t*(s))=s for all SE(-OO, 00). Next, (1.4) and the definition of W* give
Since Yo=O=xo and Ws(t)=wt, this implies that
where the second equality follows by a change of variable. Now yt=ws(t)=Wt follows from the definition of wt. The proof of the lemma is therefore complete. D We now discuss the behavior of solutions of (4.7) in the chordal coordinate system, but generalize to the case x#4. LEMMA
4.13. Let a,b>O, let Y: (-00,00)---+[-1, 1] be a solution of
and let Yt be the corresponding process, as described following (4.34). Then on any time interval which avoids {t:Ys.(t)~{±l}}, the process yt satisfies the SDE of the driving term for SLE(x;2(a-1),2(b-1)):
_ 2(a-1) dv Lt-~ yt-Xt for some Brownian motion
Bt , where
2(b-1) yt-Yt
+~
r::dB~
+v x
t
Xt and Yt satisfy (1.4) with 1135
Yt
in place of Wt·
126
O. SCHRAMM AND S. SHEFFIELD
Proof. Let (3s= J~(X) eUYu du, y;:=~((3s+eS) and x;:=~((3s-eS). Ito's formula and the definition of t* give
Now set
Ht =
j
s*(t)
J(t*)'(u) dB u ·
-(X)
Then Ht is clearly a continuous martingale. Since also (H}t= J~:;'t)(t*)'(u) du=t, we find that B is a Brownian motion with respect to t. The above formula for dw gives
~ = dyt
b-l) dt,
v'x dB~t +2 (ii-l -~-- + -~-yt-Xt
yt-Yt
2 ~
Xt-yt
,
and similarly for Yt. This concludes the proof.
D
As mentioned at the beginning of this section (§4), existence and uniqueness of solutions to the usual SDE defining SLE(x; el, (2) have not been proved beyond times when the driving term W t meets the force points. We now offer the following. ~
~
Definition 4.14. If ii, b>O, then the Loewner equation driven by the process Y of
Lemma 4.13 is called SLE(x; 2(ii-l), 2(b-l)) (starting from (0,0_,0+)).
4.6. Loewner driving term convergence
In this section, we complete the proof of Theorem 1.3. The theorem will follow quite easily from Theorem 4.1. Proof of Theorem 1.3. Fix T,c,cQ,c'>O. Let Ys and Ws be coupled as in Theorem 4.1, but with c' in place of c. Let t* (s) and s* (t) be defined from Ys as in the
Yt,
beginning of §4.5. Since the interior of the set of times for which Y s E { -1,1} is empty, 1136
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
127
it follows that there is some positive to >0 such that with probability at least I-Eo we have t*(l)-t*(O»to. Because Y s is stationary, it follows that
P[t*(s+l)-t*(s) > e2Stol ~ I-Eo. In particular, there is some So>O such that P[t*(So»T+1l~1-Eo. Fix So satisfying this and additionally e- so < bEo, Let
11
i(s):=8
s
e2u(1-W~)du,
-<Xl
which is the equivalent of t*(s) with W replacing Y. It is clear that if E'=E'(So,EO) is sufficiently small and (4.35) then for every sE[-oo,Sol the right-hand side in (4.34) differs from the corresponding quantity where W replaces Y by at most EO. Lemma 4.12 then gives
SUP{IWi(s) - Wt*(s) I: s:::;; [0, So]} < EO, where Wt is the chordal driving term for SLE(4; aj>'-l, bj>'-l) and driving term for
W is the chordal
CPD o ",(. (Here, we also use the fact that s*(t*(s)) =s.) If we assume (4.35)
with E' sufficiently small, we also get sup{lt*(s)-i(s)l:s:::;;SO}<EO' Let To be the obvious upper bound for i and t* in (-00, Sol; that is, To:= 116 e2So . Also set
Since W t is almost surely continuous, M(EO)-+O in probability as EO-+O. Now the triangle inequality IWi(s)-Wi(s)I:::;;IWi(s)-Wt*(s)I+IWt*(s)-Wi(s)1 shows that when (4.35) holds we have SUP{IWi(s) - Wi(s) I:S:::;;SO}<EO+M(EO)' Hence,
But we have seen that t* (So) is very likely to be larger than T + 1 and that
It* (So) -i(So) I < EO when (4.35) holds. By Theorem 4.1, when This concludes the proof.
TD
1137
is large (4.35) holds with high probability. D
128
O. SCHRAMM AND S. SHEFFIELD
4.7. Caratheodory convergence For K C iHi let Ne (K) denote the 10- neighborhood of K in iHi, and let iHie (K) denote the unbounded connected component of iHi\Ne(K). Set dCKC(K, K') :=inf{E > 0: KniHie(K') = 0 = K'niHie(K)}.
It is easy to see that d CKC is a metric on the collection of compact connected K clHI such that lHI\K is connected. This metric is related to the Caratheodory kernel convergence topology, which is of central importance in the theory of conformal mappings. Let K t and K~ denote the evolving hulls corresponding to two Loewner evolutions generated by continuous driving terms W t and WI, respectively (as defined in §1.4). Such evolving hulls are also sometimes called Loewner chains. We set, for T~O, d~KdK,K'):= sup dCKc(Kt,KD· tE[O,Tj
The following is a simple lemma relating uniform convergence of driving terms to d CKC convergence of the corresponding Loewner chains. LEMMA
4.15. The Loewner transform Wf-tK is a continuous map from the space
of continuous paths W with the topology of uniform convergence to the space of Loewner chains with d~KC-convergence. In other words, for every 10>0, every T>O and every W: [0, Tj--+IR continuous, there is some o=o(E,T, W»O such that if W: [0, Tj--+IR is continuous and satisfies SUPtE[O,TjIWt-Wtl
This lemma is similar in spirit to [L4, Proposition 4.47j. As is well known, KT--+KT in the Hausdorff metric does not follow from W --+ W uniformly in [0, Tj. Proof. Fix T>O. Suppose that wn--+ W uniformly in [0, Tj. Let Kn denote the
Loewner chain corresponding to wn and let g~n): lHI\Kf--+lHI denote the corresponding Loewner evolution. Fix some to E [0, Tj. Since diam K~ is clearly bounded by a function ofT and IIWnll oo [L4, Lemma 4.13], the closure of {K~:nEN+} is compact with respect
to the Hausdorff metric on non-empty compact subsets of 1HI. Consider some integer sequence nj--+oo for which the Hausdorff limit K':=limj--'tooK~j exists. If zEiHi\Kto ' then there is a neighborhood U of Z in iHi such that UnK~ =0 for all sufficiently large n, by the continuity of solutions of ODE's in the vector field specifying the ODE. It follows that z¢'.K', and hence K' cKto ' With the intention of reaching a contradiction, suppose that there is some point ZEoKto\(K'UIR). Let z' be a point in lHI\Kto satisfying Iz-z'I<~ dist(z',K'UIR). The 1138
129
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above argument shows that limj--+oogtj)(z')=gto(Z')ElliI. On the other hand, for all sufficiently large j we have dist(z',K~j»2Iz-z'l. Now the Koebe distortion theorem (e.g., [P, Corollary 1.4]) applied to the restriction of gi: j ) to the disk of radius 2lz-z'l about z' (once with z' and again with z) shows that Imgi:j)(z)~ 136 1m gi: j )(z'). However, since zEKto ' for every E>O there is some first t1 E [0, to) such that Imgtl (z) ~E. The convergence argument above shows that for all arbitrarily large n, Img~~) (z) ~2E. Since 1m g~ n) (z) decreases monotonically in t, it follows that 1m gi:) (z) ~ 2E for all sufficiently large n. This contradicts our previous conclusion (nj) ( Z ) I m gto
'3 ~ 16
I m gto (nj) (Z' ) -+
3 16
°
Im gto (') Z > ,
and proves that K'-::;{)Kto\JR.. Let K be the union of K' and the bounded connected components of iHl\K'. The above implies that K-::;Kto\JR.. Now note that Kto\JR. is dense in Kto. (This follows from the easy direction (2)::::} (1) in [LSWl, Theorem 2.6] and from the fact that (lliInKt)\Kt' is non-empty when t>t'.) Consequently, K=Kto , which implies that dCKC(Kr,Kt)-+O for every fixed t E [0, T]. Note that the above proof also gives lims--+t dCKC(Ks, Kt)=O for tE [0, T] and s tending to t in [O,T]. Thus, K t is continuous in t with respect to d CKC . Since dCKC(L,L)~ dCKc( L', L) V dCKc( L", L) when L' c L c L", the d~KC convergence easily follows from the pointwise convergence, from continuity of K t and from monotonicity of Kr in t. D
4.8. Improving the convergence topology
In this subsection we complete the proof of Theorem 1.2. There are examples showing that the convergence of the Loewner driving term does not imply the uniform convergence of the paths parameterized by capacity. (See [LSW4, §3.4].) Therefore, we will need to apply other considerations. Before embarking on the proof, we note that when a, b~'>" the trace of SLE(4; a/.>..-I, b/.>..-I) is a simple path that does not hit JR., except at its starting point. Indeed, note first that the force points are moving monotonically away from one another. By comparison with a Bessel process, for example, it is easy to see that the trace does not hit the real line at any time t>O. It also does not hit itself, since h-+gsb(t+s)) has law that is mutually absolutely continuous with the path of SLE(4).
LEMMA 4.16. Let T>O and let Wn: [O,T]-+JR. be a sequence of continuous functions converging uniformly to a function W: [0, T]-+ JR.. Suppose that each Wn is the driving term of a Loewner evolution of a path ,n: [0, T]-+iHl, and W is the driving term of a Loewner evolution of a simple path T [O,T]-+iHl satisfying ,(0,T]nJR.=0.
Then
limn--+oo SUPtE[O,T] dHbn[O, t], ,[0, t])=O, where d H denotes the the Hausdorff metric. 1139
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Proof. First note that diam,n [0, T] is bounded, because II Wn II 00 is bounded. Fix some t E [0, T], and let r denote a subsequential Hausdorff limit of [0, t]. It suffices to prove that r=,[O, t]. By Lemma 4.15, we know that for every c>O we have
,n
rnlHic:b[O, t])=0.
Since, is a simple path satisfying ,(0, T]nlPi.=0, it follows that
Uc:>o lHic:b[O, t]) =1HI\ ,[0, t], which implies that relPi.U,[O, t]. Fix some Zl ElPi.\ ,[0, t] = lPi.\ {')'(O)}. By the continuity in Wand Z of the solutions of Loewner's equation (1.3), it follows that there is a neighborhood V of Zl such that V n'n [0, T] = 0 for all sufficiently large n. This implies that Zl ~r, and hence re,[O, t]. Now let t' E [0, t]. By Lemma 4.15 again, for every c>O and every n sufficiently large, in iHi must come within distance c from ,n[O, t]. Thus, every such path must intersect r. Since re,[O, t] is closed, this implies that ,(t')Er. Therefore, r=,[O,t]; that is, ,(t')~lHic:bn[O, t]), which means that every path connecting ,(t') to
Since
,n
00
[0, t] and ,[0, t] are monotone increasing in t and ,[0, t] is continuous in t with
respect to dR, it easily follows that limn-too SUPtE[O,T] dHbn[O, t], ,[0, t]) =0.
D
Here is an outline of the main ideas going into the proof of Theorem 1.2. Let , be the path ¢o, parameterized by half-plane capacity. The main step in the proof is to show that if we fix T>O, we have SUPtE[o,T]I,(t)-,sLE(t)I-+O in probability. By Theorem 1.3 and Lemma 4.16, we get SUPtE[O,T] dHb [0, t], ,sLdO, t]) -+0 in probability (since ,SLE is a simple path). We only need to rule out the possibility that , has significant (and fast) backtracking along 'SLE. This is ruled out by invoking Lemma 3.17 and observing that the ¢-image of the place where simple random walk (starting from a vertex near ¢-l(i)) hits 8Db) can be close to any fixed segment of ,SLE(O, T]. Proof of Theorem 1.2. Let, be the path ¢o, parameterized by half-plane capacity.
Let 8, T>O and rD = rad-l (i) (D). Let W denote the Loewner driving term of 'SLE. Since ,SLE is almost surely a simple path, Lemma 4.16 implies that for every c>O there is some c'=c'(c"SLE»O such that if W is the driving term of a continuous path:r and SUPtE[o,T]IWt-Wtl
that for every co>O, if rD is larger than some function of co, 8, T, a and b, then there is a coupling of hand SLE(4;aj>"-I,bj>"-I) such that [2:= sup dHb[O,t],'SLE[O,t])
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
131
Let O
so=so(t0,t l ,t2 ,t3 ,T,8»0 such that P[Al ];?1-8, where Al is the event that (1) the harmonic measure of "(SLE[tl,t2] from i with respect to lHI\"(SLE[O, (0) is at least So, (2) dist(R, "(SLE[t O, T]) >so, (3) dist("(sLE[O, tjl, "(SLE[tj+l' T]) >So for j =0,1,2, (4) diam "(sLdO, T] < 1/ So, and (5) iEiHiso("(sLE[O, T]). (It is tedious, but straightforward, to check that Al is measurable.) Consider a simple random walk S independent of h starting at a TG-vertex closest to r/J-l(i). Let TT be the first time t when S(t)EaD(r/J-lo,,(
To prove (4.36), first observe that conditional on "(SLE such that Al holds, a 2dimensional Brownian motion S started at i has probability at least So to first hit "(SLEUR in "(SLE[tl, t2]. Moreover, if this happens, the Brownian motion is likely to stay within a compact subset LclHI before hitting "(SLE[tl, t2] and not to come arbitrarily close to "(SLE far from its hitting point. On compact subsets of 1HI, the map Tr}r/J-l distorts distances by a bounded factor, by the Koebe distortion theorem [P, Theorem 1.3 and Corollary 1.4]. Since r/J-l takes a Brownian motion to a monotonically time-changed Brownian motion, by taking TD large we may couple Tr}S and Tr}r/J-loS to stay arbitrarily close (until r/J-loS hits aD) with high probability, up to a time change. Assuming that (} is arbitrarily small and taking (5) into account, we find that on Al and given ("(SLE, "(
1141
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O. SCHRAMM AND S. SHEFFIELD
Then (3.62) reads P[A 2 ];?1-8. In conjunction with (4.36) and P[A j ];?1-8, j=O, 1, this implies that (4.37) Conditional on Sn on the event A I n{dist(¢(S7),')'SLE[tl,t2])
°
ma 3.17 with S7 translated to where the p in that lemma is chosen as ~p808 and the R is taken to be 82rD, where 82=82(80»0 is a small constant depending only on 80. Note that the assumption necessary for the lemma that dist(Sn {vo}UoD»4R holds by (2), (4) and (5) in the definition of Al and the fact that the distance distortion of r"i/¢-I is bounded on compact subsets of !HI. Let ii denote the a provided by the lemma, which is a function of 80, a, band 8. Set r=Rj(ii+1). Then the lemma together with (4.37) imply that with probability 1-0(8) there is within distance CI from ')'SLE[tl, t2] the ¢-image of a vertex VEoDh) such that ¢-lo')'[O, T] has precisely two disjoint crossings of the annulus {z:r:;:;; Iz-vi :;:;;R}. If this happens, let ZI be a point in ')'SLE [i}, t 2 ] closest to such a ¢(v). Let r' denote the lower bound we get on {1¢(v)-¢(z)I:lv-zl;?r} which follows from the bounded distortion of rD¢' We may also assume that 1¢(v)-¢(z)l:;:;;i80 when Iv-zl:;:;;R. Now take CI=~r' and let 83=83(8,r')E(0, ir') be so small that with probability at least 1-8 for every ball of radius ~r' centered at a point Zo E')'SLE [to , T] the distance outside of the ball B(zo, ~r') between the two connected components of lRu (')'SLE [0, T] \ {zo}) is at least 83. Note that when this is the case, every path connecting these two components outside of B(zo, ~r') has to intersect iHiS3 / 3 hsLElO, T]). Consequently, if additionally (!< ~83' then ')'<1>[0, T] cannot contain an arc whose endpoints are within distance ~83 of these two components, unless the arc visits the ball B(zo, ~r'). If is sufficiently small, then there is some t~:;:;;t3 such that I')'SLdt3)-')'(t~)I<~83' Now choose Zo :=ZI, for the previous paragraph. The path ')'<1>[0, t~] must pass through the ball B (zo, ~r'), and therefore ¢-I 0,),<1> [0, t~] contains two disjoint crossings of the annulus {z:r:;:;;lz-vl:;:;;R}. If we assume that ¢-lo')'[O,T] has no more than two disjoint cross{!
ings of this annulus (which happens with probability at least 1-0(8)), it follows that for tE[t3,T] the point ')'(t) is closer to ')'SLE[to,T] than to ')'SLE[O,tO]UlR. Since in the above 8, to and t3 are arbitrary subject to the constraint 0
A:={zEiHi:rl
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
133
¢-l(A), 8j :=¢-1( {zEiHi: JzJ=rj}), j=1,2, L:=dist(81, 8 2; A') and L' :=dist(8+, 8_; A'). By conformal invariance of extremal length, it follows that L' / L---+O as r2---+00, uniformly in D. (Otherwise, the metric which is equal to the Euclidean metric in the ball of radius 3L centered on a point in an arc of length at most 2L from 8 1 to 8 2 in A', and is zero outside this ball, contradicts the extremal length going to zero.) Let ,6cA' be an arc of diameter at most 2L' connecting 8+ and 8_. Let
L1 := dist(,6, 8 1; A')
and
L 2 := dist(,6, 8 2; A').
Since L' / L---+O as r2 ---+00, we have L 1;:: ~ L or L 2;:: ~ L if r2 is sufficiently large. Suppose first that L2;::~L. Let So denote the first time such that J¢o,(so)J=r2. Then there are two connected components ,61 and ,62 of ,6\,[0, sol such that ,[0, sol U,61 U,62 separates ¢-l(B(O, r1)) from Ya in D. Now the proof of Theorem 3.22 shows that
if L2/ L' is sufficiently large. (Hence when L 2 ;:: ~ Land r2 is sufficiently large.) A similar estimate holds with ,62. Thus, with probability at most 0(8), Ie? contains two disjoint crossings of the annulus r1 ~ JzJ ~r2. On the other hand, if L2 < ~L and L 1;:: ~L, then we may apply the same argument to the reversal of , (or else slightly modify the way the analog of Theorem 3.22 is proved) to reach the same conclusion. This completes the proof. D
5. Other lattices In this section we describe the modifications necessary to adapt the proofs of Theorems 1.3 and 1.2 to the more general framework of Theorem 1.4. Before we go into the actual proof, a few words need to be said about the properties of the weighted random walk on <5 and its convergence to Brownian motion. Fix some vertex Vo in <5, and let Vo denote its orbit under the group generated by the two translations T1 and T2 preserving <5. If the walk starts at vo, then a new Markov chain is obtained by looking at the sequence of vertices in Vo that the walk visits. A simple path reversal argument shows that for this new Markov chain the transition probability from v to u is the same as the transition probability from u to v, for every pair of vertices v, UEVo. Also observe that the JR.2-length of a single step has an exponential tail. This is enough to show that the Markov chain on Vo, rescaled appropriately in time and space, converges to a linear image of Brownian motion (and it is not hard to verify that the linear transformation is non-singular). Moreover, the few properties of the simple random walk on TG that we have used in the course of the paper are easily verified for this Markov chain on Vo and easily translated to the weighted walk on <5. 1143
o.
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SCHRAMM AND S. SHEFFIELD
Proof of Theorem 1.4. Very few changes are needed to adapt the proof. Let
IB* denote the planar dual of lB.
The statement and proof of Lemma 3.1 requires some changes, because in the more general setup it is not true that every vertex adjacent to an interface on the right has a
Every vertex at
where c< 1 is some constant depending only on the lattice and its edge weights. We can certainly drop the trailing additive 0(1). Induction on j now gives
where Qj=l+(l-c)+ ... +(l-c)j-l=(l-(l-c)j)/c. When j=m, this reads M(l-c)'" ~ m
'"
O(l)Q", M(l-c)'"
0'
Clearly, Mo~eX, and the bound M m =Om,X(1) follows. A corresponding bound clearly also holds for E[e-h(v) 11C] when vEV_. The remainder of the proof of the analog of Lemma 3.1 proceeds without difficulty. The proof of Lemma 3.2 needs to be similarly adapted, but essentially the same argument works. The next point which requires adaptation is the definition of zg in §3.5. Let
~
denote
the set of pairs (v, e*), where v is a vertex in
e* E"(. Let
~'
be a collection of
elements of~, one from each orbit under the group generated by the translations Tl and T2 preserving
CONTOUR LINES OF THE TWO-DIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
135
References
[A]
[B] [BPZ]
[BEF]
[CN]
[Cal [Co]
[DMS] [DoSn]
[D] [DS1] [DS2]
[F] [FS]
[Ga]
[Gi]
[GJ] [HK]
[I] [Ka] [KN] [KS]
AHLFORS, L. V., Conformal Invariants: Topics in Geometric FUnction Theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, 1973. BEFFARA, V., The dimension of the SLE curves. Ann. Probab., 36:4 (2008),1421-1452. BELAVIN, A. A., POLYAKOV, A. M. & ZAMOLODCHIKOV, A. B., Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B, 241 (1984), 333-380. BRICMONT, J., EL MELLOUKI, A. & FROHLICH, J., Random surfaces in statistical mechanics: roughening, rounding, wetting, .... J. Stat. Phys., 42:5-6 (1986), 743798. CAMIA, F. & NEWMAN, C. M., Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys., 268 (2006), 1-38. CARDY, J., SLE for theoretical physicists. Ann. Physics, 318 (2005), 81-118. CONIGLIO, A., Fractal structure of Ising and Potts clusters: exact results. Phys. Rev. Lett., 62:26 (1989), 3054-3057. DI FRANCESCO, P., MATHIEU, P. & SENECHAL, D., Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York, 1997. DOYLE, P. G. & SNELL, J. L., Random Walks and Electric Networks. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984. DUPLANTIER, B., Two-dimensional fractal geometry, critical phenomena and conformal invariance. Phys. Rep., 184 (1989), 229-257. DUPLANTIER, B. & SALEUR, H., Exact critical properties of two-dimensional dense self-avoiding walks. Nuclear Phys. B, 290 (1987), 291-326. - Winding-angle distributions of two-dimensional self-avoiding walks from conformal invariance. Phys. Rev. Lett., 60:23 (1988), 2343-2346. FOLTIN, G., An alternative field theory for the Kosterlitz-Thouless transition. J. Phys. A, 34:26 (2001), 5327-5333. FROHLICH, J. & SPENCER, T., The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Comm. Math. Phys., 81 (1981), 527602. GAW§DZKI, K., Lectures on conformal field theory, in Quantum Fields and Strings: a Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 727-805. Amer. Math. Soc., Providence, RI, 1999. GIACOMIN, G., Limit theorems for random interface models of Ginzburg-Landau '17¢ type, in Stochastic Partial Differential Equations and Applications (Trento, 2002), Lecture Notes in Pure and Appl. Math., 227, pp. 235-253. Dekker, New York, 2002. GLIMM, J. & JAFFE, A., Quantum Physics. Springer, New York, 1987. HUBER, G. & KONDEV, J., Passive-scalar turbulence and the geometry of loops. Bull. Amer. Phys. Soc., DCOMP Meeting 2001, Q2.008. ISICHENKO, M. B., Percolation, statistical topography, and transport in random media. Rev. Modern Phys., 64:4 (1992), 961-1043. KADANOFF, L. P., Lattice Coulomb gas representations of two-dimensional problems. J. Phys. A, 11:7 (1978), 1399-1417. KAGER, W. & NIENHUIS, B., A guide to stochastic Lowner evolution and its applications. J. Stat. Phys., 115:5-6 (2004), 1149-1229. KARATZAS, 1. & SHREVE, S. E., Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, 113. Springer, New York, 1988. 1145
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[M] [NS] [Nl] [N2] [dN]
[P] [RY] [RS] [SD] [Sch] [SS]
O. SCHRAMM AND S. SHEFFIELD KENYON, R., Dominos and the Gaussian free field. Ann. Probab., 29:3 (2001), 11281137. KONDEV, J., Du, S. & HUBER, G., Two-dimensional passive-scalar turbulence and the geometry of loops. Bull. Amer. Phys. Soc., March Meeting 2002, U4.003. KONDEV, J. & HENLEY, C. L., Geometrical exponents of contour loops on random Gaussian surfaces. Phys. Rev. Lett., 74:23 (1995), 4580-4583. KONDEV, J., HENLEY, C. L. & SALINAS, D. G., Nonlinear measures for characterizing rough surface morphologies. Phys. Rev. E, 61 (2000), 104-125. KOSTERLITZ, J. M., The d-dimensional Coulomb gas and the roughening transition. J. Phys. C, 10:19 (1977), 3753-3760. KOSTERLITZ, J. M. & THOULESS, D. J., Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C, 6:7 (1973), 1181-1203. LAWLER, G. F., Intersections of Random Walks. Probability and its Applications. Birkhiiuser, Boston, MA, 1991. Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J., 4 (1998), Paper 5, 67 pp. An introduction to the stochastic Loewner evolution, in Random Walks and Geometry, pp. 261-293. de Gruyter, Berlin, 2004. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, 114. Amer. Math. Soc., Providence, RI, 2005. LAWLER, G. F., SCHRAMM, O. & WERNER, W., Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187 (2001), 237-273. - Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincare Probab. Statist., 38 (2002), 109-123. Conformal restriction: the chordal case. J. Amer. Math. Soc., 16 (2003), 917-955. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32:1B (2004), 939-995. MANDELBROT, B., How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156 (1967), 636-638. NADDAF, A. & SPENCER, T., On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys., 183 (1997), 55-84. NIENHUIS, B., Exact critical point and critical exponents of O(n) models in two dimensions. Phys. Rev. Lett., 49:15 (1982), 1062-1065. - Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys., 34:5-6 (1984), 731-761. DEN NIJS, M., Extended scaling relations for the magnetic critical exponents of the Potts model. Phys. Rev. B, 27:3 (1983), 1674-1679. POMMERENKE, C., Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften, 299. Springer, Berlin-Heidelberg, 1992. REVUZ, D. & YOR, M., Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften, 293. Springer, Berlin-Heidelberg, 1999. ROHDE, S. & SCHRAMM, 0., Basic properties of SLE. Ann. of Math., 161 (2005), 883-924. SALEUR, H. & DUPLANTIER, B., Exact determination of the percolation hull exponent in two dimensions. Phys. Rev. Lett., 58:22 (1987), 2325-2328. SCHRAMM, 0., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221-288. SCHRAMM, O. & SHEFFIELD, S., Harmonic explorer and its convergence to SLE4. Ann. Probab., 33:6 (2005), 2127-2148.
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[W]
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SHEFFIELD, S., Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139 (2007), 521-541. SMIRNOV, S., Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 239-244. SPENCER, T., Scaling, the free field and statistical mechanics, in The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994), Proc. Sympos. Pure Math., 60, pp. 373-389. Amer. Math. Soc., Providence, RI, 1997. WERNER, W., Random planar curves and Schramm-Loewner evolutions, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1840, pp. 107-195. Springer, Berlin-Heidelberg, 2004.
ODED SCHRAMM was at the Theory Group of Microsoft Research One Microsoft Way Redmond, WA 98052-6399 U.S.A.
SCOTT SHEFFIELD Courant Institute New York University 251 Mercer Street New York, NY 10012 U.S.A. [email protected] [email protected]
Received October 18, 2006
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Commun. Math. Phys. 288. 43---53 (2009) Digital Object ldemifier ( 001 ) 10. 1007/sOO220-009-0731 ·6
Communications in
Mathematical Physics
Conformal Radii for Conformal Loop Ensembles Oded Schramm I. Scotl Shefficld 2. 3. • , David B. Wilson I t Microso ft Rcsean;h. One Microsoft Way. Redmond. WA 98052. USA 2 Courant Institule. New York University. 251 Mercer Slre~1. New York . NY 1002 1. USA 3 1)ep.1rtmcnt of Mathematic s. M. l. T.. 77 Massachu seus Ave .. Cambridge. MA 021 39. USA Received: 20 September 20071 Accepted: 10 Nove mber 2008 Publisllcd online: 26 February 2009 - © Springer-Verlag 2009
Abstract: The conformal loop ensembles CLE" , defined for 8/ 3 ::: /( ::: 8, arc random collections of loops in a planar domain which arc conjectured scaling limits of the O(n ) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic poi nt. Our resu lts agree with pred ictions made by Cardy and Ziffand by Kenyon and Wilson for Ihe 0(1/) model. We also compute the expectation dimension of the CLE" gasket, which consisls of po in IS nOI surrounded by any loop, to be
2- (8-K)(3K-8). 32K
which agrces with the fractal dimension given by Duplamier for the O(n) model gasket. I . Introduction
The conformal loop ensembles CLE..-, defined for all 8/ 3 ::: /( ::: 8, are random colkctions of loops in a simply connected planar domain D ~ C. They were defined and constructed from branching variants of SLE" in [She06], where they were conjectured 10 be the scaling limits of various random loop models from statistical physics, including Ihe so-called 0(/1) loop modcls with /I
= - 2cos(4JT/ K).
(I)
see e.g. IKN04 1 for an exposition. This paper is a sequel to IShe06J. We wi ll Slate Ihe results about CLE" from IShl-{)6 llhal we nced for Ihis paper (namely Propositions I and 2), but we will not repeal the definition of CLE" here. When 8/ 3 < K < 8, CLE" is almost surely a countably infinitc collection of loops. CLEs is a single space- fil ling loop almost surely and CLEs/3 is almost surely emply. • Partially supported by NSF grant DMS0403IS2.
I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_33, 1149
O. Schramm. S. Sheffield,
44
I).
S. Wi l.on
CLE6 is the scaling limit of the cluster boundaries of critical site percolation on the triangular lanice [CN06,SmiOl ,CN07 1. We will henceforth assume 8/3 < K < 8.
q
For each zE D, we inductively defin e to be the outermost loop surrounding z when the loops L ~ , .... Lk_ l aTC removed (provided such a loop exists). For each deterministic zE D , Ihe loops L i exist for all k ~ I with probability one. Define = D and let Af be the component of D\q that contains z. The conformal gasket is the random closed scI r consisting o f points that afC not surrounded by any loop of an instance o f CLEI(, i.c. the SCI of points for whic h L i does nOl exi st. If 0 is a simply con nected planar do main and ZE D , the conformal r adius of D viewed from z is defined to be CR(D, z) := 19' (z) I- I, where g is any conformal map from D to the unit disk ][Il that sends z to O. The follow ing is immediate from the construction in [She06 j:
Ab
Proposition 1. Let D be a simply connected bounded planar domain, and consider a CLEK on D for some 8/3 < K < 8. Theil r is almost surely fh e closure of fhe sef of points that lie 011 all outermost loop (i,e" a loop of the form L ~ for some z), Conditioned 0 11 the outermost loops, the law of the remain ing loops is given by an independent CLE K in each compOllent of D\ r, For zE D and k = 1,2,3, ' .. , define B~ := log CR (Ak_ I' z) - logCR (Ak' z).
For any fixed z , the
Bf 's are i.i.d. random variables.
Vari ous authors in the physics literature have used heuristic arguments (based on the so-called Coulomb gas method) LO calculate properties of the scaling limi ts o f stati stical physical loop models, including the O (n ) models, based on certain conformal invariance hypotheses of these limits. Although the scaling limi ts of the 0 (1/) models have not been shown to cx ist, there is strong evidence that if they do ex ist they must be CLEK • (For example, there is heuristic evidence that any scaling limit of the 0(11) models sho uld be in some sense eonformally invariant; it is shown in LShe06,SW08b, SW08aj that any random loop e nsemble satisfying certain hypotheses including conformal invariance and a Markov-type property must be a CLEK . ) It is therefore natural to interpret these calcu lat ions as predictions about the behavior of the CLEK • Cardy and ZilT LCZ03 j predicted and experimentally verified the e)(peeted number o f loops surrounding a point in the 0(11) model, which, in light of (I ), may be interpreted as a prediction of the expectation o f B~: --~
IE[ Bf l
(K/4 - l)cot(1f(1 - 4/K)) 1f
Ke nyon and Wil son [KW04 j went further and predicted the di stribution of its moment generating function IE[ exp(A
Bt, g iving
-COS(41f/K)
BDj =
-,(~ri-=#""=;;c=;=\ cos 1fJ(l 4/K)2 + 8A / K)'
(3)
jI, and de nsity function
for A satisfy ing RCA < I - ~ d
(2)
,
dx PrlBk < xl oo
=
- KCOS(41f / K) L: 41f
j
.
[
(- I) (J + 1/2) exp -
j=O
1150
(J +I / 2)2_( 1 - 4/K)2] x . 8/K
(4)
45
COllfomlaJ Radii for Conformal Loop Ensembles
The main result of thi s paper is Theorem I, which confirms Jhese predictions. In the special case I( = 6, th is prediction for the law of 13: was independently confirmed by OubCdat [Oub05 [. Theorem I. Let fJ( denote the density fUllction for the first time that a standard BrowlI ian motioll started at 0 exits tile iI/terral (~2JT / ./K. 2JT / ./K). Then for 8/3 < I( < 8, the density function for is
B: -d Pr[Bt:. dx
<x ] =~fJ«x)eos(4JT I I()exp
[(K - 4)' x ] . 81(
(5)
The equivalence of the form ulations (4) and (5), and the fact that they imply (3), follows from a calculation of Ciesie lski and Taylor, who showed that the exit-time di stribution of a Brownian motion from the center of a l ~dim e n s i onal bal l of radius r has a mo ment gcncrating funct ion of 1/ cos J2r 2).., and who gavc two serics cxpansions (onc in powcrs of e- x and thc othcr in powcrs of e- 1/ x ) for its dcnsity func tion [CT62 , Thcorcm 2 and Eq. 2.22J (sce also [8 S02 , Eqs. 1.3.0. 1 and 1.3.0. 2J). Since thc Fourier transform is invertible o n L 2(1R) , the equivalence of (3) and (4) follows by considering the moment generating fun ction restricted to the imaginary line RCA = O. Ouplantier [Oup90] predicted thc fractal dimension of thc gasket r associated with the 0 (II) model to be 31(
2
32
K
-+ I +-. where as usualn = -2 cos(4JT/ I(). We partially confirm this predictio n by giving the cxpectation dime nsion of the gaskct associatcd with CLEJ(' Thc expectation dimension of a rando m bounded set A is defined to be D E( A )
log !E[mini mal number of bal ls of radius £ required to cover A [ , £ ..... 0 I log £[
= lim
providcd the limi t ex ists. Thc expectation dimension upper bounds the Hausdorff dimension. Theorem 2. Let
r
be rhe gasket of a CLEo: ill/he ullit disk with
!E[minimalllumber of balls of radius £ required to COFer /11 particular. the expec/atioll dimellsion of the gasket
r
I(
E
(8/3.8). Theil
(I) ll·'·i
rJ ::::: -;;
is ~ + I +
r
Hcre, ::::: denotes cquivalence up to mul tiplicative constants. Lawler, Schramm, and Wcrncr [LSW02 ] studicd thc percolation gasket (associatcd with CLE6), effectively proving Thcorcm 2 in the casc I( = 6. More generally, they studied how long it takcs for a radial SLEo: to surround the origi n whe n I( > 4, and thc ir results impli citly imply Theorcm 2 whcn I( > 4; see the remark in Sect. 2.1 for furth er discussion. We conclude our introduction by no ting that the gasket dimension described above plays an important role in the physics literature, where it is related (at least heuristicall y) to the expone nts of magnctization and multipoint corrclation func tions in critical Ian ice models. We bricfl y describe thi s connect ion in the case of the q-state Pons model on the sq uare lanicc. More dctails and refcrenccs are found in [Shc06,Car07, Gri06 ]. 1151
O. Sch.ramm. S. Shcffleld. D. B. Wilson
46
A sample from the q-statc POliS model on a connected planar graph G is a random function (1 : V --7 {I, 2, ... ,q), where V is the sct o f vertices of G and the image values 1,2, .... q are often called spins. I f the boundary vertices (those on the boundary of thc unbounded face) of G arc all assigned a particu lar value (say b), then using the standard FK random cl uster decomposition [FK72 1, one may construct a sample from the Potts model as follows: I.
Sample a random subgraph G' of G contain ing all boundary edges (edges on the boundary of the unbo unded face), wi th probability proport ional to qilComponl'ntsOfG' (
2.
_ P_ I - p)
lIedgeSOfGI
whcreO < p < lisa paramclcr.Thc law ore ' is called Ihe FK random cluster model with parameters p and q. Call the component of G ' whieh eontains the bou ndary vertices of G the FK gasket. Set a(v) = b for each v in the FK gasket, and inde pendently assign one of the q states uniformly at random 10 each o f the remaining connected components of G ' (assigning all vertices in that component the corresponding state).
T he " magnet ization" at an interior vertex v of G (i.e., the probability that a(v) = j minus l l q ) is proportional to the probability that v is in the F K gasket. Given distinct vertices v and w, the covariancc of a (v) and a (w) is proportional to the probability that both v and w lie in the same component o f G/. We now restrict to the case in which G is a finit e piece of the square grid in the plane and the parameter p satisfies p/( I - p) = .jq. (Wi th this choice of p , the F K model is self-dual and believed to be crit ical, see e.g., [Gri06, Chap. 6J.) 11 is shown in [She06] that if q ::::: 4 and certain other hypotheses including conformal invariance hold , then the scaling limit of the set o f boundaries between clusters and dual cl usters in thecrhical FK rando m cluster models discussed above must begivcn by CLEK for the K satisfying q = 4 cos 2 (4JT I K) and 4 ::::: K ::::: 8. Assuming these hypotheses, the scaling limit of the d iscrete gasket is precisely the conti nuum CLEK gasket. A heuristic ansatz is that thc law of the critical FK gasket shou ld havc similar properties as the law of the set of squares in a fin e grid which intersect the continuum gasket. If this heurist ic holds, then when G is a bounded domai n intersected wit h asq uare grid wi th spac ing e, the magnetization at a vertex v of macroscopic distance from the bou ndary in the discrete model will be on the order of e2- d , where d is the limiting expectat ion dimension of the continuum gasket. Si milarly, the covariance between a( v) and a(w), for two macroscopically separated vert ices v and w, shou ld be on the order o f €2(2 - d j (si nce in the conti nuum model , the set of pairs v and w which lie in the same conti nuum spin cluster has di mension 2d; see [She06]). 2. Diffusions and Martingales
2.1. Reduction to a diffusion problem. Let BI [0. 00) --7 R. be a standard Brown ian motion and let BI : [0, 00) -+ [0, 2JT] be a random continuous process on the interval [0, 2JT] that is instantaneously reflecting at its endpoints (i.e., the set It : BI E 10, 2JT}) has Lebesgue measure zero almost surely) and evolves accordi ng 10 the SDE
K -4
dBr = -2- cot(B,f2) dl + jK d B/
1152
(6)
Confonnal Radii for Conformal Loop Ensembles
47
on each interval of time fo r which 0, r;. to. 2JT) . In other words, 0, is a rando m continuous process adapted to the fi ltration of Ht which almost surely sati sfi es
for ali i for which the right hand side is well defined. The law of this process is un iquel y determined by 00 [She06 ], and we also have the following from [Shc06 ]: Proposilion 2. When 8/3 < I( < 8. the law of H~ is the same as the law of in f{ t : Ot = 2JT I for the diffusion (6) started at 00 = O. It is convenient to lift the process O( so that , rather than taking values in [0, 2JT], it takes values in all of IR. Let R : lR --'I- [0. 2JT [ be the piecewise affine map for which R(x) = Ixl when x E [-2JT , 2JT] and R(4JT + x) = R(x) for all x. Gi ven 81> we can generate a continuo us process 0, with R(O,) = 01 in such a way that for each component (ft. t2) of the set It: 01 r;. 2JT Z}, we independently toss a fair coi n to dec ide whethe r 0, > Oil or 0, < Oil on that component. The 8, together wit h these coin tosses (for each interval of II : 0, r;. 2JTZ)) determine 0, uniquely. This 0, is still a solution to (6) provided we modify H, in such a way that dB, is replaced with -d B, on those intervals for which dO, = - dOl' (Th is modi fi cat io n does not change the law of 8,.) In the fCmain der of the text, we will drop the 0, notation and write 01 for the lifted process on JR.
Remark. A very simi lar diffu sion process was slUdied by Lawler, Schramm, and Werner [LSW02 ], name ly (7)
which is the same as (6) but without the factor of (I( - 4) / 2, and they too studied the time for the diffusion to reach 0, = 2JT when slarted at 00 = O. Diffusions 6 and 7 arc ide ntical when I( = 6, and Diffusion 6 (for 4 < I( < 8) is given by Diffu sio n 7 (for 4 < I( < (0) upon substituting I( --'I- (21() /(1( - 4) and scaling time by I ---;. 1(1( - 4) /2 . However, Diffusion 6 is a more si ngular Bessel-type process when I( :s: 4, req uiri ng additional tec hnical analysis to deal with Ihe times at which process is at 0 (see e.g. Lemma 3). Furthermore, o nly the large-time asymptotic decay rate of the hitting time di stribution (used in the proof of Theorem 2) is given in fLSW02], and additio nal effon is requ ired to obtain the prec ise hill ing time distribution provided in Theorem I.
2.2. Local martingales. Recall the hypergeornetrie fu ncti on defined by 00
b . . ) _ ' " (a)"(b),, "
F (a, . c. Z - L...
,,:(I
(c)"n1
z ,
where a, b, c E C are parameters, c r;. -N (where N = to, I. 2. . . . )), and (e)" denotes - 1). This defi nition holds for Z E C when [zl < I, and it may be
e(e+ 1) ··· (e+ 11
1153
48
O. Sc hramm. S. Sheffie ld . D. B. Wil son
de fined by analytic continuation else where (though il is then not always si ng le-valued). We defin e for A E C,
, Md.(fJ)= F ( l -/i4 + M:.)..{O) = F ( I _
j( I -I(4)2" 4)'" 3 4 ., , ) +"K, I - i4 ( - j( 1 -;( + -;:-:1-K<sm-'4
~ + j (! _ ~)2 + :A,
~ _ JH_ ~)2 + ~A: ~; cos2 ~ )
I_
i,
'
cos ~
... ).
( where the formul a for M : .A(0) makes sense whenever I( f~ . ~, There i s some ambiguity in the choice of sq uare root, but since F(b. a : c: z) = F(a. b; c: z) , as long
as the same choice of square root is made for both occurrences, there is no ambigui ty in these defin itions.
Lemma I. For the diffllsion (6) with K > 0, lei T be the fi rst rime at which (), E 2:7r Z. alld leI t = min(r. T). For allY).. E C, both cxp[ AtlM: .A(0;) (Uld cxppJj M:. A(8;) are local martingales p(l ramererized by t. Proof Given these formulas, in princ iple it is straightforward to verify that the d t term o f the Ito expansion o f
d [el./ M:~~(8/ )] is equal to zero (where elo is either e or 0 ). This term can be e xpressed as
,,[ AM....,'e.l..(e,)+ (M....,'e;.. )' (8/ ) -2-cot ,-4 ' ( M.:.). ,'e)"(8/)]dr. (9,/2)+ "2
e
Since Mathematiea does not simplify thi s to zero, we show how 10 do this for M:.i\ ; the case for .l.. is simi lar. We abbreviate M :.;.. wi th M , leI F de note the hypergeometric fun ction in the definition o f M:.;.., let a. b, and c de note the parameters of F in M , and change variables 10
M:.
y = y(O) = S;"2(014) = ( I - oos(012)) /2, M(O) = F (y (9 )) .
M ' (9) =
~ F'(Y(0))S;" (012).
M"«():= F"(y «()))
/6 sin2(O/2) + F'(Y(O»)~COS«()/2)
2 I J' '= F " ( Y )Y-Y - 4- + F' (y )-4-·
so that ,"
[
:=
4,
K I K F " (y) ( Y-Y J. F (Y)--4F (Y)(Y-')+g
el./ [AF(y) + (1 - 3K/8) F'(y)
J.+ ( I _ [
K +-
8
(Y -
!) + i
2) +g F (Y) K
F"(y)(y -
,
,-y)]
('
i )]
3'/8 ) (" _ ~ ",-(a "",,)(b:.. :.+,,, -,)) - ]
(a+ +
2
lI )(b +/! )
c
1154
II
.:..:: +
c +n
/I - II(n- l)
)
Confonnal Radii for Confonnal Loop Ensembles
49
and we define Ell to be C+II times the expression in brackets. (Note that indeed c ¢ -N.) We may write E/I as a polynomial in II: Ell = [( I - 3K/ S)(I - 1/ 2) + (K/ SH I - c+a +b)] x 1/ 2
+ [A + ( I - 3K / S)(C - a / 2 - b/ 2) + (K / S)(C + a b)] + rCA - ( I - 3K / S)a b/ 2].
X"
By our choices of a , band c, E" = 0 for each II , which proves the clai m for M:.A.
0
2.3. Expected first hitting of 2H Z. In this subsection we obtai n asymptotics for the fu nction
where ()I is the diffusion (6) and T is the first ti me t ~ 0 at whic h ()I E 2JT Z . (Recall that T is fi nite a.s. when K < S.) Whenever () E 2H Z, trivially L «() = () . Lemma 2. For the diffusioll (6) with S/ 3 <
K
< S, L«(),) is a martingale.
Proof Since L(O) is defined in tenns of expected values, L(OI) is a local mart ingale whenever 01 ¢ 2JT Z, and the stopped process L (Olllill (f.T) is a martingale. Since the di ffu sion behaves sy mmetrically around the points 2H Z and the number of intervals of R·,,2H Z crossed before some determin ist ic time has expone ntially decaying tai ls (whic h implies integrability), L(8, ) is a marti ngale. 0 Next, we express L «(), ) in terms of the A = 0 case of the local martingales expIA/] M:.,, «(),) and exp[At] M: .,,«(),) . Because M : .o (O ) = I, this local martingale is uni nformative, but
We have M:.o (O) = - M: .o(2JT) = j1ir(4 / K - 1/ 2)/ (2r(4/ K» , a nd since M:.o is bounded (when K =f: S), the stopped process M:. O«()lllin (l. T) is also a martingale. This delerm incs L. namely, L(O) = JT-
2 for (i) F (3 _.:! 1. 3·cos2 (1 ) cos (/ r (~ _ -1) 1 1(' l' l ' ! l' I(
f)
E [0, 2JTI
(8)
and L(8 + 2JT) = L(O) + 2JT. (It is also possible to derive (8) from the formula for Pr rSLE trace passes to left of x + i y ] [SchOl l after applying a Mobius transformation and suitable hypergeometric identities.) We wish to understand the behavior of L near the points in 2JT Z, and to this end we use the form ula (see rEMOT53, p. lOS, Eq . 2. 10. 1[) F(a , b:c:z) =
r (c)r (c - a - b) F(a , b;a+b-c+l; I - z) r (c - a)r(c - b) r(c)r (a +b - c ) c- a- b + (I -z ) F(c- a.c - b · c- a - b+ l·l -z ) r(a) r (b) "
(9) 1155
O. Schram m. S. Sheffield . D. B. Wilson
50
whic h is valid when I - c, b - a, and c - b - a are not integers and I arg( I - z)1 < Jf . In our case the noni ntcgralit), condit ion is satisfied whenever 8/ K ~ N. For the range of K Ihat we arc interested in, this rules out K = 4, for which we already know L«(}) = (J, and therefore do not need asymptotics. The endpoints o f the range, K = 8/3 and K = 8 are al so ru led out , but for the remaini ng
F G-~,~ : ~;cos2~)=
have
r (') r (± -1) ~(~) ;(1 )- F G-~ · !;~-~ : s in 2~) +
~
K ' S we
r (') r (l - ±) '1 _ Isin ol''-I F (.1 l' i+ l's in20) r (l-~) r (~)" ""'" J(
fo"r (~-j) 2r (') K
r (~-~). , 1- 1
1
-ICOs- 'I'1 + 2r ('i -
4 4 1 '29 ) x F ( ; ' l: :;:- + "2: Sin "2
4) K
I,,, ,I
.
and by (8),
L(e)~cK (Sin 2"O )~ - 1 F (.1K'
1'.1 + 2' l'si n2 2"0)cos. O • /( 2'
(} E (O . JT ).
for some con stant CK > O. (O ne can use the Legendre duplicat ion form ula to show that Co; = 281/( - 1r(~)2 / r ( ~), but we do not need this.) Since L ( -8) = - L (8), we concl ude that
e E (-n , n). where Ao is a analytic function (depending on K) satisfying Ao(O) > O. Thi s implies
e' ~ A (IL(O)I'
( 10)
near 8 = 0, for some analytic A.
2.4. Starting at 80 = O. We wil[ event ually need to start the di ffu sion at 80 = 0, but Lemma I o nly covers what happens up until the fi rst time that OJ E 2JrZ. In this subsection we show
Lemma 3. For the diffllsion (6) with 8/3 < K < 8, let T be the first time (jf which 91 E 2JrZ'AJrZ, and let i = min(t, T) . For all)' A E C. the process exp[ Ar]M:." (Or) is
a marrillgale.
Proof Let M abbreviate M;.", and let us assume without loss o f generality that -2Jr < < 2Jr. Let us define WI = L (Od, whic h is a martingale and may be interpreted as a time-c hanged Brownian motion. We wish to argue that e"r M ( L -1 (Wi» = e"i M(9i) is a local martingale. (The definitio n of L implies that it is strictl y monoto ne, and he nce L - 1 is well defin ed .) Note that by Lemma I it is a local marti ngale when ()f ¢ 2 Jr Z. To exte nd this to a ne ig hborhood o r (); = 0, one could try to use Ito's formu la. To do this, it would be necessary that f := M o L -I betwice d iffe rcntiablc . We have M (9) = A I (8 2) in ( - 2 Jr . 2 Jr), where A I is analytic. Conseque ntly, ( 10) gives for 8/3 < K < 8 and for W in a ne ighborhood o f 0,
80
( 11 )
1156
51
Conformal Radii for Con fomlal Loop Ensembles
for some anal ytic A 2. Though thi s is not necessary for the proof, one can check that A2(0) :F 0 and there fore f"(0) is not fi nile when K < 4. To circumvent the problem of f = M o L - \ not being twice d ifferentiable, we use the 110- Tanaka Theore m ([ RY99, Theorem 1.5 o n p. 223 D. The expone nt -(!:; in ( I I ) ranges from I to 00 as K ranges from 8/3 to 8. In particular, f'( O) = 0 and !' is continuous ncar O. Since A2 is analytic, ncar 0 the function f may be ex pressed as the d ifference o f two convex functio ns, namely, few) = (f(w) - f(O» Iw>O + (f(w) - f( O» Iw
F(a, b: c: I) = ~~'-c=-~ r (c a)r(c b )
( 12)
provided -c ~ N and Rec > Re(a +b ) (see e .g. [EMOT53, p. 104, Eq. 46[). Therefore, M(±2rr} is fini te . Thus, e)..i M (Oi) is bounded for bounded t, and we may concl ude that it is a martingale. 0 For future reference, we now caleulate M:.)..(±2rr). Observe that the parameters a, b. c of the hypergeometric function in the defi nition of M:. A satisfy 2 c - a - b = I. Conseque ntly, the identity r( z)r(1 - z) = rr/sin (rrz) and ( 12) give
( 13)
3. Proofs of M ain Results
We now restate and prove T heorem I. Theorem 3. SUP/Jose the diffusioll process (6) (with 8/3 < K < 8) is slarled at 00 = 0, alld T is the first lillie at which Of = ± 2:rr. If RcA ~ O. thell IE [
)..T
e
I
]
cos(:rr(l - 4/K» 00= 0 = cos(:rr J (l-4/ K)2+ 8A/ K) '
(This is equ ivalent to T heore m I by Proposition 2 and the remarks following the statement of Theore m I ,)
Proof Since M:.).. (OT) = M;,;.,C±2 rr) = M:. A (2rr) a.s. and exp[),.i'I M:. A (0;) is a martingale, the optional sampli ng theorem gives
and the proof is completed by appeal to ( 13). 1157
0
O . Schramm, S. Sheffield. D. B. Wilson
52
Proof of Th eorem 2. Fix some E: > 0 and let zED. Set ro := I -lzl,'1 := dist(z, L t), and suppose that f; < roo We seek to estimate the probabilit y that the open disk of radius e about z intersects the gasket; that is. the probability that r1 < e. By the Koebe = 1/4 theorem, 1'1 :::: CR ( D l, z) ::: 4 r l . Likewise, ro ::: CR(D, z) ::: 4 ro_ Th us, log CR(J[]I,z) - log CR (D [,z ) = log(ro/ r [) + 0 ( 1) . Referring to the density function of (4 ), we see thai
Bf
Bt
Pr[r) < E:] x e xp[-a log(ro/e)] = (E/ ro)«, where
•
~
l j4- (1 -4jK)2 31< 2 (8 -K)(3K -8) "----0;';--'-"-- = - - + I - - = .
8/1(
32
I(
321(
Fo r each j = I , ... , p/el, we may cover the annulus /z : (j - J) t ::: I -izi s j cI by D CI Ie) disks of radius e. TIle total expected number of these di sks that intersect the gasket is at most [11£1
L
O( l /s) x O(s/( js » U = O (su - 2).
j= 1
(Here we made usc of the fact that ex < I.) Th us o n average O(su - 2) disks of radius s surfiee to cover the gasket. On the other hand, we may pack into II) at least 9 ( I /s2) points so that every two of the m are more than di stance 4s apart, and each of them is at least distance 1/2 fro m the bou ndary. For each such point z there is a 8 (sU) chance that the disk or radius s cenlered at Z is not surrounded by a loop, i.e., that that the gasket r contains a point z' that is within distance s of z. Since the points z are suffi cie ntly far apart , the poi nts z' must be co vered by di stinct disks in any covering of r by disks of radius s. Thus the expected number of disks of radius s required to co ver the gasket r is at least 9 (su - 2) . 0 4. Open Problems Kenyon and Wilson [KW04] also predicted the large-k limiting distribution of anOlher quantity, the "electrical thickness" of the loops L ~ when k ~ 00. The electrical thickness of a loop compares the conformal radius of the loop to Ihe conformal radius of the image of the loop under the map mew) = l /(w - z), and more precisely it is
t1:( LD = - log CR (q.z) - 10g CR (m( q ). z) . Kenyon and Wil son [KW041 predicted that the large-k moment generating func tion of t1 z(q ) is ( 14)
or equivale ntl y that the li miti ng probabil ity density function is given by the density functio n of the exit time of a standard Brownian eXCll rsi on started in the middle of the interval (- 21f I JK. 21f I JK), reweigh ted by a facto r of const x exp[ (K - 4)2 x 1(8K) [. (llli s equivalence follows from [BS02, Eq. 5.3.0.1 1.) Recall that the density function 1158
Conformal Radii for Confomlal Loop Ensembles
53
of Bf is given by the density function of the ex it time of a standard Brownian motion started in the middle of the interval (- 2Jr I./K, 2JT I./K), also reweighted by a factor of const x exp[(K _ 4)2 X/(8K)). These forms are highly suggestive, but currently wedo not know how to calc ulate the e]ectrical thickness using CLE... nor do we have a conceptual explanation for why these di stributions take these forms, References [BS02] [Carl}7] [CN06] [CN07] [Cf62] [CZ031 [Dub05] [Dup90] [EMOT53] [ FK 72]
[Gri06] [KN04] IKW04) [LSW02) [RY99j [ScIlOI]
[She06] [SmiO l j [SW08a] [SW08b]
Sorodin. A.N .. Salminen. p,: H(llldbu(lk of Bro"'Jli(lll Mo/iml-F(lc/s(llld Formu/(le, Probabi lity and its Applications. Basel: Birkhliuser Verlag. 2nd edition. 2002 Cardy. 1.: ADEand SLE. 1. Phys. A 40(7).1427-1438 (2007) Camia. F.. Ncwman. CM.: Two-dinlCnsional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1). 1-38 (2006) Camia. E. Newman. CM.: Critical percolation exploration path and SLEt,: a proof of convergencc. Probab. Theory Relatcd Fields 139(3-4). 47 3-5 19 (2007) Ciesiclski. Z .. Taylor. S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of thc samplc path. Tran s. AnlCr. Math. Soc. tOJ(3).434450(1962) Cardy. 1.. Ziff. R,M.: Exact results for the un iversal area distri bution of clustcrs in percolation. [sing. and POliS models. 1. Stat. Phys. 11 0(1 -2). 1- 33 (2003) DubMat.1.: 2005. Persona[ communication Dup[antier. B.: Exact fractal area of two-dinlCnsional vesicles. Phys. Rev. Lett. 64(4). 493 (199Q) Erd~lyi. A .. Magnus. W.. Oberheuinger. F.. Tri comi. F.G.: Higher Tr(lnscendell/(l{ FJIIlc/io",', Vol. L. New York: McGraw -Hili Book Co mpany. 1953. based. in pan. on notes left by Harry Bateman Fonuin. CM .. Kaste[cyn. P.W.: On the random-cluster model. I. Introduct ion and relation to Olher models, I'hysica 57. 536--564 (1972) Grimmett. G.: The R(llldom-CIr,sler Model. Volume J.n of Grundlehren der Mathematischen Wissenschaften [Fundamenta[ Princip[es of Mathematical Sciences]. Berlin: Spri nger- Verlag_ (2006) Kager. W.. Nienhu is. B,: A guide to stochastic L6wner evolution and it s applications, 1, Stat, Phys. t 15(5-6). I [49- 1229 (2004) Kenyon. R,W.. Wilson. D.B.: COllformal radii Of loop models. 2004. Manuscript Lawler. G. F.. Schramm. 0 .. Werner. W.: One-ami eXf'Onelllfor crilical 2D perCOllllion. Elcctron.1.l'robab. 7 : Paper No.2. 13 pp. (2002) Revuz. D.. Yor. M,: Coli/iII/lOllS Marlingaiesond Browniall MOli(JII. Vol ume 29J ofGrundlehren der Mathematischen Wissenschaften [FundanlCntal l'rinciples of Matllcmatical Sciences], Berlin: Springer-Verlag. third edition. 1999 Schramm. 0.: A pen:ol(liiollf(}rmllill. E[ectron. Cornm. I'robab. 6. 115- 120 (2001) Sheffield. S.: Exp/(JrOliOll lrees (llld "ollform(llloop en.•emble.f. hnp:llarxiv.orgfabslmath. PRI 0609167.2006 Duke Math. L to appear Smirnov. S.: Critical percolation in the plane: conformal invariance. Cardy's fonnula. scaling limit s. C R. Acad. Sci. Paris S~r. 1 Math. 3J3(3). 239-244 (2001) Sheffield. S .. Werner. W.: Confoml(l/ loop ensembles: COIIS/rllc/ion l'i(l/ooIJ-SOlIl's. 2008. in preparation Sheffield. S .. Werner. W.: COllforlll(l/loop ensembles: The M(lr/u",i(ln ch(lr(lc/eri:Plioll . 2008. in preparation
Communicated by M. Ai7.en man
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Conformally invariant scaling limits: an overview and a collection of problems Oded Schramm
Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented. Mathematics Subject Classification (2000). Primary 60K35; Secondary 82B20, 82B43, 30C35. Keywords. Statistical physics, conformal invariance, stochastic Loewner evolutions, percolation.
1. Introduction In the past several years, many predictions from physics regarding the large-scale behavior of random systems defined on a lattice in two dimensions have become accessible to mathematical study and proof. Of central importance is the asymptotic conformal invariance of these systems. It turns out that paths associated with these random configurations often fall into a one-parameter family of conformally invariant random curves called stochastic Loewner evolutions, or SLE. We start by motivating SLE through a simple mathematical model of percolation. After giving the definition of SLE, we present a narrative of recent developments. However, since there are good surveys on the subject in the literature [94], [36], [21], [52], [95], this introductory part of the paper will be short and cursory. The rest of the paper will consist of an annotated list of open problems in the subject.
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society
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1.1. Motivation and definition of SLE. To mot ivate SLE, we now discuss percolat ion. More specifi cally, we define one particul ar model of percolation in two di mensions. Fix a number p E [0, I]. Let w be a random subset of the set of vertices in the triangular grid TG , where for vertices v E V(TG ) the events v E It) are independent and have probability p. In percolation theory one srudies the connected components (a.k.a. cl usters) of the random subgraph of TG whose vertex sel is w and whose edges are the edges in TG connecting two e lements of w. Equivalent ly, one may study the connected components of the sel of white hexagons in Figure 1, where each hexagon in the hexagonal grid dual to TG represents a vertex of TG and hexagons corresponding to vertices in w are colored white. The reasons for considering this dual representatio n are that the fi gures come out nicer and that it makes some important definitions more concise.
Figure I. Si te percolation on (he triangular grid as re presented by colored hexagons.
The above percolation model is site (or vertex) percolation on the triangular grid. Likew ise, there is a bond (or edge) model, where o ne considers a random subgraph of a grid whose vertex set is the set of all vertices of the grid, but where each edge of the grid is in the percolati on subgraph with probability p, independently. Additionall y, there are various percolation models which are not based o n a Ian ice. Some of these wi ll be discussed in later sections. There is an important value Pc of the parameter p . which is the threshold for the ex istence of an unbounded cluster and is called the critical value of p . The actual value of Pc varies depending on the parti cular percolation model. For site percolati on on the triangular grid, as well as for bond percolation on the square grid, we have Pc = 1/2. This is a theorem of Kesten [43]. based o n earlier work by Harris [33], Russo l7 8] and Seymour and We lsh [84]. The underlying reason for this nice value of Pc is a duality which these two models have, though the precise form of the duality they exhibit is 1162
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different. For bond percolation on the triangular grid Pc = 2 sin (1l/1S) [98], whi le for site percolation on the square grid there is not even a prediction for the value of Pc, though rigorous and experimental estimates exi st. As P increases beyond Pc . the large scale behavior of percolation undergoes a rapid change. This is perhaps the mathematically simplest model of a phase transition. From now o n, we wil l foc us our attention on critical percolation, that is, percolat ion with p = Pc. which is in many ways the most interesting value of p. We now define and di scuss the percolation interface curve indicated in Figure 2. Consider a bou nded domain D in the plane ]R2 = C whose boundary is a simple closed curve. Lei a+ c aD be a proper arc on the boundary of D. Given e > 0 we may consider the collection of hexagons in a hexagonal grid of mesh e which intersect D. Each of these hexagons which meets 8+ we color white, each of the hexagons which meet aD but not a+ we co lor black, and each of the hexagons contained in D we color white or black with probabi lity 1/ 2, independently. In addition to whjte clu s~ ters (con nected com ponents of whi te hexagons) sometimes, black clusters are also considered. Percolat ion theory is the study of connected components of random sets, such as these clusters.
Fi gure 2. The interface associated with percolation.
For simpl icity, we assume that aD is sufficiently smooth and e is suffi ciently small so that the union of hexagons intersecting aD but not a+ is connected. There is a un ique (random) path {3, which is the common boundary of the white cl uster meeting a+ and the black cluster meeting aD. The law ILD. 3+.e of {3 is a probabi lity measure on the space of closed subsets of
i5
wi th the Hausdorff metric. Smirnov [87] proved that as e \.O the measure 1163
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J-L D,o+,£ converges weakly to a measure JLD J)+ , and thai J-L is conformally invariant,
in the following sense. If f: i5 --+ Df C )R.2 is a homeomorphi sm thai is analytic in D, then the push forw ard of /.LD. a+ under f is J-Lf( D)./(fJ+) . In other words, if E is small, then /({3) is a good approximation for the corresponding path defi ned using a hexagonal grid of mesh e in I(D) = 0 ' , This type of conformal invariance was believed to hold for many "critical" random systems in two dimensions. However, the o nly previous resu lt establishing conformal invariance for a random scaling limi t is Levy's theorem [64] stating that for two-dimensional Brownian motion, the scaling lim it of simple random wal k on Z} , is conformally invariant up to a time-change. Smimov's proof is very beautifu l, and the result is important, bUI describing the proofw il llhrow us too far off course (since forthi s paper percolation is just an exampl e model, nOI the primary topic). The interested reader is encouraged to consu lt [12], [30], [70J , [451 for backgrou nd in percolation and highl ights of percolation theory. An elegant simplification of parts of Smimov's proof has been discovered by Vincelll Beffara [II ]. A more detai led version of other parts of Smirnov's argument appears i" [19). Though th is was not the original inspiration, we w ill now use Smirnov's resu lt to motivale the defi nition of SLE. By conformal invariance, we may venture 10 understand fJ in Ihe domain of our choice. The si mplest situation turns oul to be when D = JH[ is the upper half plane and a+ is the positive real ray R+, as in Figure 3.
Figure 3. The percolation interface in the upper half plane.
(Though thi s domain D is unbounded, that does not cause any problems.) We may consider the di screte path fJ as a si mple path fJ: [0, T) ---+ !HI starting near 0 and satisfy ing Iimt_T IfJ(t )1 = 00 (where T is finite or infin ite) . We would like to learn about fJ by understanding the one-parameter fami ly of conformal maps g, mapping 1HI \ fJ([D, t]) onto 1HI. To faci litate th is, we must firs t 1164
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recall a few basic facts and discuss Loewner's theorem. At this point, assume only that {J is a simple path in H with {J(O) E JR and {J(t) f/. JR for t > O. The existence of the conformal maps gt: H \ {J([O, t]) -+ H is guaranteed by Riemann's mapping theorem. However, gt is not unique. In order to choose a specific gt for every t, we first require that gt (00) = 00. Schwarz reflection in the real axis implies that gt is analytic in a neighborhood of 00, and therefore admits a power series representation in liz, gt(z) = al z + ao + a-I Z-I + a-2 Z-2 + ... , valid for all z sufficiently large. Since gt maps the real line near 00 into the real line, it follows that aj E JR for all j, and because gt: H \ {J ([0, t]) -+ H, we find that al > O. We now pick a specific gt by imposing the so-called hydrodynamic normalization at 00, namely al = 1 and ao = O. This can clearly be achieved by post-composing with a map of the form z J-+ a Z + b, a > 0, b E R The coefficients aj of the series expansion of gt are now functions of t. It is not hard to verify that a-I (t) is a continuous, strictly increasing function of t. Clearly, go(z) = z and hence a-I (0) = O. We may therefore reparametrize {J so as to have a_I (t) = 2 t for all t > O. This is called the half-plane capacity parametrization of {J. With this parametrization, a variant of Loewner's theorem [65] states that the maps gt satisfy the differential equation
2 gt(z) - W(t)'
(1)
where W (t) := gt ({J (t)) is called the Loewner driving term. A few comments are in order. 1. Although gt is defined in H \ {J([O, t]), it does extend continuously to {J(t), and therefore Wet) is well defined. 2. If z = {J(s) for some s, then (1) makes sense only as long as t < s. That is to be expected. The domain of definition of gt is shrinking as t increases. A point z falls out of the domain of gt at the first time T = T z such that liminft,l'r gt(z) - W(t) = o. 3. The main point here is that information about the path {J is encoded in Wet), which is a path in JR. 4. The proof of (1) is not too hard. In [53, Theorem 2.6] a proof (of a generalization) may be found. We now return to the situation where {J is the percolation interface chosen according to fLlHI,lR+,e, parametrized by half-plane capacity. It is easy to see that in this case T = 00. Fix some s > O. Suppose that we examine the colors of only those hexagons that are necessary to determine {J ([0, s]). This can be done by sequentially testing the hexagons adjacent to {J starting from {J(O) as follows. Each time the already 1165
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determined arc of fJ meets a hexagon whose color has not yet been examined, we test the color (which permi ts us to extend the determi ned initial arc of f3 by at least one segment), unti l fJ([O, s]) has been determi ned. See Figure 4.
Figure 4. Initial segment of interface.
Now comes the main point. Let Ds be the unbounded component of the collection
at
of hexagons of undetermined color in lHI, and let be the subset of aDs lying on the boundary of hexagons of determined color whjte. Then the di stribution of the continuation P([s, (0)) of the interface given .8([0, s]) is f.LD,.a~.t . By Smirnov's theorem, if G: Ds --+ 1HI is the conformal map sati sfying the hydrodynamic normal ization, then the image under G of J.LDj.a~.t is close to I-LH .G(j~ ).t. (Actually, to j ustify th is. one needs a slightly stronger "uniform" vers ion of Smirnov's theorem. But here we want to convey the main ideas, and do not bother about being entirely precise.) Now, since D J, approximates IHI \ ,8([0, s]), it fo llows that G is very close to 8s and G(a~J is close to [W(s) , (0) = [gs(,8(s» , (0). Therefore, in the limit as e '\" 0 , we have for,8 sampled according to /-LH.R+ that given J3([0, s]) the distribution of gs 0 ,8([s, 00» (which is the conformal image of the continuation of the path) is /-LH.R+ translated by We,). The Loewner driving term of the path t ~ gs 0 J3(s + 1) is W(s + 1), because gs+t 0 g;1 maps IHI \ gs(J3([s, s + tD) onto illI. The conclusion of the previous paragraph therefore implies that g iven (W(t) t E lO, sD the distribution of the continuation of W is identical to the original distribution of W translated to start at W(s). This is a very strong property. Indeed, for every II E Nand t > 0 we may write Wet) = :L = 1(WU t / /I) - W eu - 1)//11»), which by the above is a sum of II independent identicall y distributed random variables. If we assume that the variance of Wet) is fini te, then it is also the sum of the variances of the summands. By the central limit theorem, Wet) is therefore a Gaussian random variable. By symmetry, Wet) has the same distribution as - W(t), and so W(t) is a centered Gaussian. It now eas il y follows that there is some constant K ::: 0 such that Wet) has the same distribution as 8(K t), where 8 is one-dimensional Brownian motion starting at 8(0) = O. Us ing results from the theory of stochastic processes (e.g., the characterization of cont inuous martingales as time-changed Brownian motion), the same conclusion can be reached
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while replacing the assumption that Wet) has finite variance with the continuity of Wet) in t. We have just seen that Smirnov's theorem implies that the Loewner driving term of a sample from /L1HI,lE.+ is B(K t) for some K ::::: 0. This should serve as adequate motivation for the following definition from [79]. Definition 1.1. Fix some K ::::: 0, and let gt be the solution of Loewner's equation (1) satisfying go(z) = z with Wet) = B(K t), where B is standard one-dimensional Brownian motion starting at B(O) = 0. Then (gt : t ::::: 0) is called chordal stochastic Loewner evolution with parameter K or SLEK • Of course, SLEK is a random one-parameter family of maps; the randomness is entirely due to the Brownian motion. It has been proven [76], [59] that with probability 1 there is a (unique) random continuous path y (t) such that for each t ::::: the domain of definition of gt is the unbounded component of 1HI \ y([O, tD. The path is given by yet) = g;-l(W(t)), but proving that g;-l (W (t)) is well defined is not easy. It is also known [76] that a.s. yet) is a simple path if and only if K :::: 4 and is space-filling if and only if K ::::: 8. Sometimes the path y itself is called SLEK • This is not too inconsistent, because gt can be reconstructed from y([O, tD and vice versa. If D is a simply connected domain in the plane and a, bEaD are two distinct points (or rather prime ends), then chordal SLE from a to b in D is defined as the image of y under a conformal map from 1HI to D taking to a and 00 to b. Though the map is not unique, the choice of the map does not effect the law of the SLE in D. This follows from the easily verified fact that up to a rescaling of time, the law of the SLE path is invariant under scaling by a positive real constant, as is the case for Brownian motion. The reason for calling the SLE "chordal" is that it connects two boundary points of a domain D. There is another version of SLE, which connects a boundary point to an interior point, called radial SLE. Actually, there are a few other variations, but they all have similar definitions and analogous properties.
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1.2. A historical narrative. In this subsection we list some works and discoveries related to SLE and random scaling limits in two dimensions. The following account is not comprehensive. Some of the topics not covered here are discussed in Wendelin Werner's [97] contribution to this IeM proceedings. We start by very briefly discussing the historical background. In the survey paper [49], Langlands, Pouliot and Saint-Aubin present a collection of intriguing predictions from statistical physics. They have discussed these predictions and some simulation data with Aizenman, which prompted him to conjecture that the critical percolation crossing probabilities are asymptotically conformally invariant (see [49D. This means that the probability Qs(D, ai, a2) that there exists a critical percolation cluster in a domain D C
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invariant, namely, Q(D, aI, a2) = Q(j(D), f(al), f(a2)) if f is a homeomorphism from D to feD) that is conformal in D. This led John Cardy [20] to propose his formula (involving hypergeometric functions) for the asymptotic crossing probability in a rectangle between two opposite edges. The survey [49] highlighted these predictions and the role of the conjectured conformal invariance in critical percolation, as well as several other statistical physics models in two dimensions. Prior to SLE there were attempts to use compositions of conformal slit mappings and even Loewner's equation in the study of diffusion limited aggregation (DLA). DLA is a random growth process, which produces a random fractal and is notoriously hard to analyse mathematically. (See [100], [4] for a definition and discussion of DLA.) Makarov and Carleson [22] used Loewner' s equation to study a much simplified deterministic variant of DLA, which is not fractal, and Hastings and Levitov [34] have used conformal mapping techniques for a non-rigorous study of more realistic versions of DLA. Given that the fractals produced by DLA are not conformally invariant, it is not too surprising that it is hard to faithfully model DLA using conformal maps. Harry Kesten [44] proved that the diameter of the planar DLA cluster after n steps grows asymptotically no faster than n 2/ 3 , and this appears to be essentially the only theorem concerning two-dimensional DLA, though several very simplified variants of DLA have been successfully analysed. The original motivation for SLE actually came from investigating the Loop-erased random walk (a.k.a. LERW), which is a random curve introduced by Greg Lawler [51]. Consider some bounded simply-connected domain D in the plane. Let G = G(D, s) be the sub graph of a square grid of mesh s that falls inside D and let Va be the set of vertices of G that have fewer than 4 neighbors in D. Suppose that 0 ED, and let 0 be some vertex of G closest to O. Start a simple random walk on G from 0 (at each step the walk jumps to any neighbor of the current position with equal probability). We keep track of the trajectory of the walk at each step, except that every time a loop is created, it is erased from the trajectory. The walk terminates when it first reaches Va, and the loop-erased random walk from 0 to Va is the final trajectory. See Figure 5, where D is a disk. The LERW is intimately related to the uniform spanning tree. In particular, if we collapse Va to a single vertex Va and take a random spanning tree of the resulting projection of G, where each possible spanning tree is chosen with equal probability, then the unique path in the tree joining 0 to Va (as a set of edges) has precisely the same law as the LERW from 0 to Va [74]. This is not a particular property of the square grid, the corresponding analog holds in an arbitrary finite graph. In the other direction, there is a marvelous algorithm discovered by David Wilson [99] which builds the uniform spanning tree by successively adding loop-erased random walks. The survey [66] is a good window into the beautiful theory of uniform spanning trees and forests. Using sophisticated determinantcalculations and Temperley's bijection between the collection of spanning trees and a certain collection of dimer tilings (which is 1168
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Figure 5. The LERW in a disk.
special to the planar setting), Richard Kenyon [39], [41], [40] was able to calculate several properties of the LERW. For example, it was shown that the variance of the winding number ofthe above LERW in G(D, 8) is (2 + 0(1)) log(I/8) as 8 \.. 0, and that the growth exponent for the number of edges in a LERW is 5/4. In [79] it was shown that if the limit of the law of the LERW as 8 \.. 0 exists and is conformally invariant, then it is a radial SLE2 path, when parametrized by capacity. (See subsection 2.1 for a description of two alternative topologies on spaces of probability measures on curves, for which this convergence may be stated.) In broad strokes, the reason why it should be an SLE path is basically the same as the argument presented above for the percolation interface. Two important properties of the percolation interface scaling limit were crucial in the above argument: conformal invariance and the following Markovian property. If we condition on an initial segment of the path, the remainder is an instance of the path in the domain slitted by the initial segment starting from the endpoint of the initial segment. Conformal invariance was believed to hold for the LERW scaling limit, while the Markovian property does hold for the reversal of the LERW. The identification of the correct value of the parameter IC as 2 is based on Kenyon's calculated LERW winding variance growth rate and a calculation of the variance of the winding number of the radial SLEK path truncated at distance 8 from the interior target point. The latter grows like (IC + 0(1)) log(1/8). It was also conjectured in [79] that the percolation interface discussed above converges to SLE6. The identification of the parameter IC as 6 in this case was based on Cardy's formula [20] and the verification that the corresponding formula holds for SLEK if and only if IC = 6. The percolation interface satisfies the following locality property. The evolution of the path (given its past) does not depend on the shape of the domain away from 1169
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the current location of the endpoint of the path. Though this is essentially obvious, it should be noted that other interesting paths (such as the LERW scaling limit) do not satisfy locality. Another process that clearly satisfies locality is Brownian motion. Greg Lawler and Wendelin Werner [61], [62] studied the intersection exponents of planar Brownian motion and the relations between them. An example of an intersection exponent is the unique number ~ (l, 1) such that the probability that the paths of two independent Brownian motions started at distance 1 apart within the unit disk U and stopped when they first hit the circle R au do not intersect one another is R-W,I)+o(l) as R -+ 00. At the time, there were conjectures [27] for the values of many of these exponents, which were rational numbers, but only two of these could be proved rigorously (not accidentally, those had values 1 and 2). These exponents encode many fundamental properties of Brownian motion. For example, Lawler [50] showed that the dimension of the outer boundary of planar Brownian motion stopped at time t = 1, say, is 2 (l - a) for a certain intersection exponent a. Lawler and Werner [61], [62] have proved certain relations between the intersection exponents, and have shown that any process which like Brownian motion satisfies conformal invariance and a certain version of the locality property necessarily has intersection exponents that are very simply related to the Brownian exponents. Since SLE6 was believed to be the scaling limit of the percolation interface, it should satisfy locality. It is also conformally invariant by definition. Thus, the Brownian exponents should apply to SLE6. Indeed, in a series of papers [53], [54], [55] Lawler, Werner and the present author proved the conjectured values of the Brownian exponents by calculating the corresponding exponents for SLE6 (and using the previous work by Lawler and Werner). Very roughly, one can say that the reason why the exponents of SLE are easier to calculate than the Brownian exponents is that the SLE path, though it may hit itself, does not cross itself. Thus, the outer boundary of the SLE path is drawn essentially in chronological order. Later [58] it became clear that the relation between SLE6 and Brownian motion is even closer than previously apparent: the outer boundary of Brownian motion started from 0 and stopped on hitting the unit circle au has the same distribution as the outer boundary of a variant of SLE6. Lennart Carleson observed that, assuming conformal invariance, Cardy's formula is equivalent to the statement that Q(D, aI, a2) = length(a2) when D is an equilateral triangle of sidelength 1, al is its base, and a2 c aD is a line segment having the vertex opposite to al as one of its endpoints. Smirnov [87] proved Carleson's form of Cardy's formula for critical site percolation on the triangular lattice (that is, the same percolation model we have described above) and showed that crossing probabilities between two arcs on the boundary of a simply connected domain are asymptotically conformally invariant. As a corollary, Smirnov concluded that the scaling limit of the percolation interface exists and is equal to the SLE6 path. This connection enabled proving many conjectures about this percolation model. For example, the prediction [24], [73] that the probability that the cluster of the origin has diameter larger 1170
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than R decays like P[the origin is in a cluster of diameter ~ R]
= R- 5/ 4S +o (1)
(2)
as R --+ 00 was proved [56]. This value 5/48 is an example of what is commonly referred to as a critical exponent. Building on earlier work by Kesten and others, as well as on Smirnov's theorem and SLE, Smirnov and Werner [88] were able to determine many useful percolation exponents. Julien Dubedat [25] has used SLE to prove Watts' [92] formula for the asymptotic probability that in a given rectangle there are both a white horizontal and vertical crossing (for the above percolation model at P = Pc = 1/2). The next process for which conformal invariance and convergence to SLE was established is the LERW [59]. Contrary to Smirnov's proof for percolation, where convergence to SLE was a consequence of conformal invariance, in the case of the LERW the proof establishes conformal invariance as a consequence of the convergence to SLE2. More specifically, the argument in [59] proceeds by considering the Loewner driving term of the discrete LERW (before passing to the limit) and proving that the driving term converges to an appropriately time-scaled Brownian motion. The same paper also shows that the uniform spanning tree scaling limit is conformally invariant, and the Peano curve associated with it (essentially, the boundary of a thickened uniform spanning tree) converges to SLEs. Another difference between the results of [87] and [59] is that while the former is restricted to site percolation on the triangular lattice, the results in [59] are essentially lattice independent. Figure 5 above shows a fine LERW, which gives an idea of what an SLE2 looks like. Likewise, Figure 6 shows a sample of an initial segment of the uniform spanning tree Peano curve in a rectangular domain. Note that the curve is space filling, as is SLEs.
Figure 6. An initial segment of the uniform spanning tree Peano path. 1171
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Meanwhile, Gady Kozma [47] came up with a different proof that the LERW scaling limit exists. Although Kozma's proof does not identify the limit, it has the advantage of generalizing to three dimensions [48]. There are two discrete models for which convergence to SLE4 has been established by Scott Sheffield and the present author. These models are the harmonic explorer [80] and the interface ofthe discrete Gaussian free field [81]. The discrete and continuous Gaussian free fields (a.k.a. the harmonic crystal) play an important role in the heuristic physics analysis of various statistical physics models. The discrete Gaussian free field is a probability measure on real valued functions defined on a graph, often a piece of a lattice. If, for example, the graph is a triangulation of a domain in the plane, an interface is a curve in the dual graph separating vertices where the function is positive from vertices where the function is negative. See [85] or [81] for further details, and see Figure 7 for a simulation of the harmonic explorer, and therefore an approximation ofSLE4.
Figure 7. The harmonic explorer path.
Sheffield also announced work in progress connecting the Gaussian free field with SLEK for other values of K. The basic idea is that while SLE4 may be thought of as a curve solving the equation h = 0, where h is the Gaussian free field, for other K, the SLEK curve may be considered as a solution of c winding(y [0, t])
= h(y(t)),
(3)
where c is a constant depending on K. When K = 4, the corresponding constant c is zero, which reduces to the setting of [81]. Alternatively, (3) can be heuristically written as c y'(s) = exp(i h(y(t)), where s is the length parameter of y. However, we stress that it is hard to make sense of these equations, for the Gaussian free field 1172
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is not a smooth function (in fact, it is not even a function but rather a distribution). Likewise, the SLE path is not rectifiable and its winding at most points is infinite. As mentioned above, there are several different variants of SLE in simply connected domains: chordal, radial, as well as a few others, which we have not mentioned. These variants are rather closely related to one another [83]. There are also variants defined in the multiply-connected setting [101], [6], [5], [7]. One motivation for this study comes from statistical physics models, which are easy to define on multiply connected domains. Since one can easily vary the boundary conditions on different boundary components of the domain, it is clear that there is often more than one reasonable choice for the definition of the SLE path. Finally, we mention an intriguing connection between Brownian motion and SLEK fOrK E (8/3,4]. There is the notion of the Brownian loop soup [63], which is a Poisson measure on the space of Brownian motion loops. According to [93], the boundaries of clusters of a sample from the loop soup measure with intensity c are SLEK-like paths, where K = K(C) E (8/3,4]. The proof is to appear in a future joint work of Sheffield and Werner. The above account describes some of the highlights of the developments in the field in the past several years. The rest of the paper will be devoted to a description of some problems where we hope to see some future progress. Some of these problems are obvious to anyone working in the field (though the solution is not obvious), while others are borrowed from several different sources. A few of the problems appear here for the first time. The paper [76] contains some additional problems.
2. Random processes converging to SLE As we have seen, paths associated with several random processes have been proved to converge to various SLE paths. However, the list of processes where the convergence is expected but not proved yet is longer. This section will present questions of this sort, most of which have previously appeared in the literature. The strategy of the proofs of convergence to SLE in the papers [59], [80], [81] is very similar. In these papers, a collection of martingales with respect to the filtration given by the evolution of the curve is used to gain information about the Loewner driving term of the discrete curve. Although such a proof is also possible for the percolation interface (using Cardy's formula), this technique was not available at the time, and Smirnov used instead an argument which uses the independence properties of percolation in an essential way and is therefore not likely to be applicable to many other models. Thus, it seems that presently the most promising technique is the martingale technique from [59].
2.1. Notions of convergence. To be precise, we must describe the meaning of these scaling limits. In fact, there are at least two distinct reasonable notions of convergence, which we now describe. Suppose that Yn are random paths in the closed upper half 1173
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plane 1HI starting from O. Consider the one-point compactification t = c U {(X)} of C = JR2, which may be thought of as the sphere S2. The law of Yn may be thought of as a Borel probability measure on the Hausdorff space of closed nonempty subsets of t. Since that Hausdorff space is compact, the space of Borel probability measures on it is compact with respect to weak convergence of measures [26]. We may say that Yn converges in the Hausdorff sense to a random set Y C 1HI U {(X)} if the law of Yn converges weakly to the law of y. A similar definition applies to curves in the closure of a bounded domain DeC. The above stated instances of convergence to SLE hold with respect to this notion. However, in the case of convergence to SLEs, this does not mean very much, for SLEs fills up the domain. The second notion of convergence is stronger. Suppose that each Yn is a.s. continuous with respect to the half-plane capacity parametrization from 00. (This holds, in particular, if Yn is a.s. a (continuous) simple path.) If d is a metric on t = c U {(X)} compatible with its topology, then we may consider the metric d*(fh, (h) :=
sup d(th(t),{h(t)) tE[O,ex:!)
on the space of continuous paths defined on [0, (0). We may say that Yn converges to a random path Y weakly-uniformly if the law of Yn converges weakly to the law of Y in the space of Borel measures with respect to the metric d*. This implies Hausdorff convergence. Since d* is finer than the Hausdorff metric, there are more functions from the space of paths to JR that are continuous with respect to d* than with respect to the Hausdorff metric. Consequently, weakly-uniform convergence is stronger than Hausdorff convergence. In all the results stated above saying that some random path converges to SLE, the convergence is weakly-uniform when the paths are parametrized by capacity or half-plane capacity (depending on whether the convergence is to radial or chordal SLE, respectively). In the following, when we ask for convergence to SLE, we will mean weaklyuniform convergence. However, weaker nontrivial forms of convergence would also be very interesting. 2.2. Self avoiding walk. Let G be either the square, the hexagonal or the triangular grid in the plane, positioned so that 0 is some vertex in G. For n E N consider the uniform measures on all self avoiding n-step walks in G that start at 0 and stay in the upper half plane. It has been shown in [60] that when G = 7L,2 the limiting measure as n -+ 00 exists. (The same proof probably applies for the other alternatives for G, provided that G is positioned so that horizontal lines through vertices in G do not intersect the relative interior of edges of G which they do not contain.) Problem 2.1 ([60]). Let Y be a sample from the n -+ 00 limit of the uniform measure on n-step self avoiding paths in the upper half plane described above. Prove that the limit as s ~ 0 of the law of s Y exists and that it is SLEsj3. The convergence may be considered with respect to either of the two topologies discussed in subsection 2.1. 1174
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In [60] some consequences of this convergence are indicated, as well as support for the conjecture. There are some indications that the setting of the hexagonal lattice is easier: the rate of growth of the number of self avoiding paths on the hexagonal grid is predicted [72] to be (2 + -J2 + 0(1) r/2; no such prediction exists for the square grid or triangular lattice. In dimensions d > 4 Takashi Hara and Gordon Slade [32] proved that the scaling limit of self-avoiding random walk is Brownian motion. This is also believed to be the case for d = 4. See [67] for references and further background on the self-avoiding walk. 2.3. Height models. There is a vast collection of height model interfaces that should converge to SLE4. The one theorem in this regard is the convergence of the interface of the Gaussian free field [81]. This was motivated by Kenyon's theorem stating that the domino tiling height function converges to the Gaussian free field [42] and by Kenyon's conjecture that the double domino interface converges to SLE4 (see [76] for a statement of this problem). The domino height function is a function on Z2 associated with a domino tiling (see [42]). Its distribution is roughly (ignoring boundary issues) the uniform measure on functions h: Z2 ---+ Z such that h(O, 0) = 0,
0
h (x, y ) mo d 4 =
1 I
2
3
x, y even, x odd y even, x, y odd, x even y odd,
and Ih(z) - h(z')1 E {I, 3} if Iz - z'l = 1, z, Z' E Z2. Let D be a bounded domain in the plane whose boundary is a simple path in the triangular lattice, say. Let a+ and a_ be complementary arcs in aD such that the two common endpoints of these arcs are midpoints of edges. Consider the uniform measure on functions h taking odd integer values on vertices in D such that h = 1 on a+,h = -1 ona_, and Ih(v)-h(u)1 E {O, 2}forneighborsv, u. We may extend such a function h to D by affine interpolation within each triangle, and this interpolation is consistent along the edges. There is then a unique connected path y that is the connected component of h -1 (0) that contains the two endpoints of each of the two arcs a±. Problem 2.2. Is it true that the path y tends to SLE4? Does the law of h converge to the Gaussian free field? The convergence we expect for h is in the same sense as in [42]. Note that if we restrict in the above the image of h to be {I, -I}, we obtain critical site percolation on the triangular grid, and the limit of the corresponding interface is in this case SLE6. 1175
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Now suppose that (D, 3+, 3_) is as above. Let A E (0, 1/2] be some constant and consider the uniform measure on functions h taking real values on vertices in D such that h = A on 3+, h = -A on 3_ and Ih(v) - h(u)1 :s 1 for every edge [v, u].
Problem 2.3. Is it true that for some value of Athe corresponding interface converges to SLE4? Does the law of h converge in some sense to the Gaussian free field? In the case of the corresponding questions for the discrete Gaussian free field, there is just one constant A such that the interface converges to SLE4. For other choices of A the interface converges to a well-known variant of SLE4 [81]. There are some restricted classes of height models for which convergence to the Gaussian free field is known [71]. It may still be very hard to prove that the corresponding interface converges to SLE4. One interesting problem of this sort is the following.
Problem 2.4 ([81]). If we project the Gaussian free field onto the subspace spanned by the eigenfunctions of the Dirichlet Laplacian with eigenvalues in [-r, r] and add the harmonic function with boundary values ±A on 3±, does the corresponding interface converge to SLE4 as r --+ 00 when A is chosen appropriately? The problem is natural, because the Gaussian free field is related to the Dirichlet Laplacian. In particular, the projections of the field onto the spaces spanned by eigenfunctions with eigenvalues in two disjoint intervals are independent. 2.4. The Ising, FK, and 0 (n) loop models. The Ising model is a fundamental physics model for magnetism. Consider again a domain D adapted to the triangular lattice and a partition 3D = 3+ U 3_ as in subsection 2.3. Now consider a function h that take the values ± 1 on vertices in D such that h is 1 on 3+ and -Ion 3_. On the collection of all such functions we put a probability measure such that the probability for a given h is proportional to e- 2{3k, where f3 is a parameter and k is the number of edges [v, u] such that h(v) i= h(u). This is known as the Ising model and the value associate to a vertex is often called a spin. It is known that the critical value f3e (which we do not define here in the context of the Ising model) for f3 satisfies e 2{3 = J3 (see [69], [35]). Again, the interface at the critical f3 = f3e is believed to converge to an SLE path, this time SLE3. For f3 E [0, f3e) fixed, the interface should converge to SLE6. Note that when f3 = 0, the model is again identical to critical site percolation on the triangular grid, and the interface does converge to SLE6.
Problem 2.5. Prove that when f3 = f3e, the interface converges to SLE3 and when f3 E (0, f3e) to SLE6. When f3 > f3e we do not expect convergence to SLE, and do not expect conformal invariance. The interface scaling limit is in this case a straight line segment if the domain is convex [75] (see also [29]). The Fortuin-Kasteleyn [28] (FK) model (a.k.a. the random cluster model) is a probability measure on the collection of all subsets of the set of edges E of a finite 1176
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=
529
(V, E). In the FK model, the measure of each weE is proportional to
(p/(l - p) )Iwl qC, where q > 0 and p E (0, 1) are parameters, Iwl is the cardinality of w, and c is the number of connected components of the subgraph (V, w). The FK model is very closely related to the well known Potts model [8], which is a generalization of the Ising model. Many questions about the Potts model can be translated to questions about the FK model and vice versa. On the grid 7l 2 , when p = .j7j/ (1 + .j7j) the FK model satisfies a form of selfduality.
Problem 2.6 ([76]). Prove that when q E (0,4) and p = .j7j/(1 +.j7j), the interface of the FK model on 712 with appropriate boundary conditions converges to SLEK , where K = 4 n / cos -1 ( - .j7j/2). (See [76] for further details.) The O(n) loop model on a finite graph G = (V, E) is a measure on the collection of subgraphs of G where the degree of every vertex in the subgraph is 2. (The subgraph does not need to contain all the vertices.) The probability of each such sub graph is proportional to x e n C , where c is the number of connected components, e is the number of edges in the subgraph, and x, n > 0 are parameters. When n is a positive integer, the 0 (n) loop model is derived from the 0 (n) spin model, which is a measure on the set of functions which associate to every vertex a unit vector in ]Rn. In order to pin down a specific long path in the 0 (n) loop model, we pick two points on the boundary of the domain and require that in the random subgraph the degrees of two boundary vertices near these two points be I, while setting the degrees of all other boundary vertices to 0, say. Then the measure is supported on configurations with one simple path and a collection of loops. 1/ 2 , Now we specialize to the hexagonal lattice. Set xc(n) := (2 + which is the conjectured critical parameter [72].
,J2=11r
Problem 2.7 ([36]). Prove that when n E [0,2] and x = xc(n) (respectively, x > xc(n» the scaling limit of the path containing the two special boundary vertices is chordalSLEK , whereK E [8/3,4] (respectively, K E [4,8]) andn = -2 cos(4n/K). When x < xc(n), we expect the scaling limit to be a straight line segment (if the domain is convex). The fact that at n = 1 we get the same limits as for the Ising model is no accident. It is not hard to see that the 0 (1) loop measure coincides with the law of boundaries of Ising clusters. See [36] for further details. Similar conjectures should hold in other lattices. However, the values of the critical parameters are expected to be different. 2.5. Lattice trees. We now present an example of a discrete model where we suspect that perhaps conformal invariance might hold. However, we do not presently have a candidate for the scaling limit. Fix n E N+, and consider the collection of all trees contained in the grid G that contain the origin and have n vertices. Select a tree T from this measure, uniformly at random. 1177
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Problem 2.8. What is the growth rate of the expected diameter of such a tree? If we rescale the tree so that the expected (or median) diameter is 1, is there a limit for the law of the tree as n ---+ oo? What are its geometric and topological properties? Can the limit be determined? It would be good to be able to produce some pictures. However, we presently do not know how to sample from this measure.
Problem 2.9. Produce an efficient algorithm which samples lattice trees approximately uniformly, or prove that such an algorithm does not exist. See [86] for background on lattice trees and for results in high dimensions and [17] for an analysis of a related continuum model.
2.6. Percolation interface. It is natural to try to extend the understanding of percolation at Pc to percolation at a parameter p tending to Pc. One possible framework is as follows. Fix a parameter q E (0, 1). Suppose that in Figure 3 with small mesh c: > 0 we choose p (c:) so that the probability to have a left to right crossing of white hexagons in some fixed 1 x 1 square in the upper half plane is q at percolation parameter p = p(c:). The corresponding interface will still be an unbounded path starting at 0, but its distribution will be different from the interface at p = 1/2 if q t= 1/2. Thus, it is natural to ask Problem 2.10 (Lincoln Chayes (personal communication)). What is the scaling limit of the interface as c: '\i. 0 and p = p (c:) if q t= 1/2 is fixed? This problem is also very closely related to a problem formulated by F. Camia, L. Fontes and C. Newman [18]. Site percolation on the triangular lattice is only one of several different models for percolation in the plane. Among discrete models, widely studied is bond percolation on the square grid. As for site percolation on the triangular lattice, the critical probability is again Pc = 1/2. At present, Smimov's proof does not work for bond percolation on the square grid. The proof uses the invariance of the model under rotation by 2 n /3. Thus, the following problem presents itself.
Problem 2.11. Prove Smimov's theorem for critical bond percolation on Z2. Some progress on this problem has been reported by Vincent Beffara [9]. There are other natural percolation models which have been studied. Among them we mention Voronoi percolation and the boolean model. In Voronoi percolation one has two independent Poisson point processes in the plane, Wand B, with intensities p and 1- p, respectively. Let Wbe the closure of the set of points inlR2 closer to W than to B, and let Bbe the closure of the set of points closer to B. The set Wis a sample from Voronoi percolation at parameter p. Some form of conformal invariance was proved for Voronoi percolation [14], but the version proved does not imply convergence 1178
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to SLE. It is neither stronger nor weaker than the conformal invariance proved by Smirnov. Notable recent progress has been made for Voronoi percolation by Bollobas and Riordan [15], who established the very useful Russo-Seymour-Welsh theorem, as well as Pc = 1/2 for Voronoi percolation. Problem 2.12. Prove Smirnov's theorem for Voronoi percolation.
The boolean percolation model (a.k.a. continuum percolation) can be defined by taking a Poisson set of points W C ]R2 of intensity I and letting W be the set of points in the plane at distance at most r from W. Here, r is the parameter of the modeL (Alternatively, one may fix r = 1, say, and let the intensity of the Poisson process be the parameter, but this is essentially the same, by scaling.) The RussoSeymour-Welsh theorem is known for this model [1], [77], but the critical value of the parameter has not been identified. A nice feature which the model shares with Voronoi percolation is invariance under rotations. Problem 2.13. Prove Smirnov's theorem for boolean percolation.
3. Critical exponents The determination of critical exponents has been one motivation to prove conformal invariance for discrete models. For example, den Nijs and Nienhuis predicted [24], [73] that the probability that the critical percolation cluster of the origin has diameter larger than R is R- 5/ 48 +o (l) as R ---+ 00. Likewise, the probability that a given site in the square [- R, R]2 is pivotal for a left-right crossing of the square [-2 R, 2 R]2 was predicted to be R- 5/ 4+o (l). (Here, pivotal means that the occurrence or nonoccurrence of a crossing would be modified by flipping the status of the site.) These and other exponents were proved for site percolation on the triangular grid using Smirnov's theorem and SLE [56], [88]. The determination of the exponents is very useful for the study of percolation. Richard Kenyon [39] calculated by enumeration techniques involving determinants the asymptotics of the probability that an edge belongs to a loop-erased random walk. The probability decays like R- 3/ 4+o (1) when the distance from the edge to the endpoints of the walk is R. However, Kenyon's estimate is much more precise; he shows that, in a specific domain, R 3/ 4 times the probability is bounded away from zero and infinity, and in fact estimates the probability as f R- 3/4 (1 + 0(1») as R ---+ 00, where f is an explicit function of the position of the edge. Thus, it is natural to ask for such precise estimates for the important percolation events as well. Namely, Problem 3.1. Improve the estimates R- 5/ 48 +o (1) and R- 5/ 4+o (1) mentioned above (as well as other similar estimates) to more precise formulas. It would be especially nice to obtain estimates that are sharp up to multiplicative constants. 1179
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In addition to the case of the loop-erased random walk mentioned above, estimates up to constants are known for events involving Brownian motions [57]. The difficulty in getting more precise estimates is not in the analysis of SLE. Rather, it is due to the passage between the discrete and continuous setting. Consequently, the above problem seems to be related to the following. Problem 3.2. Obtain reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE. There are still critical exponents which do not seem accessible via an SLE analysis. For example, we may ask Problem 3.3. Calculate the number ex such that on the event that there is a left-right crossing in critical percolation in the square [0, R]2, the expected length of the shortest crossing is Ra+o(l). Ziff [102] predicts an exponent which is related to this ex, but it seems that there is currently no prediction for the exact value of ex.
4. Quantum gravity Consider the uniform measure ILn on equivalence classes of n-vertex triangulations of the sphere, where two triangulations are considered equivalent if there is a homeomorphism of the sphere taking one to the other. One may view a sample from this measure with the graph metric as a random geometry on the sphere. Such models go under the name "quantum gravity" in physics circles. One may also impose statistical physics models on such random triangulations. For example, the sample space may include such a triangulation (or rather, equivalence class of triangulations) together with a map h from the vertices to {-I, I}. The measure of such a pair may be taken proportional to a k , where k is the number of edges [v, u] in the triangulation for which h(v) i= h(u) and a > is a parameter. Thus, we are in effect considering a triangulation weighted by the Ising model partition function. (The partition function is in this case the sum of all the weights of such functions h on the given triangulation.) Likewise, one may weight the triangulation by other kinds of partition functions. In some cases it is easier to make a heuristic analysis of such statistical physics models in the quantum gravity world than in the plane. The enigmatic KPZ formula of Knizhnik, Polyakov and Zamolodchikov [46] was used in physics to predict properties of statistical physics in the plane from the corresponding properties in quantum gravity. Basically, the KPZ formula is a formula relating exponents in quantum gravity to the corresponding exponents in plane geometry. To date, there has been progress in the mathematical (as well as physical) understanding of the statistical physics in the plane as well as in quantum gravity [2], [3], [16]. However, there is still no mathematical understanding of the KPZ formula. In fact, the author's understanding of KPZ is too weak to even state a concrete problem.
°
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However, we may ask about the scaling limit of /Ln. There has been significant progress lately describing some aspects of the geometry of samples from /Ln [2], [23]. In particular, it has been shown by Chassaing and Schaeffer [23] that if Dn is the graphmetric diameter of a sample from /Ln, then Dn / n -1 /4 converges in law to some random variable in (0, (0). However, the scaling limit of samples from /Ln is not known. On the collection of compact metric spaces, we may consider the Gromov-Hausdorff distance dGH(X, Y), which is the infimum of the Hausdorff distance between subsets X* and Y* in a metric space Z* over all possible triples (X*, Y*, Z*) such that Z* is a metric space, X*, Y* C Z*, X is isometric with X* and Y is isometric with Y*. Let Xn be a sample from /Ln, considered as a metric space with the graph metric scaled by n- 1/ 4 , and let /L~ denote the law of X n .
Problem 4.1. Show that the weak limit limn ---+ oo /L~ with respect to the GromovHausdorff metric exists. Determine the properties of the limit. Note that [68] proves convergence of samples from /Ln, but the metric used there on the samples from /Ln is very different and consequently a solution of Problem 4.1 does not seem to follow.
5. Noise sensitivity, Fourier spectrum, and dynamical percolation The indicator function of the event of having a percolation crossing in a domain between two arcs on the boundary is a boolean function of boolean variables. Some fundamental results concerning percolation are based on general theorems about boolean functions. (One can mention here the BK inequality, the Harris-FKG inequality and the Russo formula. See, e.g. [30].) Central to the theory of boolean functions is the Fourier expansion. Basically, if I: {-I, l}n ---+ lR is any function of n bits, the Fourier-Walsh expansion of I is I(x)
=
L
j(s) Xs(x),
Sc[n]
where Xs(x) = njESXj for S C [n] = {I, ... , n}. When we consider {-I, l}n with the uniform probability measure, the collection {XS : S C [n]} forms an orthonormal basis for L2({ -1, l}n). (Often other measures are also considered.) If we suppose that II I 112 = 1 (in particular, this holds if I: {-I, 1}2 ---+ {-I, I}), then the Parseval identity gives LSc[n] j(S)2 = 1. Thus, we get a probability measure /LIon 2[n] = {S : S C [n]} for which /L/({S}) = j(S)2. The map S t-+ lSI, assigning to each S C [n] its cardinality pushes forward the measure /LI to a measure iLion {O, 1, ... ,n}. This measure iLl may be called the Fourier spectrum of I. The Fourier spectrum encodes important information about I, and quite a bit of research on the subject exists [37]. For instance, one can read off from iLl the sensitivity of I to noise (see [13]). 1181
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When I is a percolation crossing function (i.e., 1 if there is a crossing, -1 otherwise), the corresponding index set [n] is identified with the collection of relevant sites or bonds, depending if it is a site or bond modeL Though there is some partial understanding of the Fourier spectrum of percolation [82], the complete picture is unclear. For example, if the domain is approximately an e x e square in the triangular lattice, then for the indicator function Ie of a crossing in critical site percolation it is known [82] that for every a < 1/8 (4)
as e ---+ 00. This is proved using the critical exponents for site percolation as well as an estimate for the Fourier coefficients of general functions (based on the existence of an algorithm computing the function which is unlikely to examine any specific input variable). On the other hand, using the percolation exponents one can show that (4) fails if a > 3/4. This is based on calculating the expected number of sites pivotal for a crossing (i.e., a change of the value of the corresponding input variable would change the value of the function) as well as showing that the second moment is bounded by a constant times the square of the expectation. The expected number of pivotals is known [88] to be e3 / 4+0 (1) as e ---+ 00. It is reasonable to conjecture that (4) holds for every a < 3/4.
ProblemS.I. Is it true thatlime--+ooJLJe([eal, ea2 ]) = I ifa1 < 3/4 < a2? Determine the asymptotic behavior of JLIe ([e al , ea2 ]) as e ---+ 00 for arbitrary 0 ::: a1 < a2 ::: 2. Estimates on the Fourier coefficients of percolation crossings (in an annulus) play a central part in the proof [82] that dynamical percolation has exceptional times. Dynamical percolation (introduced in [31]) is a model in which at each fixed time one sees an ordinary percolation configuration, but the random bits determining whether or not a site (or bond) is open undergo random independent flips at a uniform rate, according to independent Poisson processes. The main result of [82] is that dynamical critical site percolation on the triangular lattice has exceptional times at which there is an infinite percolation component. These set of times are necessarily of zero Lebesgue measure. A better understanding of the Fourier coefficients may lead to sharper results about dynamical percolation, such as the determination of the dimension of exceptional times. Some upper and lower bounds for the dimension are known [82]. One would hope to understand the measure f.LJ geometrically. Gil Kalai (personal communication) has suggested the problem of determining the scaling limit of f.LJ. More specifically, the Fourier index set S for critical percolation crossing of a square is naturally identified with a subset of the plane. If we rescale the square to have edge length I while refining the mesh, then f.LJ may be thought of as a probability measure on the Hausdorff space Jf of closed subsets of the square. It is reasonable to expect that f.LJ converges weakly to some probability measure f.L on Jf. We really do not know what samples from f.L look like. Could it be that f.L is supported on singletons? Alternatively, is it possible that f.L[S = entire square] = I? Is S a Cantor set f.L-a.s.? 1182
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Problem 5.2 (Gil Kalai, personal communication). Prove that the limiting measure J-t exists and determine properties of samples from J-t. Kalai suspects (personal communication) that the set S is similar to the set of pivotal sites (which is a.s. a Cantor set in the scaling limit). This is supported by the easily verified fact that P[i E S] = P[i pivotal] and P[i, j E S] = P[i, j pivotal] hold for arbitrary boolean functions. Examples of functions where the scaling limit of S has been determined are provided by Tsirelson [90], [91]. One may try to study a scaling limit of dynamical percolation. Consider dynamical critical site percolation on a triangular lattice of mesh £, where the rate at which the sites flip is A > O. We choose A = A(£) so that the correlation between having a leftright crossing of a fixed square at time 0 and at time 1 is 1/2, say. Noise sensitivity of percolation [13] shows that lime\.,o A(£) = 0 and the results of [82] imply that A(£) = £0(1) and £ = A(£)0(1) for £ E (0, 1]. It is not hard to invent (several different) notions in which to take the limit of dynamical percolation as £ ~ O. Problem 5.3. Prove that the scaling limit of dynamical critical percolation exists. Prove that correlations between crossing events at different times t1 < t2 decay to zero as t2 - t1 ---+ 00 and that a change in a crossing event becomes unlikely if t2 - t1 ---+ O. Since the correlation between events occurring at different times can be expressed in terms of the Fourier coefficients [13], [82], it follows that the second statement in Problem 5.3 is very much related to strong concentration of the measure fife, in the spirit of Problem 5.1. Because of the dependence of A on £, it is not reasonable to expect the dynamical percolation scaling limit to be invariant under maps of the form I x identity, where I: D ---+ D' is conformal and the identity map is applied to the time coordinate. In particular, in the case where I (z) = a Z, a > 0, one should expect dynamical percolation to be invariant under the map I x (t 1---+ af3 t), where f3 := -lime\.,ologA(£)/log£ (and this limit is expected to exist). It is not too hard to see that f3 = 3/4 ifthe answer to the first question in Problem 5.1 is yes. This suggests a modified form of conformal invariance for dynamical percolation. Suppose that F(z, t) has the form F(z, t) = (f(z), g(z, t)), where I: D ---+ D'is conformal and g satisfies 8t g(z, t) = II' (z) 1f3, with the above value of f3. Is dynamical percolation invariant under such maps? If such invariance is to hold, it would be in a "relativistic" framework, in which one does not consider crossings occurring at a specific time slice, but rather inside a space-time set. It is not clear if one can make good sense of that.
6. LERW and UST The loop-erased random walk and the uniform spanning tree are models where very detailed knowledge exists. They may be studied using random walks and electrical 1183
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network techniques, and in the two-dimensional setting also by SLE as well as domino tiling methods. (See [66] and the references cited there.) However, some open problems still remain. One may consider the random walk in a fine mesh lattice in the unit disk, which is stopped when it hits the boundary of the disk. The random walk converges to Brownian motion while its loop-erasure converges to SLE2. It is therefore reasonable to expect that the law of the pair (random walk, its loop-erasure) converges to a coupling of Brownian motion and SLE2. (If not, a subsequential limit will converge.) Problem 6.1. In this coupling, is the SLE2 determined by the Brownian motion? It seems that this question occurred to several researchers independently, including Wendelin Werner (personal communication). Of course, one cannot naively loop-erase the Brownian motion path, because there is no first loop to erase and there are cases where the erasure of one loop eliminates some of the other loops. It is also interesting to try to extend some of the understanding of probabilistic statistical physics models beyond the planar setting to higher genus. The following problem in this direction was proposed by Russell Lyons (personal communication). Consider the uniform spanning tree on a fine square grid approximation of a torus. There is a random graph dual to the tree, which consists of the dual edges perpendicular to primal edges not in the tree. It is not hard to see that this random dual of the tree contains precisely three edge-simple closed paths (i.e., no repeating edges), and that these paths are not null-homotopic.
Problem 6.2. Determine the distribution of the triple of homotopy classes containing these three closed paths. The problem would already be interesting for a square torus, but one could hope to get the answer as a function of the geometry of the torus.
7. Non-discrete problems In this section we mention some problems about the behavior of SLE itself, which may be stated without relation to any particular discrete model. The parametrization of the SLE path by capacity is very convenient for many calculations. However, in some situations, for example when you consider the reversal of the path, this parametrization is not so useful. It would be great if we had an understanding of a parametrization by a kind of Hausdorff measure. Thus we are led to Problem 7.1. Define a Hausdorff measure on the SLE path which is a-finite. We would expect the measure to be a.s. finite on compact subsets of the plane. 1184
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That the Hausdorff dimension ofthe SLE path is min{2, 1 + K 18} has been established by Vincent Beffara [10]. When K < 8 (in which case the path has zero area), we expect the a-finite Hausdorff measure to be the Hausdorff measure with respect to the gauge function
Problem 7.2. Fix K < 8. Find necessary or sufficient conditions on a deterministic compact set K c 1HI to satisfy p[ K n y 1= 0] > 0, where y is the SLEK path. The case K c lR is of particular interest. When K = 8/3 and 1HI \ K is simply connected, there is a simple explicit formula [58] for p[y n K 1= 0]' It is not clear if such formulas are also available for other values of K. Wendelin Werner [96] proved the existence of a random collection of SLEs/3-like loops with some wonderful properties. In particular, the expected number of loops which separate two boundary components of an annulus is conformally invariant, and therefore a function of the conformal modulus of the annulus. However, this function is not known explicitly. Many of the random interfaces which are known or believed to converge to SLE are reversible, in the sense that the reversed path has the same law as the original path (with respect to a slightly modified setup). This motivates the following problem from [76].
Problem 7.3. Let y be the chordal SLEK path, where K :::: 8. Prove that up to reparametrization, the image of y under inversion in the unit circle (that is, the map z t-+ 1I z) has the same law as y itself. The reason that we restrict to the case K :::: 8 is that this is false when K > 8 (as will be proved in a forthcoming joint paper with Steffen Rohde). Indeed, there are no known models from physics that are believed to be related to SLEK when K > 8. Sheffield (personal communication) expects that at least in the case K < 4 Problem 7.3 can be answered by studying the relationship between the Gaussian free field and SLE. 1185
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Acknowledgments. Greg Lawler, Wendelin Werner and Steffen Rohde have collaborated with me during the early stages of the development of SLE. Without them the subject would not be what it is today. I wish to thank Itai Benjamini, Gil Kalai, Richard Kenyon, Scott Sheffield, Jeff Steif and David Wilson for numerous inspiring conversations. Thanks are also due to Yuval Peres for useful advice, especially concerning Problem 7.1.
References [1] Alexander, K. S., The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6 (2) (1996),466-494. [2] Angel, 0., Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (5) (2003), 935-974. [3] - , Scaling of Percolation on Infinite Planar Maps, I. Preprint, 2005; arXiv:math.PrI 0501006. [4] Barlow, M. T., Fractals, and diffusion-limited aggregation. Bull. Sci. Math. 117 (1) (1993), 161-169. [5] Bauer, R. 0., and Friedrich, R. M., On radial stochastic Loewner evolution in multiply connected domains. 1. Funct. Anal. 237 (2) (2006), 565-588. [6] - , Stochastic Loewner evolution in multiply connected domains. C. R. Math. Acad. Sci. Paris 339 (8) (2004), 579-584. [7] - , On Chordal and Bilateral SLE in multiply connected domains. Preprint, 2005; arXiv:math.Pr/0503178. [8] Baxter, R. J., Kelland, S. B., and Wu, F. Y., Equivalence of the Potts model or Whitney polynomial with an ice-type model. 1. Phys. A 9 (1976), 397-406. [9] Beffara, v., Critical percolation on other lattices. Talk at the Fields Institute, 2005; http://www.fields.utoronto.ca/audio/05-06/. [10] - , The dimension of the SLE curves. Preprint, 2002; arXiv:math.Pr/0211322. [11] - , Cardy's formula on the triangular lattice, the easy way. Preprint, 2005; http://www.umpa.ens-lyon.frrvbeffara/files/Proceedings-Toronto.pdf. [12] Beffara, v., and Sidoravicius, 0507220.
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[39] Kenyon, R., The asymptotic determinant of the discrete Laplacian. Acta Math. 185 (2) (2000), 239-286. [40] - , Conformal invariance of domino tiling. Ann. Probab. 28 (2) (2000), 759-795. [41] - , Long-range properties of spanning trees. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. 1. Math. Phys. 41 (3) (2000), 1338-1363. [42] - , Dominos and the Gaussian free field. Ann. Probab. 29 (3) (2001),1128-1137. [43] Kesten, H., The critical probability of bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74 (1) (1980),41-59. [44] - , Upper bounds for the growth rate of DLA. Phys. A 168 (1) (1990), 529-535. [45] - , Some highlights of percolation. In Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. I, Higher Ed. Press, Beijing 2002, 345-362. [46] Knizhnik, V. G., Polyakov, A. M., and Zamolodchikov, A. B., Fractal structure of 2Dquantum gravity. Modern Phys. Lett. A 3 (8) (1988), 819-826. [47] Kozma, G., Scaling limit of loop erased random walk - a naive approach. Preprint 2002, arXiv:math.Pr/0212338. [48] - , The scaling limit of loop-erased random walk in three dimensions. Preprint 2005, arXiv:math.Pr/0508344. [49] Langlands, R., Pouliot, P., and Saint-Aubin, Y., Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 (1) (1994), 1-61. [50] Lawler, G. F., The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab. 1 (5) (1996), 29-47 (electronic). [51] - , A self-avoiding random walk. Duke Math. 1. 47 (3) (1980), 655-693. [52] - , Conformally invariant processes in the plane. Math. Surveys Monogr. 114, Amer. Math. Soc., Providence, RI, 2005. [53] Lawler, G. F., Schramm, 0., and Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 (2) (2001), 237-273. [54] - , Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 (2) (2001),275-308. [55] - , Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincare Probab. Statist. 38 (1) (2002),109-123. [56] - , One-arm exponent for critical2D percolation. Electron. 1. Probab. 7 (2) (2002), 13 pp. (electronic). [57] - , Sharp estimates for Brownian non-intersection probabilities. In and out of equilibrium (Mambucaba, 2000), Progr. Probab. 51, Birkhauser Boston, Boston, MA, 2002, pp.I13-131. [58] - , Conformal restriction: the chordal case. 1. Amer. Math. Soc. 16 (4) (2003), 917-955 (electronic). [59] - , Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (1B) (2004), 939-995. [60] - , On the scaling limit of planar self-avoiding walk. In Fractal geometry and applications: ajubilee of BenOIt Mandelbrot, Part 2, Proc. Sympos. Pure Math. 72, Amer. Math. Soc., Providence, RI, 2004, 339-364. 1188
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Microsoft Corporation, Redmond, Washington, U.S.A.
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ON THE SCALING LIMITS OF PLANAR PERCOLATION
By
ODED SCHRAl\IM .. ,t AN D S T AN ISLAV SMIRNOVt ,§
WITH AN Ap PEND I X BY C H RISTOP H E GARBA N
Microsoft Researcht and Universiti de Geneveli CNRS, ENS Lyon \ Ve prove T sirelson's conjecture that any scaling limit of the crit· ical planar pe rcolation is a black noise. Our theorems apply to anum· ber of percolation models, including s ite percolation on the triangular grid a nd any s ubsequential scaling limit of bond percolation o n the square grid. We also suggest a natural cons truction for the scaling limit of planar percolation , and more generally of any discre te planar model describing connectivity properties.
Contents. Introduction. 1.1 Motivation 1.2 Percolat ion basics and notation 1.3 Definition of the scaling limits . 1.4 Statement of main results 2 The Discrete Gluing Theorem 3 The space of lower sets . 4 Uniformity in the mesh size . 5 Factorization . . . . . . . . . A Continuity of crossing events B Multi·scale bound on the four·arm event on Z2 (by Christophe Garban) . . . . References . Aut hor 's addresses 1
2 2
7 13 15 23 27
32 40 42
49 53 55
• Decembe r 10, 1961 Septe mber I, 2008 tS upported by the Euro pean Research COllnci l AG CO NF RA, the Swiss National Sci· ence FOllndation, and by t he C hebyshev Laboratory ( Faculty of Mathema t ics and l\'lechanics, St. Petersburg State University) under the grant of the government of the Russian Federation AMS 2000 subject classifications; Prima ry 60K35; secondary 28C20, 82B43, 6OG60 Keywords and phmses: percolation, noise, scaling limi t
1
I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_35, 1193
2
ODED SC HRAM l,,1 AND ST AN ISLAV Si'vIlRNOV
1. Introduction. 1.1. Motivation.
This paper has a two~ fold motivation: to propose a new construction for the (subsequent ial) scaling limits of the critical and near-critical percolation in the plane, and to show that such limits are twodimensional black noises as suggested by Boris Tsirelson. Percolation is perhaps t he simplest statistical physics model exhibi ting a phase transition. We will be interested in planar percolation models , t he archetypical examples being site percolation on the t riangular lattice and bond percolation on t he square lattice. More generally, our theorems apply to a wide class of models on planar graphs, satisfying the assumpt ions outlined below. Consider a planar graph and fix a percolation probability 1) E [0, Ij. In site percolation, each site is declared open or closed with probabilities p and I - p , independently of others. In the pictures we represent open and closed sites by blue and yellow colors correspondingly. One t hen studies open clusters, which are maximal connected subgraphs of open sites. In bond percolation, each bond is similarly declared open or closed independently of others, and one studies clusters of bonds. Percolation as a mathematical model was originally introduced by Broadbent and Hammersley to model the flow of liquid t hrough a random porous medium. It is assumed that the liquid can move freely between the neighboring open sites, and so it can How between two regions if there is an open path between them . Fixing a quad Q (= a topological quadrilateral) , we say t hat t here is an open crossing (between two distinguished opposite sides) for a given percolation configuration , if for its restriction to t he quad , t here is an open cluster intersecting both opposite sides. One then defines the crossing probabil ity for Q as the probability that such a crossing exists. "Ve will be interested in t he (subsequential) scaling li mits of percolation, as t he lattice mesh tends to zero. It is expected t hat for a large class of models t he sharp thr-eshold phenomenon occurs: there is a critical value Pc such t hat for a fixed quad Q t he crossing probabili ty tends to 1 for p > Pc and to 0 for p < Pc in the scaling limit. For bond percolation on the square lattice and site percolation on the t riangular lattice, it is known that Pc = 1/2 by t he classical t heorem of Kesten [Kcs80]. In the critical case p = Pc t he crossing probabilities are expected to have non-trivial scaling limits. Moreover, those are believed to be universal , conformally invariant, and given by Cardy's formula [LP SA94 , Car92j. For the models we are concerned with , t his was established only for site percolation on the triangular lattice [SmiOlb, Smi01aj. However, the Russo-Seymour1194
ON TH E SCA LI NG LIM ITS
or PLANAR PERCOLATI ON
3
Vvelsh theory provides existence of non-trivial subsequential limi ts, which is enough for our purposes. Our theory also applies to near-critical scaling limits like in [NoIOS], when the mesh tends to zero and t he percolation probabili ty to its critical value in a coordinated way, so as the crossing probabilities have non-trivial limi ts, different from the critical one. The Russo-Seymour\ ¥elsh t heory still applies below any given scale. Having a (subsequential) scali ng limit of the (near) cri tical percolation, we can ask how to descri be it . T he discrete models have finite a-fields, but in the scaling limit t he cardinali ty blows up, so we have to specify how we pass to a limit. First we remark that we rest rict ourselves to t he crossing events (which form t he original physical motivation for the percolation model), disregardi ng events of t he sort "number of open sites in a given region is even" or "number of open sites in a given region is bigger than the number of closed ones." \¥hile such events could be studied in another context, they are of little interest in t he framework motivated by physics. Furt hermore, we restrict ourselves to the macroscopic crossing events, i.e. crossings of quads of fixed size. Otherwise one could add to the a-field crossing events for microscopic q uads (of t he lattice mesh scale or some intermediate scale), which have infini tesimal size in the scaling limit. The resulting construction would be an e.xtension of ours, simila r to a non-standard extension of t he real numbers . But because of the locality of t he percolation, different scales are independent, so it won 't yield new non-trivial information (as our results in fact show). However, we would like to remark that in dependent models such extensions could provide new information. For example, Kenyon has calculated probabilities of local configurations appearing in domino tilings and loop erased random walks [Ken09J, while such information is lost in describing the latter by an SLE curve. Summing it up, we restrict ourselves only to the scaling limit of t he macroscopic crossing events for percolation - it describes the principal physical properties (though adding microscopic events can be of interest). The question then arises, how to describe such a scaling limit, and several approaches were proposed. We list some of them along with short remarks:
(1). Random coloring of a plane into yellow and blue colors would be a logical candidate, since in the discrete setting we deal with random colorings of graphs. However, it is difficult to axiomatize the allowed class of "percolation-induced colorings," and connectivity properties are hard to keep t rack of. In particular, in the scaling limi t almost every point does not belong to a cluster, but is rather surrounded by an infinite number of nested open and closed clusters, and so has no well-defined color. 1195
4
ODED SC HRAMM AN D STA NI SLAV Si\'IIRNOV
(2) . Collection of open clusters as random compact subsets of t he plane would satisfy obvious pre-compactness (for Borel probability measures on collections of compacts endowed with some version of the Ha usdorff distance)) guaranteeing existence of the subsequential scaling limi ts . However) crossing probabilit ies cannot be extracted: as a toy model imagine two circles) 0 and C) intersecting at two points, x and y. Consider two different configurations: in the fi rst one, 0 \ {x} is an open crossing a nd C \ {y} is a closed one; in t he second one, 0 \ {y} is an open crossing and C \ {x } is a closed one. In bot h configurations the cluster structure is t he same (wit h 0 being t he open and C being the closed cluster), while some crossing events differ. Thus one has to add the state of t he "pivotal" points (like x and y) to the description) making it more complicated . Constructing such a limit, proving its uniqueness, and studying it would require considerable technical work . (3). Aizenman's web represents percolation configuration by a collection of a ll curves (= crossings) running inside open clusters. Here precompactness follows from t he Aizenmal1-Burchard work [AB99J, based on t he Russo-Seymour-Welsh t heory, and crossing events are easy to extract. Moreover , it turns out t hat the curves involved are almost surely Holder-continuous, so this provides a nice geometric description of t he connectivi ty structure. However , there is a lot of redundancy in taking all the curves inside clusters, and there are deficiencies similar to those of t he previous approach . (4) . Loop ensemble - instead of clusters themselves, it is enough to look at the interfaces between open a nd closed clusters, whicll form a collection of nested loops. Such a limit was constructed by Camia and Newman in [CN06] from Schramm's SLE loops. However , it was not shown t hat such a limit is full (i.e. contains all the crossing events) , and in general, extracti ng geometric information is far from being stra ight-forward. (5) . Branching exploration tree - there is a canonical way to explore all the discrete interfaces: we follow one, a nd when it makes a loop: branch into two created components. T he continuum analogue was studied (in relation with t he loop ensembles) by Sheffield in [She09J, and this is perhaps t he most natural approach for general random cluster models with dependence, see [KSIOj. Construction is rather technical, however the scaling limit can be described by branching SLE curves and so is well suited for calculations. (6). Height function - in the discrete setting one can randomly orient in~ terfaces, constructing an integer-valued height function, which changes by ±l whenever a n interface is crossed. This stores all the connectivity 1196
ON T I-I E SCA LI NG L IJ\HTS OF PLANAR PERCO LATI ON
5
information (at least in our setup of interfaces along the edges of trivalent graphs) , but adds additional randomness of the height change. It is expected that after appropriate manipulations (e.g. coarse-graining and compactifying - i.e. projecting mod 1 to a unit interval) t he height function would converge in the sca ling limit to a random distribution, which is a version of t he free field. This is the cornerstone of the "Coulomb gas" method of Nienhuis [NicS4], which led to many a prediction. It is not immediate how to connect this a pproach to t he more geometric ones, retriev ing the interfaces (= level curves) from t he random distribution, but SLE theory suggests some possibilities. Also it is not clear whether we can implement this approach so that the extra random ness disappears in the limit, while the connectivity properties remai n. (7). Correlation functions - similarly to the usual Conformal Field Theory approach , instead of dealing with a random field as a random distribution, we can restrict our attention to its n-point correlations. For example, that would be t he usual physics framework for fields arising in approach (6), but to implement it mathematically we would need to reconstruct height lines of a field, based on its correlations, which is perhaps more difficult than developing (6) by itself, and possibly requires some extra assumptions on t he field. A more geometric al>proach would be to st udy for every finite collection of points {ZJh,k the probability Fr (zj) j ,k that for every k all the balls {B(zj, r) }j are connected by an open cluster. To obtain the correlation funct ion, one takes t he double limit, passing to the scaling limi t fi rst and sending r _ 0 afterwards. Since the probability of a radius r ball bei ng touched by a macroscopic cluster (conjecturally) decays as 1"5/48, one has to normalize accordingly, considering F as a n j ,k Idzjls/48_ form . The geometric approach seems more feasi ble, but along the way one would have to further develop a version of CFT corresponding to such "connectivity" correlations. (S). Collection of quads crossed - in the discrete setting one can describe a percolation configuration by listing the finite collection of all discrete quads (Le. topological quadrilaterals) having an open crossing. We propose a continuous analogue, described in detail below. The percolation configuration space t hus becomes a space of quad collections, satisfyi ng certain additional properties. (9). Continuous product of probability spaces, or noise - t his approach was advocated by Tsirelson [T si04a, Tsi04b, Tsi05]. In the discrete setting 1197
6
ODED SCHRAMJ'd AN D ST AN ISLAV Sl\'I I RNOV
t he site percol ation (011 a set V of vertices) is given by t he product probability space (fl v = v {open, closed} , Fv , {tv ). Note that when V is decomposed int o two d isjoint s ubsets, we obviously have
n
(1.1 ) Straight-for ward generalization of t he product space t o t he continuous case does not work: the prod uct a -field for t he space Dc {open , closed} won 't contain crossing events. Instead Tsirelsol1 proposed to send the percolation meas ure space to a sca ling limit , described by a noise or a homogeneous conti nuous product of pro bability spaces. Namely, for every smooth domain D t he percolation scaling limit inside it would be given by a probabili ty space (S1 D , Fo ,I"'D) , which is t ranslat ion invariant, cont inuous in D , and sat isfies t he property (1.1 ) for a smoot h domain V cut into two smooth domai ns VI, V2 by a curve. There is a well-developed theory of such spaces, but establishing the lat ter property was problematic [T si05].
To most constructions one can also add the information a bout t he closed clusters, though in principle it should be retrievable from t he open ones by d uality. This list is not exhaustive, but it shows t he variety of possible descriptions of t he sam e object. Indeed , in the discrete set t ings the approaches above contain all the information about macroscopic open clusters, a nd a re expected to keep it in the scaling limi t; on the ot her hand wi t h an appropri ate set up (one has to be careful , especially wit h (6)) we do not expect to pick up any ex tra information Oil t he way. However , showing t hat most of t hese a pproaches lead to equivalent results, and in particula r to isomorphic a -fields, is far fmm easy. \Ve decided to proceed with t he a pproach (8) , since it fo llows t he original physical mot ivat ion for the percolation model, and also provides us wi t h enough pre-compactness to establish existence of subsequent ia l scaling limits before we a pply any percolation techniques. It a lso serves well our second purpose: to provide escript ion of t he percolation (subsequential) scaling limits, followin g T sirelson 's approach (9), a nd show t hat it is a noise, i.e. a homogeneous continuous (param et eri zed by an appropria te Boolean a lgebra of piecewise-smooth planar domains) product of probability spaces . The question whether crit ical percolation scaling limit is a noise was posed by T sirelson in [T si04a , T si04b, T si05], where he has a lso noted that such noise must be non-classical, or black (compared to t he classical white noise) . The t heory of black noises was started by T sirelson and Vershik in [TV98]. In part icular, t hey constructed the fi rst examples of zero and one-dimensiona l 1198
ON THE SCALING
LI ~HT S
or
P LANA R PERCO LATION
7
black noises (and more of those were found since); we provide the first genuinely two-dimensional example. To establish that percolation leads to noise it is enough to prove t hat its scaling limit, restricted to two adjacent rectangles , determi nes the scaling limit in their union . Following is a version of t his result , perhaps more tangible (though stated slightly informally here). It is proved in P roposition 4.1 , which is t he most technical result of our paper. Consider a rectangle, and a smooth path 0' cutting it. We show that for even) E > 0 theTe is a finite number of percolation crossing events in quads disjoint from Ct, from which one can reliably predict whether or not the rectangle is crossed, in the sense that the probability for a mistake is less than E. The essential point is that t he set of crossing events t hat one looks at does not depend on the mesh of the lattice, which allows us to ded uce t he needed result, t hus showing that any subsequential scaling limit of the critical percolation is indeed a black noise.
1.2. Per-colation basics and notation. For completeness of exposition we present most of the needed background below, interested readers can also consult t he books [Kes82, Grigg, BRaG, WerOg]. 'We start with a rather general setup, which in particular includes all planar site and bond models (as well as percolation on planar hypergraphs, cf. [BRlO]). Since we will be interested in scaling limits, it is helpfu l from the very beginni ng to work wit h subsets of the plane, rather than graphs. V!le consider a locally finite tiling H of the plane (or its subdomain) by topologically closed polygonal tiles P. Furthermore, we ask it to be trivalent, meaning t hat at most three tiles meet at every vertex and so any two tiles are either disjoint or share a few edges. A percolation model fixes a percolation probability p(P) E [O,lJ for every tile P , and declares it open (or closed) with probability p(P) (or I - p(P)) independently of the others. In principle, we can work with a random tiling H , with percolation probabilities p(P, H ) satisfying appropriate measurability conditions (so t hat crossi ng events are measurable). More generally, we can consider a random coloring of tiles into open and closed ones, given by some measure J-L on the space fl pEH {open, closed} with a product a-field (which contains a.Il events concerned with a fi nite number of tiles). P ercolation models correspond to product measures J-L. vVe define open (closed) clusters as connected topologically closed subsets of t he plane - components of connect ivity of the union of open (closed) tiles taken with boundary. All site and bond percolation models on planar graphs can be represented 1199
8
ODED SCHRAMM AN O STAN ISLAV Sr-.WlNOV
in t his way. Indeed , given a planar site percolation model , we replace sites by t iles, so that two of t hem share a side if and only if two corresponding sites are connected by a bond. It is easy to see that such a collection always exists (e.g . tiles being the faces of t he dual graph) . Each tile is declared open (or closed) with the same probability p (or 1 - p) as the corresponding site.
FIG 1. 1. Critical site percolation on the triangular lattice. Each site is represented by a hexagonal tile, which is open (blue) or closed (yellow) with probabilities 1/ 2 independently of others . Int erfaces between open and closed clusters are pictured in bold.
If our original graph is a t riangulation, t hen the dual one is trivalent and only three tiles can meet at a point. However, when our graph has a face with four or more sides, two non-adjacent sites can correspond to tiles sharing a vertex (bu t not an edge) . To resolve the resulting ambiguity we insert a small tile P around every tiling vertex, where four or more tiles meet , and set it to be always closed, taking p(P) = O. After the modification, at most three tiles can meet at a point, as required. Bond percolation can also be represented by partitioning the plane into ti les. T here will be a t ile for every bond , site, and face of t he original bond graph, chosen so that two different face-tiles or two different site-tiles are disjoint. IVIoreover , a bond-tile shares edges with two site-tiles and two facetiles, corresponding to adjacent sites and faces. T he bond-t iles are open (or closed) independently, with the same probability p (or 1 - p) as the corresponding bonds. The site-tiles P are then always open (p(P) = 1) , while the face-tiles P are always closed (p(P) = 0). 'Vith such construction, no ambiguities arise, and at most three tiles can meet at a point. A possible construction for t he bond percolation (with octagons being the bond-tiles) is illustrated in Figure 1.2. Following the original motivation of Broadbent and Hammersley, we rep1200
ON '1'1-1 E SCALING LIl\H TS OF PLA NA R PERCOLAI'ION
9
FIG! .2. Left: critical bond percolation on the squan~ lattice, every bond is open (blue)
closed (yellow) with pmbabilities 1/2 independently of others. Right: n~presentation by critical percolation on the bathroom tiling, with direction 01 original bonds marked on the corresponding octagonal tiles. Rhombic tiles alternate in color, with blue (always open) corresponding to the sites 01 the original square lattice, while yellow (always closed) - to the laces. Octagonal tiles correspond to the bonds, and are open or closed independently, with probabilities 1/ 2. Of'
resent holes in some random porous medium by open t iles. A liquid is assumed to move freely through the holes, t hus it can flow between two regions if there is a path of open tiles between them. More precisely, we say t hat t here is an open (closed) crossing between two sets 1<1 and 1<2 inside the set U, if for t he percolation configuration restricted to U, there is an open (closed) cluster intersecting both 1<\ and ](2. Vile will be mostly interested in crossings of quads (= topological quadrilaterals). Note the following duality property: there is an open crossing between two opposite sides of a quad if and only if theTe is no dual (i. e. closed) crossing between two otheT sides. Exchanging t he notions of open and closed, we conclude t hat, on a given tiling, percolation models with probabilities p(P) and p*(P) = 1 - p(P) are dual to each other. In particular, the model with p(P) 1/2 is self-dual, which can often be used to show t hat it is critical, like for t he hexagonal lattice. The bathroom-tile model of the square lattice bond percolation in Figure 1.2 is dual to itself t ranslated by one lattice step (so that always open rhombi are shifted to always closed ones), which also leads to its criticality, [Kes80]. There are however dualities (e.g. arising from t he star-triangle transformat ions) with more complicated relations between p and p* , and so t here are critical models with p away from 1/2, see e.g. [WicS1 , BRlO[. Though we have in mind critical models, we do not use criticality in
=
1201
10
ODED SC HRMMvl AND STAN ISLAV Sr-.'IIRNOV
the proof, rather some crossing estimates, which are a part of the RussoSeymour-Welsh t heory. Those also hold (below any given scale) for t he nearcritical models, like those d iscussed in [NoI08]. In a seDse the estimates we need should hold whenever the crossing probabilities do not tend to a trivial limit , see Remark 1.3. We work with a collection of percolation models 1-41 (and sometimes more general random colorings) on Wings H q , indexed by a set {1]} (e.g. square latt ices of different mesh) , and denote by /-trj also the corresponding meas ure on percolation configurations (= random coloring of tiles). By 1171 E IR+ we will denote a "scale parameter" of the model J-Lr/" \Ne assume that all tiles have diameter at most 1771 (for a percolation on random tiling, our proof would work under the rela.xed assumption that for any positive T , the probability to find in a bounded region a tile of diameter bigger than T tends to zero as 117 1 0) . This parameter a lso appears in t he RussoSeymour-Welsh- type estimates below, which have to hold on scales larger than 117 1. Note that in most models 1111 can be taken to be t he supremum of tile diameters (or the lattice mesh for t he percolation on a lattice). We will be interested in scaling limits as the scale parameter tends to zero , looking at sequences of percolation models with 1171 - 0, which do not degenerate in the limit. 'rVe will show in Remark 1.17, that in our setting working with sequences and nets yields t he same results. As discussed above, all critical percolation models are conjectured to have a universal and confonnally invariant scaling limit . Moreover, its SLE description allows to calculate exactly many scaling exponents and dimensions and we will use this conjectured picture in informal discussions below. However, to make our paper a pplicable to a wider class of models , we will work under much weaker assumptions. The famous Russo-Seymour-'Velsh theory provides universal bounds for crossing probabilities of the same shape, superimposed over lattices with different mesh. Originally established for the critical bond percolation on the square lattice, it has been since generalized to many critical (and nearcritical, when the percolation probabilities tend to their critical values at appropriate speeds as t he mesh tends to zero) models with several different approaches, see [KesS2, Grigg, BROG, \·VerOg]. We will use one of t he recurrent techniques: conditioning on t he lowest crossing. If a quad is crossed hori zontally, one can select t he lowest possible crossing (the closest to t he bottom side), which will t hen depend only on the configuration below. If we condition on the lowest crossing, the configuration above it is unbiased, allowing for easy estimates of events there. Besides using this neat trick, we will assume certain bounds on the crossing probabilities for t he annuli. 1202
ON T HE SCAL ING L Ii\I(JTS OF PLANA R PERCOLATI ON
11
A prominent role is played by the so-called k - arm crossi ng events in annuli , si nce they control dimensions of several important sets. In particular , for a given percolat ion model J..I.-ri we denote by p~ (Z,1\ R) the probability of a one-arm event, that there is an open crossing connecting two opposite circles of the annulus A(z, T, R ). T hen this is roughly the probability that the disc B(Z,1') intersects an open cl uster of size ~ R , which is expected for 1171 < l' to have a power law like
p,:(z ,r,R) =
(r/R)a,+a(l)
as
l,/R~O,
morally meaning (modulo some correlation estimates) that percolation clusters in the scaling li mit have dimension 2 - al. Similarly, denote by P~( z, r, R) the probability of a four-arm event, that
there are alternating open- closed- open - closed crossings connecting two opposite circles of the annulus A(z, 1', R). Then this is roughly t he probabili ty that changing the percolation configuration on the disc B(z, r) changes the crossing events on t he scale:::::: R. Indeed , if all the ti les in B(z, r) are made open, t hat connects two open arms; whi le making them closed connects two closed arms, destroyi ng the open connection. For 1171 < l' a power law P,i(z,r,R)
=
(r/R)",+a(l),
as r/R~O,
is expected, roughly meaning that t he pivotal tiles (such that altering their state changes crossi ng events on large scale) have dimension 2 - a4 in the scaling limit. \-\7 henever the RS\V theory applies, the probabilities for k arm events have power law bounds from above and below as r --+ O. In fact, the probabilities are conjectured to satisfy universal (independent of a particular percolation model) power laws with rational powers, predicted by t he Conformal Field Theory, see discussion in [S\'V01 j. T hose predictions have been so far proved fo r t he critical site percolation on triangular lattice only [S\-\701 , Ls \~r 02 ], giving at = 5/ 48 and a4 = 5/ 4. Other k-arm probabilities are also of interest, but estimating just the mentioned two allows to a pply our methods. Houghly speaking, we need to know that the percolation clusters have dimension smaller than 2 and t hat pivotal points have dimension smaller than 1, or, in other words, that al > 0 and a4 > 1. Note that we do not actually need the power laws, but rather weaker versions of t he corresponding upper estimates. Namely, we need the fo llowing two estimates to hold uniformly for percolation models under consideration: 1203
12
ODED SCI'IRAi\HI'I AND ST AN ISLAV Si'dIRNOV ASSUM PT IONS 1.1.
There exist positive functions 6 [ (1', R) and 6. 4 (1', R),
such that lim t.j(T, R) = 0 for any fixed R
,-0
< Ro ,
and the following estimates hold whenever z E C and 0 < 117 1< r < R < F4J. The p10bability of one open arm event and the probability of a similar event
for one closed arm satisftJ (1.2) while the pmbability of a four ann event satisfies
(1.3)
For r ~ R, when the annulus is empty, we set .6..;{1', R) := 1, so t hat the function is defi ned for all positive arguments. Without loss of generality we can assume that functions .6.. j {r , R ) are increasing in r and decreasing in R .
(A STRONGER R SW ESTI MATE ) . For most percolation models where the RSW techniques work, their application would prove a stronger estimate, with D. J(r , R) replaced in (1.2) by C(rjR)a j with aJ > O. H oweveT we can imagine situations where only OUT weaker version can a priori be established. RE I'.'IARK 1.2
1.3 ( RSvV ESTIMATES AN D SCA LI NG L l l'IHT S). For most percolation models where the RSW techniques work, the estimate (1.2) would be equivalent to saying that crossing probability for any given quad (01' faT all quads) is uniformly bounded away from 0 and 1. So morally our noise characterization would apply whenever one can speak of non-trivial subsequential scaling limits of crossing probabilities. REMARK
The first Assumption (1.2) is used several times throughout t he paper. In a stronger form it was one of t he original Russo-Seymour- ,~relsh results, and has since been shown to hold for a wide range of percolation models, see [BR06, BRlO, Gri99 , Kes82J. For site percolation on the triangular lattice, one can take t. ,(,·, R) '" (rI R )5/48, see [LSW02]. The second Assumption (1.3) is used only once, namely in the proof of Theorem 1.5. It is known (in a sharp form with D.4 (1', R ) ~ (Tj R)I /4 ) for site percolat ion on the t riangular lattice, see IS\:VOl]. Christophe Garban ki ndly allowed us to include as Appendix B his proof for bond percolation on the square lattice, which is partly inspired by t he noise-sensitivity arguments in 1204
ON THE SCA LI NG Lli\HTS OF PLANAR PERCOLATION
13
[BKS99J. It seems possible to extend the proof to a wider range of percolation models (by changing the values of the Cj random variables so they become centered, rather than using symmetry arguments).
For which models can (1.3 ) be proved? Can it be deduced from (1.2)? Is there a geometric argument? QUESTION 1.4 (EST lfI'IATI NG P IVOTALS) .
Summing up t he discussion above, we study collections of percolation models satisfyi ng Assumptions 1.1. In part icular, our results apply to critical and near-critical site percolation on the triangular lattice and bond percolation on the square lattice.
1.3. Definition of the scaling limits . Every discrete percolation configuration (or a coloring of a tiling) is clearly described by t he collection of quads (= topological quadrilaterals) which it crosses (and for a fixed lattice mesh in a bounded domain a finit e collection of quads would suffice). 'vVe introduce a setup , which allows to work independently of the lattice mesh and pass to t he scaling limit, but t he idea remai ns the same: a percolation configuration is encoded by a collection of quads - t hose which are crossed by clusters. Our setup is inspired by the Dedekind's sect ions and uses the metric and ordering on the space of quads. The ordering comes from the obvious observation that crossing of a quad automatically contains crossings of "shorter" sub-quads, and so possible percolation configurations (= quad collections) should satisfy certain monotonicity properties. \Ve will work in a domain De e = "[R2. A quad in D is a topological quadrilateral , i. e. a homeomorphism Q : [0, IF -+ Q([O, IF) cD. We introduce some redundancy when considering quads as parameterized quadrilatends , but we avoid certai n technicalities. The space of all such quads will be denoted by Q = QD. It is a metric space under t he uniform metric
d(QI ,Q,)
~
sup IQI(z) - Q,(z)l ·
zE[O,i]2
A crossing of Q is a connected compact snbset of [QI ,~ Q( [O, I]') t hat intersects both opposite sides BoQ, ~ Q({O) x [O,ID and iJ,Q' ~ Q({I} x [0, IIll. We also denote the remaining two sides by IlrQ , ~ Q([O , II x (OJ) and fJ,Q , ~ Q([O, I] x (Il) , see Figure 1.3. The whole boundary of Q we denote by 8Q. In t he discrete setti ng t here is no difference between connected and path-connected crossings, but one can imagine lattice models where the probabilities of the two would be different in the scaling limi t. However, for the models under consideration, every crossing in t he scaling limit almost 1205
14
ODED SC HRAM!'.'I AND ST AN I SLAV SM IRNOV
Q(O,I)
r::---~a,~Q:"'- _ _ _~Q(I,I)
7
[Q]
L
Q(O,O)~---------CQ(I ,O) f iG 1.3. A quad with a crossing.
surely can be realized by a Holder continuous curve [AB99J, so t he two notions are essentially t he same. In addition to t he structure of Q D as a metric space, we will also use the follow ing partia l order on Qv . If QI , Q2 E Qv, we write Q J ~ Q2 if every crossing of Q2 contains a crossing of Q I. The simplest example is seen in Figure 1.4. Also , we write Q1 < Q2 if there are open (in t he uniform metric) neighborhoods Ui of QJ and U2 of Q2 in Qo, such t hat for every Q E UI and Q' E U2 we have Q :::; Q' . Thus, t he set of pairs (Q, Q') E Q2 satisfying Q < Q' is the interior of {( Q, Q') E Q2 : Q ::; Q'} in the product topology. A subset S c Qo is a lower set if, whenever Q E Sand Q' E Q D satisfies Q' < Q, we also have Q' E S. The collection of all closed lower subsets of QD will be denoted H D. Any discrete percolation configuration w in C is naturally associated with an element Sw of H D: t he set of all quads for which w contains a crossing formed by an open cluster. Thus a percolation model on t he tili ng H,. , induces a probabili ty measure 11-,.., on He (and more generally on H D for any domain D contained in its doma in of definition). A topology is defined on H D by specifying a subbase. If U c QD is topologically open and Q E QD, let
Vu
{S E 'li D oSnU7'0), and
V Q .- {S E 'liD 0 Q ¢ S}. If we regard S as a collection of crossed quads, then Vu VQ
-
{S: some quads from U are crossed}, and {S: quad Q is not crossed} . 1206
O N THE SCALI NG LIi\'IITS
or PLA NAR PERCOLATIO N
Ql(O,I) ~
[QI]
Q2(0,1)
15
.
J Q2(1,1)
[Q2]
I
f
Q2(0,0)
/
/
Q2( 1,0)
Ql (O,O)
Ql(I,O)
F IG 1.4. Any crossing of Qz contains a crossing of Q\ J so we write Q\ :S Qz .
Our topology of choice on 1to will be t he minimal topology To which contains every such Vu and V Q. As we will see, (H o, To) is a metrizable compact Hausdorff space. The percolation scaling limit configuration will be defined as an element of He , and the scaling limit measure is a Borel proba bility measure on He. Then the event of a percolation configurat ion crossing a quad Q C01Tesponds to 8Q , ~ ~ VQ ~ (S E 'Ii, Q E S) c 'Ii. We will prove that if QO c Q is dense in Q , then the events E3Q for Q E QO, generate the Borel a -field of H .
1.4 . Statement o/ m ain results. Construction of the (subsequential) scaling limits is accomplished for all random colorings of t rivalent tilings wi thout additional assumptions. Other results of this pa per apply to all percolation models on trivalent tilings, satisfiJing Assumptions 1.1, i.e. the RSW estimate (1.2) and the pi vot al estimate (1.3). As discussed a bove, we fix a collection of such percolation models indexed by a set {1]}, and denote by /1,,, the (locally) discrete proba bili ty measure on percolation configurations for the model corresponding to parameter 1]. By 1171 E ~+ we denote the scale parameter of the model (which enters in the 1207
16
ODED SCH RAM J\·! A N D STAN ISLAV S;vIIRNOV
RS"V estimates} . Later we will show that 1-£1/ has subsequential scaling limits as 1771 ---+ 0, and wi ll work wit h one of t hose, denoted by Po . First we present a simplified discrete version of the gluing t heorem. Informally, t he statement is t hat if you fix a quad Qo and a finite length path a cutt ing it into two pieces, t hen the discrete percolation configuration outside a small neighborhood of Ct reliably predicts t he existence or non-existence of a crossi ng of Qo with probability close to one, uniformly in t he mesh size. Following is the precise version, estimating t he random variable which is t he conditional probabili ty of observing an open crossing given the configuration s-away from o. As a regularity assumption , we need a constant C(o) such that for every s > 0 the set 0 can be covered by at most C(O')js balls of radius s . T his is t he case if and only if 0 has fin ite I-dimensiona l upper Minkowski content M *'(o) = lim suPS--4o+ (area{z : dist (z,o) < s}) .
Consider a collection of percolation models satisfying A ssumptions 1.1 . Let Qo E Q be some quad, and let 0 C [Qol be a finite union of finite length paths, or more generally a set of finite I -dimensional upper Minkowski content. Let BQo denote the event that Qo is crossed by w, and for each s > 0 let Fs denote the a-field generated by the restriction of w to the complement of the s-neighborhood of o . Then for every to > 0, THEORE f\·1 1.5 ( DI SCRETE G L UI NG) .
lim
sup JLo('
s","o Itlle (o,s)
< JLo(BQ,
I F ,)
< 1- ,)
=
o.
Theorem 1.5 essentially states that event BQo for any given 11 can be reconstructed with high accuracy by sampling crossing events away from o . This does not lead to the noise characterization of the scaling limit, since a priori the number of events we need to sample can grow out of control as 1111 --+ O. We address this by providing a unijorm bound on the number oj events to be sampled in Proposition 4.1 . However, we do not give a recipe, and in general the gluing procedure may still depend on 1] . We deal with this possibility by assuming that there is a (su bsequential) scaling limit, then, since the number oj possible outcomes oj sampling a finite number oj crossing events is finite, we can choose a sub-subsequence along which the same procedur'e is used. The downside is that, in the current jonn, our results alone cannot be used to show the uniqueness oj the percolation scaling limit. REMARK 1.6 ( GLU I NG I S NON-CONST RUCTIVE) .
The previous Remark leads to the following 1208
ON THE SCA LI NG Ll M I T S OF PLA NA R PERCOLATI O N
17
Can one show that there is an independent of'rJ and preferably constructive procedure to predict BQo by sampling crossing events away from a 'l Q UESTION 1.7 (CONSTRUCTI VE GLU I NG) .
A positive answer would provide an elegant proof of the uniqueness of the scali ng limit (provided Cardy's formu la is known), make the universality phenomenon more accessible, and lay foundation for t he renonnalization theory approach to percolation, similar to one discussed by Langlands, see [Lan05] and the references therein. A particularly optimist ic scenario is suggested in Conjecture 1. 11. Theorem 1.5 is a toy version of our main gluing resul ts, and t he logic behind it was folklore for a long time (but strangely enough appeared few times in print, as was pointed out in [Ste09]) . The proof proceeds by consequently resampling radius 2s balls covering the s-neighborhood of a and using the pivotal estimate ( 1.3). For this to work , the bounds have to sum up to something small , and so wi th better pivotal estimates we can relax assumptions on a (and the number of balls needed to cover it).
a). When applying the pivotal estimate ( 1.3), we cannot relax the regularity assumption that a has finite I-dimensional upper Minkowski content. However, as discussed above, most percolation models are expected to satisftj stronger estimates, and then the regularity assumption can be considerably weakened (though it cannot be dropped) . The set of pivotal points conjecturally has Hausdorff dimension 3/ 4 in the scaling limit (proved on the triangular lattice in /SWOl j), so if the Hausdorff dimension of the set a is greater than 5/4, the two should intersect with positive probability (needed correlation estimates are expected to hold). In such a situation, there is little hope to reliably determine the crossing without knowledge of a. To be more rigorous, instead of pivotals one has to work with the percolation spectrum, but it has the same dimension (see /GPSIOaj for the definition and discussion), so a has to be of dimension at most 5/ 4. On the other hand, our proof works for sets 0' of dimension smaller than 2 minus dimension of the pivotal points, provided also that an 8Qo is finite. So 5/ 4 seems to be the correct dimensional threshold, and this can be established for· the critical site percolation on the triangular lattice. REMA RK 1.8 ( REGUL A RITY ASSUMPT I ONS ON
Suppose that a smooth curve a cuts the quad Qo into two quads Q + and Q _ . The follOwing informal discussion concerns the critical percolation scaling REMA RK 1.9 ( CROSSINGS W I TH PI NIT E I NTERSECTION NU!"I'I BER).
1209
18
ODED SC HRAI\'I ]\'] AND STAN ISLAV Sl\mlNOV
limit, assuming its existence and conformal invariance (but it can be eas~ iiy made more precise by war-king with the limits of discrete configurations and events). First observe, that open clusters inside Q± intersect the smooth boundar1J 0: on a set of Hausdorff dimension 2/3, (and we have a control for the correlation of points inside it). Being independent, two large open clusters on opposite sides of 0: have a positive probability of touching. But if they touch at some point x E 0, we cannot have a large closed cluster passing thmugh x and separating them. Indeed, that would mean a four-arm event occurring at the pivotal point x on the smooth curve 0:, and since a4 = 5/4, the set 0/ pivotal points has Hausdorff dimension 3/4 and so almost surely does not inter"sect the smooth a. Thus the two open clusters in Q± which touch on a are in fact parts of the same large open cluster in Qo. We conclude, that given a smooth curve a, with positive probability there is a crossing of Qo which intersects a only once. The last remark says that for a smooth curve a , if we condition on the quad being crossed, the probability of fi nding a crossing intersecting a only finitely many times is positive. We can ask whether this conditional probability is one, leading to the following:
LdQ be a quad and a a smooth curve. Then in the scaling limit, given existence of a crossing/ ther-e is almost surely a crossing intersecting a only a finite number of times. CON J ECTU RE 1.10 ( FINITE INT ERSECTION PROPERTY).
Alternatively, one can form ulate a d iscrete version, which is a pflori stronger (for an equivalent one we would have to count small scale packets of intersections):
Let Q be a quad and a a smooth curve. For a discrete percolation measure flrlj let the random variable NTJ = NTJ (Q, a) be the minimal possible number of intersections with a for a crossing of Q (and 0 if Q is not crossed) . Then N,/ is tight as CON J ECTU RE 1.11 (TIG HTNESS OF INTERSECT IO NS).
Iryl ~ o.
Proving either of t hese conjectures would not only simplify our proof of the main t heorem, but also provide an easy and constructive "gluing procedure": we take all :<excursions" (= crossings in the complements of a) and check whether a crossing of Q can be assembled from finitely many of them (as discussed in Remark 1.9, the finite number of excursions will necessarily connect into one crossing). Such a procedure would give a very short proof 1210
ON THE SCA LINC LI J\HTS OF PLA NA R PERCO LATION
19
of the uniqueness of t he fu ll percolation scaling limit , provide an elegant setup for the renormalization t heory of percolation crossings, and show t hat the "finite model" discussed by Langlands and Lafortune in [LL94] indeed approximates t he critical percolation . Note t hat if Conjectures above fa il , the construct ive procedure for gluing crossings will have to deal with countably many excursions intersecti ng a on a Cantor set C. Then the criterion for possibility of gluing t he excursions into one crossing is likely to be in terms of some capacitary (or dimensional) characteristics of C. The critical percolation scaling limit is conjectured to be conformally invariant, which was est ablished for site percolation on t he t riangular lattice [SmiOla, SmiOlb]. In Remark 1.9 we mentioned t hat, for a given quad Qo , a smooth curve a has t he property that two large clusters inside two components of [Qo] \ a touch each other (on a) with positive probabi li ty. One can ask, whether an arbi t rary curve a would always enjoy t hi s property, and the answer is negative. Moreover , under conformal invariance assumption, t his property (equivalent to positive probability of finding a crossing intersecting a only fini tely many times) can be reformulated in terms of multi fractal propert ies of harmonic meaSUl'e on two sides of fr. Below we present an informal argument to this effect, note a lso similar discussions in the SLE context in [GRS08[. To simplify t he matters, we discuss a related (and probably equivalent) property t hat the expected number of such touching points for two large clusters inside two components of [Qo] \ fr is posit ive (or even infini te). Let f(a +, a_) be the t wo-sided dimension spectrum of harmonic measure on a, see [AP SlO]. Roughly speaking, f(a+ , a_} is the dimension of the set of z E a where harmonic measures w± on two sides of a have power laws a±:
Consider a set E C a such t hat harmonic measures w± have power laws a ± on E. Cover most of E by a collection of balls Bj of small radi i Tj . By conforma l invariance, the probability of a ball Bj t o be touched by a la rge open cluster on the "+ side" of fr is governed by its harmonic measure and Cardy's formula [SmiOlb] that gives the exponent 1/ 3: P+ (B j } ~ W+ (B j}I /3 ~ 1·? /3 .
Combining wit h t he similar est imate on the "- side," we wri te an estimate for t he expected number of balls B j touched by large clusters on both sides:
L
P+(Bj)P_(Bj) '"
j
L ,.;"++"-1/ 3 . j
1211
20
ODED SC I·IRAJ\U",I AND ST AN ISLAV SM IRNOV
+ a_) / 3 is bigger (smaller) im than dim(E ), since L.j r1 (E) ~ 1. \Ve conclude that the expected number
Here t he right hand side is small (large) when (a+
of points touched by large cl usters on both sides is positive if and only if we
can find E such t hat (a+ + a_ ) / 3 < dim (E ).
R EMA RK 1.12 (S HA RPNESS OF T HE FI NI TE I NTERSECTION PROPERTY ) .
It seems that positive probability of having a crossing intersecting Q a finite number of times is roughly equivalent to the existence of positive exponents
u+ and a_ such that the two-sided multi/raetal spectrum of harmonic measure on a satisfies 3/(a+ ,a_ ) > a+ + a_, see (A PSI OJ for discussion of such spectra. The latter- property seems to fail even for some non-smooth Ct of dimension close to one, so we do not expect the factorization property to be equivalent to the finite intersection property, but rather to be a weaker one. Before proving the main t heorem, we discuss a new framework for t he percolation scaling limit and show t hat collection of discrete random colorings is precompact, so in this framework the subsequential scaling limits exist for any random model. Recall that we work with H o, which is t he collection of all closed lower (i.e. mono tonicity of t he crossing events is obeyed) subsets of Qo, which one should think of as sets of quads crossed. The topology TD is t he minimal one containing all sets Vu (of S such that some quads in U are crossed) and VQ (of S such t hat Q is not crossed). We now list some important properties of Ho. THAEOREfo,'1
1.13 ( T HE SPACE OF PERCOLATI ON CONFIGU RATI ONS ).
D ee be topologically open and nonempty.
Let
1. (HD , To) is a compact m etrizable Hausdorff space. 2. Let A be a dense subset of QD. If 5 1, 5 2 E HD satisfy 51 n A = S2 n A , then 51 = 8 2. Moreover, the O'-field generated by V Q , Q E A is the Borel a ~field of ( lt ~ , To). 3. If 8 1 , 8 2 , ... is a sequence in HD and 8 E HD , then Sj -. 8 in TD is equivalent to 8 = lim SUPj Sj = lim iufj 8 j . 4· Let D' c D be a subdomain, and for S E HD let 8' := 8 n QD' . Th en S 1-+ 8' is a continuous map from H D to H 0' . In the above, lim infj 8 j is the set of Q E Qo such t hat every neighborhood of Q intersects all but finitely many of the sets Sj, and lim SUPj 5 j is the set of Q E QD such that every neighborhood of Q intersects infinitely many of the sets 8 j . 1212
ON TH E SCA LI NG LIMITS Of PLANAR PERCOLATION
or
21
The result concerns the topological properties of the space of percolation configurations (colorings of Wings), and involves no probability measures. Thus it can be applied to any random coloring of a trivalent tiling (into open and closed tiles), yielding results for dependent models, like the Fortuin-Kasteleyn percolation. R EfI'I A RK 1.14 (Ap PLI CABI LITY
T HEO REM 1.1 3 ) .
It follows from the Theorem above that t he space of continuous func-
t ions on (Jiv) Tv) is separable, and hence the unit ball in its dual space M(Jiv) of Borel measures is compact (by the Banach-Alaoglu theorem, 3.15 in [Rud73j), metrizable (by 3.16 in [Rud73 j), and obviously Hausdorff in the weak-* topology. The space of probability measures being a closed convex subset of the unit ball, we arrive at the following
or PERCOLATIO N
The space P rob(Jiv) of Borel probability measures on (Jiv , Tv ) with weak-* topology is a compact metrizable Hausdorff space. CO ROLLARY 1.15 (THE SPACE
MEASURES).
Discrete percolation measures MrI clearly belong to Prob(Hv) and hence t he existence of percolation subsequential scaling limits (or, more generally, for any random colorings) is an immediate corollary:
The collection of all discrete measures {J-L,/} (corresponding to random colorings of trivalent tilings) is precompact in the topology of weak-* convergence on Prob(Hv ) . CO ROLLARY 1.16 ( PRECOM PACT NESS).
We can make {1J} a directed set by writing 1JI :::s m whenever l1Jti ;::: Iml· Then the scaling limit 1iml'1I..... 0 141 is just the limit of the net J.L,/. Since the target space Prob(Hv) is metrizable and hence first-countable , it is enough to work with the notion of convergence along sequences, rather than nets. In particular, if limj ..... oo j.£71j = 110 for every sequence 1"Jj with limj ..... oo 111j I = 0, then limllll _o 141 = flo· R EMA RK 1. 17 (NET S VS. SEQUEN CES).
Our construction keeps information about all macroscopic percolation events, and so gives the full percolation scaling limit at or near· criticality. For the critical site percolation on triangular lattice, Garban, Pete and Schramm explain in section 2 of (GPSIOcj that the scaling limit is unique) appealing to the work {CN06j of Camia and Newman. Basically, we know the probabilities of individual crossing events, and it allows R EMA RK 1. 18.
1213
22
ODED SCHRAJ\U..·! AN D ST AN ISLAV
S~.'IIRNOV
to reconstruct the whole picture, albeit in a difficult way. It would be interesting to establish the uniqueness using an appropriate modification of Proposition 4.1 . So that our t heorems apply even to models where t he uniqueness of the scaling limit has not been proved yet , we work with one of the subsequential limits Jio - a Borel probability measure on 'He, provided by Corollary 1.16. By Theorem 1.13, the Borel O"-field of H D is 0"(8Q ' Q E QD ), i.e. is generated by the crossing events inside D. Let FD: = a{BQ: Q E QD) also denote t he corresponding subfield of t he Borel a-field of 'He. THEOREM 1.19 (FACTORI ZATIO N). Consider a collection of percolation models, satisfying Assumptions 1.1 . Let D be a domain, and let aCe be a finite union of finite length paths with finitely many double points. Denote the components of D \ 0' by D j . Then for any subsequential scaling limit
(1 .4) up to sets of measure zero. Again , we note t hat this does not hold when Q' is a sufficiently wild path. Fix a subsequential scaling limit of the cri tical percolation for the bond model on the square lattice or for the site model on the triangular lattice. By the results above for a domain D it is described by a probability space (HD , FD , ttD), which is invariant under translations of t he plane (in fact , under all conformal transformations, but it is so far established for t he triangular lattice only), depends continuously on D , and satisfies (1.4). Then in the language of Tsirelson (see [Tsi04aj, Definition 3dl ) it is a noise or a homogeneous continuous product of probability spaces with a Boolean base given by an appropriate algebra of piecewise-smooth planar domains (e.g. generated by rectangles) . There is a well developed and beautiful t heory of noises, and we refer the reader to two exposi tions [Tsi04a, T si04bj by T sirelsol1 . In particul ar, every noise call be decomposed into a Hcla.'3sical," or "stable" part and a ~; non classical," or "sensitive" part. White noise is purely classical, and purely 110n-classical noi ses are called "black" by Tsirelson . The black noises are difficu lt to construct, with t he first example given by Tsirelson and Vershik in [TV98j. As poi nted out by T sirelson in [T si05j and in Remark 8a2 of [T si04bj, percolation noise has to be non-classical, and so our paper provides the first genuinely two-dimensional example of a black noise: 1214
ON T I'IE SCA LI NG LHvllTS OF PLANAR PERCOLAT ION
23
1.20 ( P ERCOLAT I ON IS A NOISE). Thus we conclude that, in the language of Tsirelson {Tsi04a}, any subsequential scaling limit as above is a noise with a Boolean base given by an appropriate algebr-a of piecewisesmooth planar domains (e.g. generated by rectangles). Th erefore, it has to be a black noise, as explained in {Tsi04b}, Remark 8a2. CO ROLLA RY
By t he conjectured universality) a ll the critical percolation models should have the same scaling limit and so correspond to t he same black noise. The situation with near-critical noises is less clear ) so t he followi ng question was proposed by Christophe Garban (the answer is expected to be negative): Q UEST ION 1.21. Are the noises arising fmm the critical and near-critical models isomorphic, or do we get a family of different noises'?
The notion of isomorphism for noises is discussed by T sirelson in Section 4a of [Tsi04aj. R.oughly speaki ng) we ask whether one can get a near-critical noise from the critical one by a local deterministic procedure'? Note that t he current construction of the near-critical percolation scaling limit would use the critical percolat ion a long wit h some extra randomness) as in [CFN06, GPSlOb).
Our construction of t he scaling limit applies to general random colorings, and much of what we do afterwards mostly uses "anTIs estimates," which are known for a wider class of models, t han just percolation . Therefore it is logical to ask 1.22 . To what extent out results apply to the models with dependence'? One has to be more careful in formulating the results, since now the configuration inside a domain depends on the boundaT1J conditions, but this can be addressed by working with a full-plane model and stripe domains. Q UEST ION
Acknowledgments . Vve are thankful to Christophe Garban and Wendelil1 \~lerner for useful conversations. T he second author is grateful to Hugo Duminil- Copin, Clement Hongler, Gabor Pete, J eff Steif, the referee, and especially Christophe Garban and Boris Tsirelson for t heir comments on the manuscript. 2. The Discrete G luing Theorem. In t his Section , we prove Theorem 1.5. 1215
24
ODED SCHRAl'vl M A N D STA NI SLAV Srv[IRNOV
Applying several times in succession Lemma A.I : we can find a smooth (Le. given by a diffeomorphi sm) quad Q, such t hat the crossing events for Qo and Q a re a rbi tra rily close. By the same Lemma, as c --+ 0, t he crossing event fo r Q is well a pproximated by t he crossing event of its perturbation Q~ := Q ([e, 1 - £]2). On t he ot her hand , a has finite lengt h and so for almost every c t he intersection an 8Qf: is fini te . This can be deduced , e.g., from the area t heorem in t he geometric measure t heory [Fcd69, Mat95], which implies t ha t a n ort hogona l proj ection of a rectifia ble curve on a given line covers almost every point at most finitely ma ny times. Thus, a pproximating if necessary, we can assume t hat 0: intersects 8Qo a t fini tely many points Xi, i = 1, ... , k. Fix EO > 0, then by the RSW' estima te (1.2), there is an T > 0 such that , if 0 < 1171 < T , then the proba bility under /1-,., that there is a crossing of Qo which intersects a ny of the disks B (Xi , T ) is less tha n EO. Fix such a n T , remove from the curve Ct the disks, the rest will be away from t he bounda ry, simplifying fu t ure estima tes: 0'.'
.-
d
.-
a \ ui B (Xil T), and
inf{lx-yl :x E a' , y EiJQo} > O.
Fix s E (O)d/ 4). By the regula rity assumption ) we ca n choose n::; C (a)/s balls of radius s, entirely covering a. Denote their centers by W I , W 2 , ... , W n , and set B j := B (w J , 28) . Let M denote the union of the set of t iles of H T) that intersect t he sneighborhood of a' minus Ui B (Xi, T). Assume 117 1 < 8 , so that !v! would be cont ained in the 2s-neighborhood of a' . Let wand w' be samples from J1-T) t ha t agree on the complement of 1\1{ and are independent in M . For j = 0, 1, ... , n , let Wj be the configuration t hat agrees with Wi on t he tiles of H T) lying inside B J U Bu ' . . U B j and agrees with w elsewhere. Then each Wj is a fair sample from Ji-,/ . Also observe that Wo = w and W n differs from Wi only inside u i B (Xi, T) . Our goal is to estimate
which we will do by successively comparing Wj - l to Wj ' F ix j E {I , 2, . . . , n }. In order for { Wj - l E BQo } n { Wj ~ BQo } to hold ) there must be a crossing in Wj _ 1 from 8oQo to ChQ o that goes through B j a nd t here must be a dual closed crossing in Wj from OJQ o to 0 3QO t hat goes through B j , see Figure 2.2. Thus, in Wj we have the four arm event from oBj to fJQo. Since the radius of B j is 2 s and the dista nce from B j to oQo is at least d - 2 s > d/ 2, by 1216
ON THE SCA LI NG LI J\HTS OF PLA NA R PERCOLATIO N
25
[Qol
FIG 2.1 . Setup JOT the prooj oj the dis crete gluing theorem in the simplest case.
Assumption (1.3) we have
P [Wj _l E
BQo,wj
rt
BQoJ <; P,i( Z, 28, dj 2) < (J/ 2)' /',.4( 28,dj2) .
Summing over j, we conclude t hat P two E BQo' Wn
1:.
n
BQo] ::;
L P [Wj _ 1 E BQo' Wj ¢. BQo]
j=1
<; n U/2) . /',.4 (28, dj 2) C(a) 48 <; - 8- . d
<;
4C(a)
d
. /',.,(28, dj 2)
. /',., (28, dj 2) .
By Assumption 1.1 the right hand side tends to 0 with s, and so we can make sure t hat P two E B Qo' Wn ¢. BQo ] < fO by choosing s sufficiently small. This is t he only place in the paper, where we use the pi votal estimate (1.3). 1217
26
ODED SC I·IRA r-.U...] AND ST AN ISLAV SM IRNOV
[Qol
F!G 2.2. Ij resampling a part of the ball B j changes the crossing event, than there are two open crossings fro m Bi to 8oQo and thQo, and two dua l closed crossings from Bi to 8oQo and &'Qo, - a four ann event.
Now , recall that Wo = w, and Wn differs from Wi only near endpoints of curves in 0, i.e. inside ui B (Xi, T) . By our choice of T, we have the estimate P [w n E 8QOl Wi ¢. 8Qol < fO. Summing t he above, we get the bound
P [WE BQ" w'
2'0 > P [w E BQ"w' ¢ BQoJ = E
[I'<, (BQo I F ,)(l - I',,(B Qo I F ,))].
In particular , by Chebyshev inequality,
Since
fO
was an arbitrary positive number , t his completes the proof. 1218
0
ON T HE SCA L ING L IM ITS
or PLA NAR PERCOLATION
27
3. The space of lower sets. The main goal of t his Section is to provide a n abstract setup for the percolation scaling limit and to prove Theorem 1.13 . \~re discuss topology on t he space ltx of closed lower subsets of an abstract pa rtially ordered topological space X . In the percolation case, X corresponds to t he space Qo of quads, and t he space (ltx , T'H ) to the space (lto, To) of percolation configurations in a domain D c if.:. Wi t h these conventions, Theorem 1.13 directly follows from Theorem 3.10 below. The lat ter a pplies to t he space of percolation configurations (colorings of tilings), since Qo is clearly second-counta ble, (3.1) holds by defini tion and (3.2) holds since a quad Qo can be easily a pproximated by smaller quads (e.g. by quads Qq wit h q = (- f,f, 1 + f, I -f) from the proof of Lemma 5.1 ). Vve now start t he (mostly classical) abstract construction , inspired by the Dedekind 's sections. Let (X , T) be a topological space, let ~x denote the collection of topologically closed subsets of X . For a topologically open U C X and a compact J( C X let
Wu -
{F E Ox,F n U fo },
WK
{F E Ox,F n J( =O }.
Let T be the topology on ~x generated by a ll such sets WI( and Wu . It is a more convenient version of t he Vietoris topology, called Fell topology or topology of closed convergence. The following result is Lemma 1 in [FeI62J: P ROPOSITIO N
3.1.
( ~x , T ) is compact.
For completness we include a short proof, based on Ale.xander 's subbase theorem (see, e.g., theorem 5.6 in [Ke1 75]), which states that if B is a subbase for the topology on a space X a nd every cover of X by elements of B contains a fini te subcover, t hen X is compact. By Alexander 's subbase theorem it suffices to show t hat every cover of JX by sets of the form Wu (where U E T) and WI( (where J( C X is compact ) has a fini te subcover. Let P ROOP.
{Wu ' U E I} U {WK , J( E J}
:::J ~x
be such a cover. Observe t hat U UEI W U = WO, where U = UUE I U. Let F := X \ U, and note that F is topologically closed. Since F E JX c 1219
28
ODED SC HRAl\H,.f AND STAN ISLAV Sl\'I I RNOV
WOUUKEJ WJ< , we know that F E UKEJ Hr I< , namely, there is some J(o E J with F E 1V K o. In other words, F n Ko = 0; that is, [(0 c U. Since Ko is compact and {U : U E I} covers Ko , there is some finite I' c I such that {U : U E I f} covers Ko - Then {Wu : U El'} U {Hf KO} is a finite cover of OX. 0 Now suppose that "<" denotes a partial order on X such that
(3.1)
R:= {(x ,y) E X': x < y} is a topoJogicallyopen subset of X"
Define R X := {y E X: y < x} and R x := {y EX : x < V}. Then for every x E X t hese sets are topologically open. A set H eX is said to be lower if x E H implies R X c H. Let 1ix denote t he collection of all topologically closed lower sets.
Assuming (3.1 ), H x C JX is closed with respect to that is, 6X \ H x E T.
PROPOSIT ION 3.2 .
the topology PROOF'.
T;
Set U x := W {x} n yVR", and
U:=
U Ux={F:
3x,y withx
xEX
Then U is topologically open and H x = ~x \
u.
o
Let Ix be the topology of H x as a subspace of ~x. It is the topology generated by the sets Vu := WunHx and VK := H! K n Hx , for topologically open U and compact f{ subsets of X. As we will later see, in our setting it is s ufficient to restrict ourselves to one-point compact sets, working with V {x}, which we will often abbreviate V X. For si mplici ty we will also abbreviate H := H x and T := TH. Propositions 3.1 and 3.2 imply that (H , T) is compact. In general, H does not have to be a Hausdorff space, even if X is Hausdorff. Indeed, the sufficient condition would be for X to be Hausdorff and compact, but for the application we have in mind , X is not compact. However, the following result gives anot her sufficient cond it ion for H to be Hausdorff: PROPOSITION
(3. 2)
3.3.
Suppose, in addition to (3.1), that
'tx
E
X , x E R" .
Then (H , T) is a Hausdorff space. 1220
ON THE SCAL ING Ll r-.HTS OF PLA NAR PERCOLATION
29
PROOF. Let HI , H2 E 1t be different , then one of the two differences H I \ H2 and H2 \ H I is non-empty, withou t loss of generality the first one. Take some x E HI \ H2. Since X \ H2 is topologically open in X and contains x E R X, we can find some y E R X n (X \ H 2) = R X \ H2. Then VRy is disjoint from V Y. "tvloreover, HIE Vny and H2 E V Y. So there are disjoint topologically open sets in I containing H I and H 2, respectively, and t he space is Hausdorff. 0 REMARK 3.4. Let I' be the topology on ~x generated by the sets W u , U E Tx and W {x}, x EX. Since this topology is coarser than I , Proposition 3.1 implies that ( ~x , If ) is compact. Th e proof of Proposition 3.2 shows that'H is a topologically closed subset of ~x , also with respect to T'. Moreover, assuming {3.2}, the proof of Proposition 3.3 shows that 'H is Hausdorff with respect to If. It is well known (and not hard to verify, see theorem 5.8 in /K el'l5j) that if a compact topology on a space is finer than a Hausdorff topology on the same space, then they must be equal. We conclude that the topology on 1t induced by T is the same as that induced by T' whenever· (3.2) and (3.1 ) hold. L EMMA 3.5. Suppose that (3.1 ) and (3.2) hold, and that Xo is a dense subset of X. If H I) H2 E 'H and H I =j:. H2, then H I n Xo =j:. H2 n Xo.
PROOF. Take x E H I \ H2, if the latter is non-empty. Then (3.2) implies that n x \ H2 =j:. 0. Since Xo is dense and n x \ H2 is topologically open and nonempty, we have X o n n x \ H2 =j:. 0. Because n x c H I, this implies that H I n Xo =j:. H2 n Xo. A symmetric argument applies if H2 \ H I =j:. 0. 0 In t he next Lemma we characteri ze convergence of nets in 1tx. As we shall see, (1t x , Ix.) is first countable, thus convergence of sequences actually leads to the same properties, but we decided to work with the more general notion for now , since it does not lead to additional difficulties. Suppose t hat D is a directed set (i.e. part ially ordered by"::;" so that every two elements have a common upper bound), and {y"t}nE'D is a net (i.e. a sequence indexed by D) of subsets of X. We wri te lim sUPn Y,., for the set of all x E X with the property that for every topologically open U C X containing x and for every mE]) there is some n ::: m in ]) satisfying Yn n U =j:. 0. Similarly, lim infn Yn is the set of all x E X with the property that for every topologically open U C X containing x there is some mE]) such t hat y;, n U =j:. 0 fo r every n ::: m . VIle write Y = limn Y,., if Y = lim sup" Yn = lim infn Y,.,. 1221
30
ODED SC HRAMM AN D ST AN ISLAV SM IRNO V
LEMMA 3.6.
A ssume that (3.1 ) and (3.2) hold, and that Sit, nE V , a net with 5 n E 1t for even) n .
lS
1. If limn S n c X exists, then limn Sn E 'H.
2. If the net S n converges to S with respect to T , then S = limn Sn . S. Conversely, if S = limn SrI! then the net Sn converges to S with respect to T. PROOF. To prove (1), suppose t hat S = limn S n) xE S and y < x . T hen
x E R y. Therefore, t here is some m E V such t hat Sn n R y f- 0 for every n t m . Si nce S n E 1i, t his implies that y E S n for every n t m . Thus, yE S , which proves that S is a lower set. The defini t ion of limn 8 11 makes it clear that S is topologically closed. This proves (1) .
To prove (2), suppose t ha t 5 n converges to S wi th respect to T . Let xE S. If U c X is topologically open a nd contains x, the n S E Vu . The refore, there is some 1n E 'D such that STI E Vu for every n ~ 1n . But STI E Vu is equivalent to Sn n u =I 0 . Therefore, x E lim inf1l STI; that is, S c lim in fn Sn. Now suppose t hat y 1. S. By (3.2) there is some z E R Y \ S. T hen S E V Z • Consequently, t here is some 1n E 'D such t hat Sn E V Z fo r all n ~ 1n. Hence S n n R z = 0 for all n ~ 1n . Since y E R z a nd R z is topologically open, this implies tha t y rt li m SUPn 8 n . Thus, S ~ lim sUPn Sn. Summing it up, we have lim inf1l Sn ~ S ~ lim sUPn Sn. Since lim infn Sn C lim sUP/t S1I1 this proves (2). To prove (3), suppose that S = limn Su' Then we know from (1) t hat S E 1t. Suppose t hat Snj is a subnet of S n that converges to some S' E 1t. By (2), we have S' = limj S nj ' Therefore, S' = S. Thus, every convergent subnet of S1l converges to S . Since 1t is compact, this proves t hat STI converges to S, and completes the proof of the Lemma. 0 3.7. A ssuming (3. 1) an d (3.2), if X is second-countable (i.e. with a countable base), then Jt is second-countable an d hence metrizable. LEMMA
PROOF. Assume that X is second-countable. In particula r, t here is a counta ble dense set Xo e X. Let B be a counta ble basis for Tx . We claim that 6 ' := {Vu : U E 6} u {V " : x E Xo} is a subbase fo r T. Indeed, if U' E Tx , then there is a subset J c B such that U' = U UEJ U. Then Vu' = U UEJ Vu; thus, Vu ' is in the minima l topology conta ining B' . Now suppose tha t x E X , then (3.3)
Vx =
vY .
1222
ON TH E SCALING LJ ivllTS OF PLA NAR P ERCOLATION
31
Indeed, if S E VX, then x fJ- S and by (3.2) the topologically open set R X \ S is nonempty. Take some y E X 0 n R X \ S, then SEVY . This proves that V X C UyEXon R" V Y. Since y < x, we also have V Y C V X for y E R X , concluding that VX = U yEXOn R" VY. Thus V X is also in the minimal topology containing 8 1 • Now Remark 3.4 shows that 8 ' is a subbase for T. Since 8 1 is countable, T also has a countable basis (of all fini te intersections of elements of 8 1 ). By Proposition 3.3, topology T is Ha usdorff. Since a second-countable compact Hausdorff space is metrizable (which follows from t he Urysohn 's theorem 4.16 in [Kcl75]), this completes the proof. 0 Suppose that (3.1) and (3.2) hold, and that X o is a dense subset of X. Then the B orel a-field of ('H., T) is genemted by V Y, Y E Xo. LE MMA 3.8.
Observe that the Borel a-field of ('H. , T ) is generated by topologically open sets, which in turn are generated by the sets of t he form Vu and V X for topologically open U C X and x EX. So we must prove that those can be obtained from V Y, y E Xo by the operations of countable union, intersection and t aking complements. For x E X , the set V X can be obtained as V X = U yExOn R" V Y, by (3.3). For a topologically open U C X note t hat by the definitions Vu = UXEU "" V x , Therefore Vu ::J UyEUn xo""VY, On the other hand , take x E U. By (3.2) we can find y E Xo inside the topologically open set Un R X , then y < x and ...,V X c ...,V Y. Thus Vu C UyEu n x o...,V Y. Vve conclude that Vu = UyEunxo ...,V Y, and so the sets Vu can also be obtained . 0 PROOF.
Finally observe that a topologically open X' C X inherits the properties (3. 1) and (3.2) from X , and the foll owing continuity under inclusions holds: L EMMA 3.9. Let X' c X be topologically open, and for S E 1t x let iJ> (S):= Snx' . Then IP is a continuous map from 1ix to 'H.X" PROOF.
Vve have to check that preimages of topologically open sets a re
also topologically open. This follows easily since a topologically open subset U c X' is topologically open inside X and 1P- 1 (V V) = VV, while for x E XI we have 1P- 1 (V X) = V x . 0 Summing it up, we arrive at the follow ing Let (X ,T) be a partially ordered second-countable topological space, such that the ordering satisfies (3.1 ) and (3.2). Then THEORE M 3.10.
1223
32
ODED SCHRAMJ'd AND ST AN ISLAV Sl\'I I RNOV
1. ('H.x, T'H) is a compact metrizable Hausdorff space. 2. Let Xo be a dense subset of X. If 5!, 52 E 'li x satisfy 5\nXo = S2nXO, then 51 = 52 - Moreover, the cr-field generated by VY ,yE Xo is the Borel a-field of ('Hx , T,,). 3. If {Sn} is a sequence in Hx and S E 'H.x I then Sn - S in T1t is equivalent to S = lim sUPn Sn = lim inrl! Sn' 4 . Let X' c X be a topologically open subset, and for S E Hx let I}>(S ) := Sn X', Then I}:l is a continuous map from 'H.x to 1i x ',
PROOF. P art 1 follows from P ropositions 3.1, 3.2 and Lemma 3.7; part 2 follows from Lemmas 3.5 and 3.8; part 3 follows from Lemma 3.6; and part 4 follows from Lemma 3.9. 0 4. Uniformity in the m esh s ize. In this Section, we prove a mesh-independent version of t he glui ng theorem:
Let De e be some domain, and let Qo E QD be a piecewise smooth quad in D. Suppose that a C C is a finite union of finite length paths, with finitely many double points. Consider a collection of percolation models indexed by a set {17}. A ssume that I--i--rI converges to (a subsequential) scaling limit /10 as mesh 1 171 -+ O. Then for even) € > 0 there is a finite set of piecewise smooth quads Q( C QD\a: and a subset W ( C H , which is measumble with respect to the finite a-field a (8Q ' Q E Q, ), such that P ROPOSIT ION 4 .1 (MES H-INDE P ENDENT GLUING).
(4.1 )
PROOF'. This is the most technically difficult part of our paper. T he rough plan is as follows: we set up a thin strip around a, and cut it into "bays," form ing a neighborhood of a , and a part away from a , bounded by "beaches." Partition depends on the percolation configuration , and has t he properties t hat (i) percolation in t he neighborhood of a is decoupled from the rest, and (ii) the crossings outside this neighborhood can be encoded by a finite information , stable under small perturbations of the percolation configuration. \"le use a complicated construction of the neighborhood for the sake of the property (ii) . To effect ively bound the needed crossing information away from a (whose amount grows as we pass to the scali ng limit) , we further coarse-grain t his procedure to a fixed small scale, when it is described by a fi nite a-field. This will allow to write the required estimates. 1224
ON THE SCA LI NG LIM I T S OF PLA NA R PERCOLATION
33
\:Ve now star t t he proof by fixing a count able subsequence {'TJj} from our set , such t hat 111J I -----Jo 0 (an d {/.7/j converges to Jio). Recall that by Remark 1.17 in our sett ing it is sufficient to work with sequences, rather t han nets. Setup for the neighborhood of a. Reasoning like in the proof of Theorem 1.5 (the first paragraph of Section 2), we can approximate the quad Qo , by q uads whose boundaries intersect a on a finite set. Thus we can assume t hat a has fin itely many double points, intersects fJQo on a finite set. Prolonging some of the curves in a if necessary (to cut the non-simply-connected components into smaller pieces) , we can further assume that each component of [Qol \ a is simply connected. Fix small EO > O. Define two random variables: one given by t he ind icator function of the crossing event Yo := l s Qo ; and another by t he conditional probability of a crossing, given t he percolation configuration s-away from a: Y, := />ry(ElQo IF,). Here Fs , 8 > 0, denotes t he O'-field generated by t he restriction of w to the complement of the 8-neighborhood of a, as in Theorem 1.5. By that theorem, for all suffici ently small 8 > 0 ,
(4.2)
sup
11IIE(O,s)
J-L1I(€O
< Ys < 1 ~ EO) < EO ·
Denote by Xi, i = 1, ... , n, t he finitely many points where a intersects fJQo, as well as the end points and the double points of curves in a. Let d be t he minimum over i of the distances from Xi t o ot her Xj'S and t he farthest of fJoQo and ChQo · \:Ve now fix s > 0 sufficiently small so t hat (4. 2) holds, a nd sufficiently small so t hat for i = 1, ... , n, the balls B(Xi' 2s) are disjoint, and (4.3)
sup 11IIE (O,s)
l-4l3 a crossing from B (Xi' 2 8) to 8B(Xi , d/3)) < co/n .
T he latter follows for small 8 from the RSW estimate (1.2). We also require that t he intersection of a with ui8B(Xi, 8) is finite (which holds for almost every value of 8, since the total length of a is finite). For t he following construction, the reader is advised to refer to Figure 4.1 . Let {3 and {3' be finite unions of disjoint smooth simple paths in [Qol \ UiB(xiJ s), and let f( be t he closure of the union of connected components of
[QoJ \ (U, 8 (x" s) Up UP') whose bound ary intersects both {3 and {3'. We can choose these unions of paths so that 1225
34
ODED SC URAt\Hd AN D ST A NI SLAV SM IRNOV
[Qol
f3 f3' O!
Xi
Xn FIG 4.1 . Curves in 0: are between {J' which are in turn between {3 . We modify configumtion on the small discs so that long quads bounded by f3 and f3' on long sides, have open tiles all
one short side and closed on another.
• all t heir endpoints belong to U/)B(Xil S) , • {3' separates (3 fr om cr, • t he connected component of [Q oJ \ f3 t hat contains a is cont ained III t he (s/2)-neighborhood of Ct , • each component of connectivity I<j of J( is a quad with two "long" sides on {3 a nd f3' and two "short" sides on t wo of t he circles 8B(x j, s). It is immediate to verify t hat t here exist such unions of pat hs.
Bays and beaches. The following construction is illustrat ed in Figure 4.2. Let w be a sample from l.t.,., for some small 17 > O. For somewhat technical reasons it will be convenient to consider in pl ace of w the configuration w which is modified on t he t iles intersect ing d isks B {xj, s) . \Ne alt er w so t hat for every component J(i one of two short sides is covered by open tiles and another by closed. Clearly, (4.3 ) implies t hat if 1] < s, (4.4) 1226
or
ON THE SCA LI NG LHvlITS
PLANAR PERCO LATION
35
An interface
{3' An open bay
An open beach
A closed bay
A palm tree on
8.
closed beach
FIG 4.2. Int erfaces between open and closed clusters, starting from {J, cut the strips l<; into components. Bays are the components touching {J', and are alternatively open and closed. Beaches are the inner boundaries of the bays, when entered through {J' .
For this reason , studying wwill still be useful. In what follows , we start on the curve f3 and explore all the potential crossings of the strip between f3 and f3'. To be more precise, denote by ow the set of percolation interfaces - curves, separating open t iles in w from the closed ones. Let I be t he set of connected components of the intersection of ow wi~h the interior of]{ (i.e., percolation interface in the interior of ](), and let I denote the set of elements of I that have at least one end poi nt on f3 . T he definition of wguarantees that ill each component Ie t here is at least one interface of w with one endpoi nt on f! and the other Ol~ f3'. Let r denote t he union of the elements of I ; that is. r = U I = U')"E!'Y. Each connected component of J( \ r which meets f3' will be called a bay. Clearly, each point in f3' is in the closure of some bay, and for each bay its intersection with f3' is connected. Let B be some bay. The union of the tiles of H1) that intersect B as well as oB \ f3' will be called t he beach of B. It is easy to see that t he beach is connected, and all t he tiles in t he beach are either open (Le. in w) or closed (not in w) . If they are all open [respectively, closedJ , then B itself will be referred to as an open [respectively, closed] bay. Let M' = i\l[' (w) be the union of the component of [QoJ \ (J' containing cr. and of all the bays without beaches. Denote by 1\;[ = 1\;[(w) t he complement in [QoJ of 1\;[', as well as the correspondi ng set of t iles. The following property of l\tJ is essential: given 1\1 and the restriction of w to it, t he conditional law of t he restriction of w to /vI' (i.e. the complement of 1\1) is unbiased . In 1227
36
ODED SCHRM ..uI'\ AND STAN ISLAV Stvll RNOV
other words, if we take
independent from wand of t he same law, and we define W2 to agree with w on ftll and to agree with WI elsewhere, then W2 also has the law fJ.-,}' This follows directly from the fact t hat Nf(W2) = A1(w). WI
Below we will denote the restriction by wLAIf. Fix some small 8 ' E (0, s), and let a small bay B , i.e. with diam(Bn j3') end points of B n f3' are endpoints of
of a percolation configuration w to !vI
S = 58' denote the event that t here is < s' . Observe that for every bay B t he interfaces of w which meet both f3 and f3'. Thus three crossings of J{ (which are alternating, e.g. closed , open and closed) land on the arc B nf3' and Lemma A.2 applied to all components of J( implies that lim sup ",, (S) ~ O. 8' ..... 0 IT/le(o,s')
By choosing 8' sufficiently small , we t herefore assume, with no loss of generality, that 1)..,,(8) < £0 holds for every 1] E (0,8') and that 8' < 8. Observe that on -,8 the number of bays is bounded by a constant depending only on 8', {3,{3' and Qo · We will fin d t he structure of the bays t o be a convenient component in the estimation of the conditional probability (given the bays and some additional information) of w E BQo' However, as 1171 __ 0, the number of possible bay configurations increases without bound, and therefore their structure cannot be completely captured by a finite O"-field. For th is purpose, we need to introduce a discretization , and any sort of coarse-graini ng, e.g. by the square lattice, would do.
Discretization to a finite a-field by coarse-graining. ' ''''e need some tessellation of J( , and the following one is somewhat ad hoc, but suitable. Let 6 E (0, i) be small, to be chosen later. Fix some fini te set of points W C J( such that the distance between any two points in W is at least 6/ 4 and every point in J( is within distance 6/2 of some point in W. Let T be t he collection ofVoronoi tiles in J( with respect to W; that is, each Tw , w E W, is the set of points x E f{ whose distance to W is equal to Ix - wi. Then t he diameters of the t iles in the tessellation Tare all less t han 6 (but the lattice H,/ tiles are still much smaller ). Let T denote the tessellation obtained by adjoining to T t he clos ures of the (finite in number) connected components of [QoJ \ J( that do not contain Ct. '''''e think of tessell ations T a nd T as cell complexes. The vertices of T are defined as the vertices of t he Voronoi tiles together with the points in (,Bu {3') n aQo . The vertices of T are defined as those of T plus the four corners of Qo. Note that some of t he edges of Tare not necessari ly straight. Denote by [T] the union of the tiles in T. 1228
ON THE SCALING LIMITS OF PLANAR PERCOLATION
37
We say that a quad Q E QD is compatible with T if [Q] is a union of tiles of T and each of its four corners Q(O, 0), Q(O, 1), Q(l, 0), Q(1, 1) is a vertex of T. Since T is finite, it is clear that up to reparameterization preserving corners there are only finitely many compatible Q. Let Qt denote a set of representatives of equivalence classes of compatible quads up to reparameterization preserving corners. Then Qt is a finite collection of piecewise smooth quads in D \ £x. Let FT denote the O"-field generated by the events E3 Q , Q E Qt. Without loss of generality we assume that along our chosen sequence {7]j}, for each edge e of the tessellation T that does not touch 8Qo, • each edge of HT/ intersects the tessellation edge e in at most finitely many points, • the tessellation edge e does not contain any vertices of HT/' • the endpoints of the tessellation edge e are not on edges of HT/. There is no loss of generality in this assumption, because we may slightly perturb /3, /3' and T (and we don't even need T to be a Voronoi tessellation). Let B be some bay. Define T(B) to be its T-discretization, namely the union of all edges of T inside B n /3' and all tiles of T inside B, which can be connected to /3' by a path of tiles of T also contained inside B. By construction, each point in 8T(B) is within distance 5 from 8B. Note that there is a finite number of possibilities for the choices of T(B), and also observe that whenever S does not hold, all T-discretizations are non-empty and so T(B) = T(B') implies B = B'.
Connectivity in the coarse-grained model. The next objective is to show that on the complement of S the collections of discretized bays Zo .- {T(B): B an open bay}, and Zc .- {T(B): B a closed bay}, can be determined from FT. That is, there are FT-measurable random variables Z~ and Z~ such that Zo = Z~ and Zc = Z~ on the complement of S. Here and in the following, we ignore events that have zero itT/-measure for each 7] E {7]j}. Note that on the event -,s the cardinality of Zo and Zc is bounded by a constant depending on Qo, /3' and s' only. Let e be an edge of T that lies on the component /3j = /3' n K j , denote also /3j := /3 n K j . Let Qe and Q~ be the quads in Qi' that satisfy 80 Qe = 81Q~ = e, 82 Qe = 83Q~ = /3j and [Qe] = [Q~] = K j . Then the event that there is an interface 'Y E j that meets e is the same as the event E3 Qe \ E3 Q~ . 1229
38
ODED SCHRAMM AND STANISLAV SMIRNOV
If E is the set of all edges of T on f3j which meet interfaces in i, then on the event -,S, the connected components of f3j \ U E are the intersections of the elements of Zo U Zc with f3j, and since the bays alternate in color, it follows that the sets {T n f3j : T E Zo} and {T n f3j : T E Zc} can each be determined from FT, up to an event contained in S. Next, suppose that, c f3j is of the form Tonf3j for some TO E Zc (assuming -,S, such intersection is always non-empty). Let To be some tile of T. Then To C TO if and only if there is a simple path " of edges of T with the endpoints of " on , such that " separates To from f3j in K j and there is no open crossing in K j n W from " to f3j. This, along with the dual argument, implies the existence of Z~ and Z~, as claimed.
Describing the crossing structure by a finite graph. We now introduce a graph that describes the connectivity (by open crossings) structure of the various open beaches and the two boundary edges ooQo and 02QO' We start by defining a random graph G which would describe the connectivity away from a. The vertices of G are Zo U {ooQo, 02QO}, where an edge is placed between vertices v, v' E Zo if the beaches of the corresponding bays are connected by an open crossing in the restriction of W to M, denoted by wLM; while if v and/or v' are in {ooQo, 02QO}, the connectivity to the beach is simply replaced by the connectivity to the corresponding boundary edge itself. Note that there is a finite number of possibilities for the choice ofG. Suppose that TO, Tl E ZOo Let ej, j = 0, 1, be an edge ofT that is contained in f3' and has an endpoint in Tj, but ej itself is not in Tj. Then ej meets an interface on the boundary of the corresponding bay. On -,S, the two edges ej are connected by an open crossing in wL[1'l if and only if [TO, TIl is an edge of G. Thus, on the event -,S the subgraph of G induced by Zo is determined from FT. A similar argument applies also to the edges with endpoints in {ooQo, 02QO}. Thus, on the complement of S the graph G can be determined from FT. Now we define a random graph G* so that it describes the connectivity near a. Consider L = (ooQo U 02QO) \ [1'], which contains finitely many arcs on oQo. Denote by G* the graph whose vertices are Zo and the connected components of L, where v, v' E Zo are connected by an edge in G* if the two corresponding beaches of ware connected by an open crossing in WL([Qol \ M), and if v and/or v' are components of L, then the connectivity to the beaches is replaced by connectivity to v and/or v'. Clearly, W E E3 Qo if and only if there is a path from ooQo (or a subarc thereof) to 02Q2 in G U G*. Thus, on the event -,S, the graphs G and G* 1230
ON THE SCALING LIM IT S or PLANAR PERCOLATION
39
determine whether W E 8 Qo ' Denoting by wLAI{ the restriction of W to AI{, we set
Y = Y(w) := /1o,(w E BQo I wLM). V!le now show_that on -,5, the knowledge of G, Zc and Zo can be used to approximate Y . Let that is, t he complement in [1'] of t he union of T(B), where B runs over all the bays. Let Gt be t he graph on t he same vertex set as G*, where an edge appears between v, Vi E Zo if t he common boundary of v and T(AI{) is connected in WL([QOJ \ T(M)) with t he common boundary of v' and T(M) , while if v and/or Vi are components of L , t hen t he connectivity is instead to v and/or Vi, but still within WL([QO] \ T(Jv!)). VJe now show that if 6 is sufficiently small, "1 < 6 and 1} E {"1nL then
(4.5)
/1o,(Gt
# G',
~S I wLM)
< '0·
As we have mentioned, on the event -,5 t here is a fin ite upper bound on the size of Zc U Zo, depending on Qo, {31 and Sl only. Therefore, it is enough to have a good estimate for the probability that a particular pair of vertices v and Vi of G* are connected by an edge in one of the two graphs but not in the other, and we are free to impose addi tional conditions on 6. Connectivity of any pair of vertices is reduced to 6-perturbations of quads of size at least Sl, and so is easily obtained by several applications of Lemma A.I (some of them applied with the corners of t he quads appropriately permuted), proving (4.5).
Final estimates.
Yr
:=
Define
Yo =
Yo(w) :=
l WEBQo
and
!">,(fioQo and /hQo are connected by a path in Gt U G I wLM).
Then Yr is a function of the triple (Zc, Zo, G), and is therefore Fer measurable. By the above observation that on -,5 the event BQo is described by connectivity in G U G*, we have
1231
OD ED SCH RAM i'd A ND STA NISLAV Si\H RNOV
40
By taking expect ations, applying (4.5) and recalling t hat i was chosen t o guarantee /-L(S) < EO one gets
(4.6) where the norm refers to t he measure J-l-r,.
By (4.2) a nd the defini tion of Ys , we can write
and by (4.4 ), we have
(4 .7) Hence II Yo - Ysll2 < ) 3 EO + J€o < 3 Fa- Since Ys is F s-measura ble, i t is also wL.M-measurable. By its defini tion Y minimizes liVo - X II~ among wL M -meas urable random variables X. Therefore co mparison t o Ys (reca ll t hat t he set 1\11 includes t he complement of t he s-neighborhood of a) yields
Combining t his with (4.6) a nd (4.7), we concl ude that
Because :Yr is F I'-measurable and EO may he chosen a rbi t ra rily small , this proves t hat t here is an Fr-measura ble event W (which may depend on 1JL such that J-L1/(W.1.. E3Qo) < t./ 2. However, since Fr is fini te, t here are finitely many possibili t ies for t he event W , and one of t hose works for a subsubsequence T/jk' Since we work wi t h a (subsequent ial ) scaling limi t , the limit in (4.1 ) exists along t he or iginal subsequence and is equal to Jio (W.6. BQo) ' Since along a sub-subsequence t he quant ity in question is bounded by € , so is t he limit and we deduce (4.1 ). 0 5 . Fa ctorization.
In this Section , we prove t he Factorization Theorem 1.19. LEM /I.'IA 5.1. Let Jio be some subsequential scaling limit, then the boundary (i n topology T on 1i) of a crossing event has pmbability zem. Namely,
l'o(8BQo) = 0 holds for every Qo E Q D. 1232
ON T HE SCA L ING LI M ITS OF PLANA R PERCO LATI ON
41
Fix Qo E Q D and let f > O. It is easy to see (e.g. , using the Riemann map onto t\ [Qo]) that there is a continuous injecti ve 00: [- 1, C, whose restriction to [0, I]' is QQ. Let M := [- 1,1/3[2 x [2/ 3, 21 '- For every quadruple q = (xo, Yo, Xl, vI) E !vI define t he quad Q q : [0, C by PROOF.
2F-
IF _
Qq (x, y) := Qo(xo
+ (XI
- xo) x , Yo
+ (YI
- Yo) V) ·
It is a perturbation of Qo , obtained as an image of [xo, xI] x [Yo, vd by Qo. Lemma A.l implies that there is some positive 0'0 > 0 (depending on Qo), such that if q and q' are in A1 and differ in exactly one coordinate, and t he difference in t hat coordinate is at most 0'0, t hen lim sUPI'lI-+o 1l-rJ( BQQ b. 8QQ/) < f. Consequently, we have lim sUP /J.'I(BQQ b. 8 QQ ,) <4f, provided
1'1 1-0
IIq- q' lIoo::=; Jo.
Set qs := (-8, 8, 1 + 8, 1 - 8) , Q' := QQ- 60/ 2 and Q" := Q Q60 / 2 . Then Q' < Qo < Q" and
(5. 1)
li m SUPi'>I(ElQ' L'> ElQ")
17/1-+0
<
4,-
Also, Q' , Q" E Q D, if 60 is chosen sufficiently small. Recall t hat V Qo = 'H. D \ 8Qo is open in 'H. D. Hence 8Qo is closed. Let U ' be the set of quads Q E QD satisfying Qo < Q < Q". Then By passing to t he complements, we see that t here is a closed subset (the complement of VU/) t hat contains V Qo = -,8Qo and is contained in V Q" . Hence, the closure of VQo is contained in V Q", which gives o ElQ, = oV Q, C V Q, C V Q" .
On the other hand , let U be t he set of quads Q E Q D satisfying Q' < Q < Q", then Now observe t hat Vu c 8Q' while V Q" = -,8Q". We have shown t hat t he open set Vu n V Q" contains 88Qo' The portmanteau t heorem (see, e.g., [DudS9, Thm. 11.1 .1] ) therefore gives
~o(aElQq)
s: "o(Vu n V Q") s: li1111-+0 m in! 1.,(Vu n V Q") : =; li m inf IJ.,/( ElQ' b. 8Q") . 1'11---.>0
Since
f
was arbitrary, the result now follows from (5.1 ) 1233
D
42
ODED SC HRAi\Hd AND ST AN I SLAV SMIRNOV
If M C J-iD is in the Boolean algebra generated by finitely many of the events I3Q, Q E Q D (i.e., it can be expressed using finitely many E3Q and the operations of union, intersection and taking complements), then Jio(M ) = iimlIJl--+o {/1'}(M ). CO ROLLA RY 5 .2 .
PROOF'. Lemma 5. 1 implies t hat Jio(8M) = O. Thus, the corollary follows from the portmanteau t heorem [DuclS9, Thm. 11.1.1J and the weak convergence of f-J.-,/ to JioD PROOF OF TI-IEOREr.·[ 1.1 9 . Clearly, F D\I:t. = Vj :For It therefore remains to prove the left hand equali ty stated in the t heorem. Note t hat we work up to sets of /.l{J-measure zero. Take a smooth (given by a diffeomorphism ) quad Qo E Q D. By Proposit ion 4.1 and Corollary 5.2, we have
(5. 2) Since such collection of quads is dense in QD 1 Theorem 1.19 follows from Proposition 1.13.2. D APP ENDIX A, CONTINUITY OF CROSSING EVENTS In t his Section, we apply Russo-Seymour-\;Velsh techniques to deduce estimates of various crossing probabilities. vVe start by showing that crossi ng events are stable under small perturbations of quads. LEr.H... IA
A.1.
Th eTe exist a positive function .6. c(6, d), such that
lim .6.c( 6, d ) = 0 fOT any fixed d ) '_0 and the following estimates hold. Let Q be a quad. Let dj , j = 0, 1, be the infimum diameteT of any path in [Q] connecting OjQ and OJ+2Q , and define d:= max{do, d 1 }. Fix some 5 < d/ 2. Let Q' be another quad, satisfying at least one of the following conditions, see Figures A.i, A. 2 for a graphical interpretation.
(l). [Q'[ = [Q[, floQ' = floQ, {hQ' = 8 , Q , and there is a path 7 C [QJ of diameter at most 8 that sepamtes {Q(I , 1), Q'(I , I)} from floQ U &, Q inside [QJ. (2). [Q'J C [Q[, floQ' = floQ, 8 , Q' &'Q' can be connected to &,Q (3). [Q'J c [Q[, floQ' c floQ, &,Q' ChQ' can be connected to ChQ
C &,Q, &'Q' c &,Q, and each point on by a path a C [QJ with diam(a) :S 8. = &,Q, &'Q' c &,Q, and each point on by a path a C [Q] with diam(a) :::; 5. 1234
O N T H E SCAL ING LIMITS OF PLA NA R PERCOLATIO N
43
Q(O , l )~ Q'(O , l )
fhQ
[Q]
Q(O ,o) ~ Q'(O ,O )
Q( l ,O) ~ Q'(l,O)
F Ie A . 1. Case (lJ : quads Q and Q' differ by only one vertex, sepamtedjrom the other vertices by a short cut "f . Configumtions with only one oj two quads crossed have an open crossing jrom "f to fJoQ and a dual closed crossing jrom "f to PI Q, making their probability small.
The n JOT every 1111
< 6 we have
For si te percolation on the t riangular lattice, t here is a short proof based on Cardy's formul a. The proof below is a simple a pplication of the RS\~T estimate (1.2) and t he "lowest crossing" concept. First we deal wit h Case (1) in t he statement and prove the estimate with L1.. c equal to L1..1 from Assumption (1.2). Suppose first that d = do. The event B Q6. BQ' is contained in the event that there is a percol ation cluster meeting i and 8oQ. The latter has an open crossing of t he annulus A(Q(I , 1), 6, do), which by the RS\'" estimate (1.2) has probability at most L1..1 (6,d), and we are done with Case (1) . Ifd = d l , a similar argument shows the symmet ric difference between the event of a closed crossing from 01 Q to fhQ in [Q] and t he corresponding event in Q' is bounded by 6. 1 (6, dJ) . Duali ty shows that the lat ter symmetric difference is the same as B Q L1.. BQI, which proves Case (1). We now deal with Case (2), when clearly BQ 6. B Q = BQI \ B Q. \Ve start by estimating the crossing proba bility for Q' . Recall our convention that 6. (1', R) = 1 for r ~ R (i. e. when annul us in question is empty) . Let x be some point on the path T of diameter dl , connecting OIQ to fhQ. Any crossing of Q' has diameter at leas t do - 6 and passes wi thin distance 6 of T, so in particular it crosses an annulus A (x, d l + 6, (do - 6) / 2) if well defined , PROOF.
I
1235
44
ODED SC HRAMM AND STAN I SLAV SM IRNOV Q(O ,I)~ Q'(O,I )
, 1,1)
[Q]
FIG A.2. Case {2}: side fhQ' is close to the side fhQ. We take the lowest crossing "f of Q', landing at x . A dual closed crossing prevents it from landing on fhQ, making the probability small. Case (3) is symmetric.
and by t he RSW estimate (1.2) we have
(A.! ) Next we cut thQ' by a point z into two parts, Ul and (/3, so that l7j cannot be connected to f)j Q by a path of diameter less than d I/2 inside Q. If E3QI occurs, there is a crossing to at least one of 0"1 and <73. We will work with configurations with a crossing landing on (13 , the ot her ca.'3e being symmetric , wit h the lowest crossing replaced by the uppermost, so the total estimate will be double of what we obtain. Consider a percolation configuration w . On the event 8QI, let I be the "lowest" w-crossing of Q' (in t he sense that it is the closest to thQ', i.e. separates th Q' from all other w-crossings of Q' within [Q']), and let x be the endpoint of "f on thQ' . Our assumpt ion implies that x E {T3, or there would be a lower crossing. We now estimate /'.L7](-.BQ I BQI ,"f,x E (T3) . Let All be t he connected component of [Q'l \ "f which has 8 1Q' on its boundary. T he event -.B Q would imply that "f cannot be connected to thQ by a crossing inside [Ql \ M . Hence t here is a dual closed crossing from thQ to (a3 n NI ) U 8 1Q, in particula r crossing the annulus A( x, 6, d 1/ 2} inside [Ql \ M. The lowest crossing "f depends only on the configurat ion inside NI , so t he restriction of w to [Ql \ NI is unbiased. Therefore, the conditional probabi lity of the mentioned annulus crossing is can be bound by the RSW 1236
ON TH E SCA LI NG L1 l\HTS OF PLA NA R PERCOLATION
45
estimate (1.2) and we conclude t hat (A.2 ) Now we are ready to prove t he estimate, working ou t sepa rately t hree possibilities:
(i) d = d l , (ii) d = do > 6 ?: d l · (iii ) d = do > d l > 6, \¥hen (i) occurs, we majorate the proba bility of the event in question by the max imum of its condit ional probability, estimated by (A.2 ), to arrive at
When (ii) occurs, we use t he estimate (A. I }:
Finally, when (iii) occurs, we a pply t he total expectation la w, using both estimates (A.2) and (A.I ) along the way:
1',( El Q'
\
ElQ):S E [1',( ~ElQ I El Q " 'Y, x E "3)I + E [J1., (~El Q I ElQ','Y,XE " d l
:s 2 1'>1 (ElQ') ~I (6,~I )
:S 2~ I (dJ+6, do ;6) ~1(6, ~1) (A.5 )
:s 2 ~I (2dl'~) ~I
(6,d;)
To prove Lemm a in Case (2), we have to show that for fLxed d and any € > 0, the right ha nd side of t he estimates (A.3), (A.4) and (A.5 ) is smaller t han €, if t5 is small enough. For estimates (A.3 ) and (A.4 ) t his follows directly from Assumption 1.2. For t he rema ining (A.5), let p > 0 be such that for l' < p we have 6. ] < €/ 2. T hen if d 1 ::; p, then t he right hand side of (A.5) can be bounded by
(21', *)
2~ I (2dl '~)
~ I (6, ~I ) :s n 1237
l
(2d
l,
n
< "
46
ODED
SC H RAM~d
we are done. Otherwise d. bounded by
>
AND STANI SLAV sr-,mlNOV
p and the right hand side of (A.5) can be
which tends t o zero with 15 by Assumption 1.2, and we finish the proof in
Case (2). The proof for Case (3) is easily obtained by considering the dual closed crossing from to Ih and applying Case (2). The details are left to the reader. 0
o.
The following Lemma shows that it is unlikely to see three crossings approaching the same boundary point. This would make the crossing event unstable under boundary pertur bation , so in princi ple the Lemma can be deduced from t he previous one, but instead we give a self-contained proof. The Lemma would follow from considering the three ann event in half-annuli, for whose probabil ity Ai~enman proposed an argument to be comparable to (r/ R? We will use a version of his reasoning along with RSW techniques to show a weaker estimate, sufficient for our purposes.
Let Q be a quad with smooth sides, two opposite being labeled f3 and f3', and with the tiles on each of the two other sides v and v' possibly declared all open or all closed. The n there is a function 6.Q(b) , tending to zero with 15 and such that the following estimate holds. Denote by S (0) the event that there are three crossings with some prescribed a lternating order (say closed, open, and closed) inside [QJ between the set 0 C {3' and {3. Then for 1171 < 15 we have LEMl'I'I A A.2.
'" {30 c (!' , 5(0), diam(O)
< 6J :S AQ (6)
PROOP. Fix t: > O. \Ne shall prove that for sufficiently small 5 the probability of the event in question is less than (. Since crossings can in principle use tiles on the sides v and II (which are declared open or closed), we have to treat O's near the corners of Q separately. Let z\ and Z2 be the endpoints of {3'. Since tiles on each of the arcs v and v' can be used by at most one of the t hree crossings, event S (0) implies existence of at least one crossing from e to f3 inside [Q], not touching the sides. By Assumption (1.2), there is l' E (0, b) such t hat probability to have a crossing between Uj B (Zi' 21' ) and f3 is smaller than t:/2:
1'>, (30 C U.B(z., 2,.) , 5 (O)}
1238
:S
,
2
ON THE SCA LI NG L IM ITS
or PLANA R PERCO LATION
47
Set {3" := {3' \ UiB(Zi' 1'), then to prove Lemma it remains to show t hat
'"" pO c (3" , S (0), diam(O) < oj S ~ .
(A.6)
Using Assumption (1.2), choose p E (0,1'/4) such t hat /:;:" 1(2p, 1'/2) < 1/2 . Cover {3" by a finite number n = n(p, f3") of overlapping arcs OJ of diameter at most p. Let b < P be so small that /:;:"1 (46,1') < E/(12n). Since f3' is smoot h, and decreasing b if necessary, we can cover each a rc OJ by arcs 0; of length 315, overlappi ng at most t hrice, and such t hat any set 0 C f3" of diameter le'3s t han b is entirely contained wi t hin one of the arcs 0; . To deduce (A.G ) it is sufficient to show that, for a fixed j, (A. 7)
Fix j and let B(z, p) be a radius p ball, contai ning OJ. The point ZI splits &Q \ OJ into two arcs: )..' c f3' and a >. => f3 . Both start at endpoints of OJ inside 8(z , p) and end at least T-away. Observe t hat event is contained in t he event 5' t hat theTe aTe thTee alternating crossings inside [Q] between 0 and f3 U v U v'. Condition on the event 5' and, starting fro m Z2, denote t he three such crossings, closest to Z2, by /1 , /2, /3. Namely, let /1 be the closed crossing between fJ and f3 U v U v' inside [Q], closest to Z2 . Let Ml be t he component of connectivity of [Q] \ /1, not containi ng Z2 . Then /1 depends only on t he percolation configuration in [Q] \ MI. Now, take /2 be t he open crossing between 0 and {3 U v U v' inside Nh, closest to /1. Let Ah be t he component of connectivity of Nh \ /2, not containing ')'1. Then ')' 1 and ')'2 depend only on the percolation configuration in [QI \ M,. \'Ve define /3 and ftih similarly, and observe that /1, /2, /3 depend only on t he percolation con figuration in [Q] \ !vh Therefore, given A'/3 and t he restriction WL([Q ] \ fti!3) (Le. the restriction of percolation configuration W to [Q] \ !vh), t he conditional law of the rest rict ion of W to !vh is unbiased. Let 5* = 5* (ftih) be the event that theTe is a closed crossing /4 between .\ and /3 inside M3. Using t hat t he restriction of W to M3 is unbiased, we can then wri te
S(On
(0;)
(0;) ,
(A.S)
'""
[S' I M"
WL
[QJ \ M,]
~
1'"
[S' I > 1 -
P,: (z, p, ,)
>
~
.
Above we use t hat (in an unbiased percolation configuration) , if t here is no open crossing of t he annul us A (z, p, r), t hen by duality there is a closed 1239
48
ODED SCHRAMM AND STANISL AV SM IRNOV
circui t (a path going around t he annulus), whose intersection with then contain a crossing required for S*.
(On
AI{3
would
(OJ))
Note that when 8' and S* occur , t he crossing /3 from (8 and the crossing ')'4 (from S*) together give a closed crossing between fJ and .\
and so t he following event 5" (8}) occurs: there is a closed crossing '1'3 from () to A, an open crossing from ;~ from fJ to {3 u v U Vi , and a closed crossing fmm Ii jmm f) to (3 u v u v', Then the estimate (A.S) can be rephrased as
and therefore (A.9)
Consider some percolation configuration w in [Qj, and assume that 8" (8~) holds for some 0; C OJ. Choose inside Q a closed crossing 'i'3(W) between ).' and OJ; and an open crossing ')'~(w) between ,\ which are the closest to each other (and t he point Zl separating >'" from >.. ). Denote by L = L (w) the union of t iles in 12, 13, and in t he part of [QJ between them. T hen 12 and 13 depend only on t he restriction of w to L , and t he rest is unbiased. tvloreover, if 12 ends at a tile on OJ, then 13 ends at a neighboring tile on OJ - otherwise using duality reasoning we can choose two closer crossings. Denote by x the only common vertex of t hese two t iles lying on the boundary of N. Note that each configuration w has a unique such point x = x(w) and it depends only on t he restriction of w to N. Now for to occur, besides 12 and 13 we must also have a closed crossi ng ,; from 0 to (J U j) U ,;. The crossing ,; necessarily lies in [QJ \ N, which is unbiased on wLN. Note also t hat ,; contains a crossing of the annul us A (x, 2&, r), whose center x is a random point dependi ng on wL L only. Summing it up, we can estimate
S(On
"'" {w S"
(oj)
I x(w) =
x } :S P~ (x , 48, r) :S
Since every point x is covered by at most three arcs 0), we recall (A.9) and
1240
ON THE SCA LI NG L IM ITS
or PLANAR PERCO LATION
49
concl ude that
L::, 1',[S (ej) 1:s L::,
[s" (ej) 1 ~ L::L::2!',,{w s(ej) x
2/f.'7
,
:s ;;: (2.3
I~n ) 1"'7 (w
~ 2~ ~ /17I{W:
Ix(w)~x } !',,(w x(w)~xl 0
x(w)
x(w)=x} :::;
~ xl 2~ '
o
so estimate (A.7) a nd the Lemma follow.
APPENDIX Bo MULTI-SCALE BOUND ON THE FOUR-A RM EVENT ON Z2 (BY CHRlSTOPHE GARBAN) In this appendix , we give a proof of Assumption (1.3) for critical bond percolation on the square lattice. Namely: B.1. For critical bond percolation on the square lattice with mesh 1111, there is a positive € such that the pmbability of a four ann event satisfies LE1..1t'.O\A
p;; (z,
(B.I ) whenever 11}1 <
1',
R) :::; const
(R,")1+' '
1' .
The case l' = 1111 can be extracted from [KesS 7] as well as [BKS99] or [SSlO]. Note that the above multi-scale generalization is not a consequence of the so-ca lled ~'quasi-mul t i plicativity" property. If one wanted to use t his property, t he problem would more or less boil down to the existence of the four-arm critical exponent. Since its existence is still open, we believe that Lemma (B. 1) cannot be extracted directly from the above mentioned papers. In t he papers [BKS99, SSlO], the general idea which enables one to prove that points are unlikely to be pivotal is the observation t hat a macroscopic crossing event for critical percolation is asymptotically uncorrelated with the Majority Boolean function defined on the same set of bits. This asymptotic uncorrelation can be understood using an exploration path which will compute t he percolation event while giving little information on t he Majority event. A careful analysis then shows that t his decorrelation is possible only if points are unlikely to be pivotal for the percolation event. This heuristical program has been carried out in a very convenient manner in [OS07J. 1241
50
ODED
SC H RA M~d
AN D ST AN I SLAV sr-,mlNOV
The present proof has the same flavor except t hat one now looks at the correlation of a percolation event in a doma in of diameter O(R) wit h a twolayers majority function: namely t he bits of t his tvlajority function are now indexed by the O(R2 /r2) r-squares included in the domain and for each such T-square, its corresponding bit ±l depends 011 how "connected" the percolation configuration is within t his square. This setup allows t o "interpolate" the a bove program from the macroscopic scale R to t he mesoscopic scale r . PROOF'. We leave out a few details, which are easy to fill in for readers familiar with t he applications of the RSW· t heory, similar to [KesS7, SW01, GPSlOal· After rescaling and , if needed, changing the radii by bounded factors, we can assume wit hout loss of generality t hat the lat tice mesh is 1 and T, Rare posit ive integers. Denote by Q an R x R square, and cut the concentric ! Q square into TXT
un
squares, denot ed by Qj, with j = 1, .. . , 2 . Denote by X = 2·1 8Q - 1 the "crossing random variable" equa l to 1 when Q is cro5.'3ed and - 1 otherwise. Given a percolation configuration w, we say that Qj is pi votal for X ; if al tering w so that all t he bonds in Qj are open , a nd so t hat all t he bonds in Qj are closed yields two different values of X. Note t hat Q j is pivotal for X if and only if then there are two open arms connecting GQj to 80Q and (hQ , and two dual closed arms connecting GQj to G1Q and fhQ. By arm separation proper t ies s imilar to ones used in [SvV01],
p 4(r, R) '" P IQj pi votal for
(B.2)
XI ,
and so it is sufficient to estimate the latter probability, or its sum over all Qj's. Let 5j be an annulus (1 - 5)Qj \ (1 - 25)Qj where nQj denotes a copy shrank of Qj by a factor of a: (if 1· is not exactly divisible by 0:, t he copy shrank is understood modulo integer pa rts). The fixed para meter 5 > 0 will be chosen later . Let
S;
~
S; denote t he same annulus but on t he dual lattice: Le.
Sj + (1/ 2, 1/ 2).
Define a random variable OJ equal to • 1, if t here is an open circuit in Sj and no dual-closed circui t in Sj, • - 1, if there is a dual-closed circuit in Sj and 110 open circuit in Sj , • 0, otherwise. Note that by symmetry an d by the RSW t heory, up to the const ant depending 011 5,
(B.3)
E ICjl
~O, E 1242
[CJ] '"
1.
ON THE SCA LI NG LIM IT S
or PLA NA R PERCOLATION
51
Now, let us argue that if t5 > 0 is chosen small enough, one has
E [XCjJ '" P [Qj pivota l for X J ('" P' (r , R) ) ,
(B.4)
Indeed , one has
(B,5)
E [XCjJ ~ P [Qj pivotal for X J E [XCj
I Qj
pivotal for X l,
since, conditioned on the event t hat Qj is not pivotal for X , Cj is independent of X and is s uch t hat its (conditiona l) expectation is st ill O. Therefore it rema ins to bound from below E [XCj I Qj pivotal for X]. Let p = p(t5) > 0 be such that P [Cj ~ I J ~ P [C, ~ - I J > p,
E [XCj
I Qj
pivotal fa>' X l ~ P [Cj ~ IJ E [X I Cj ~ 1 & Qj pivota l fa>' X l - P [Cj ~ IJ E [X I Cj ~ - 1 & Qj pivota l fa>' X l ,
Now, again by arm separation properties similar to ones used in [s\~rOl , SSlO], and using the import ant fact t hat t he event {Cj = I} is increasing which enables to use FKG for t he given conditional law inside Qj , one can see that , if t5 is chosen small enough, , 1 E [X I Cj ~ 1 and Qj pivotal for Xl > ;) ,
One has the opposite bound for t he term conditioned on {Cj together t his gives
E [XCj
I Qj
-
- I }. All
pivotal for X l > ~ ,
which implies t he desired estimate (8.4 ) if t5 is chosen to be small enough. Consider Q with open boundary condit ions on t he side &oQ and dual closed on t he complementary t hree sides. Let r be t he interface running between the two ends of the side &oQ and separating the open cluster rooted on it from t he dual closed cluster rooted on t he t hree other sides. Denote by Yj the event that r intersects Qj, as \vell as its indicator function. Note t hat , by the RS\,y theory, (B,6)
E [Y, J:S
(R")" '
for some posit ive E. Note t hat the interface r drawn from Q(O, 0) unti l some stopping time depends only on t he percol ation configuration in the immediate neighborhood of the drawn part. Drawing r until t he first t ime it hits 1243
52
O DE D SCH RA M M A ND ST AN ISLAV Sr-.HRNOV
&Qj, we conclude t hat }j is independent from the inside of Qj and hence from Cj . Note also that, if Q j is pivotal for X , then t here a re four alternating arms connecting DQj to four sides of Q which forces t he interface, to intersect Q j) and Yj to occur. T herefore, similarly as for the estimate (B.5) above, onc can check t hat (B .7)
E
[X C,] - E [X C,Y; J,
which combined wit h (B A ) gives
(B.8)
P [Qj pivotal for X ] " E [XC, Y; ] .
Summing over all Qj's, we use Cauchy-Schwarz ineq uali ty to write
,
l:E [X C) Y; ] (B.9)
In t he last sum , nOll-diagona l terms vanish. Indeed, let i t- j and stop t he interface 'Y t he first t ime it t ouched bot h squa res Q i a nd Qj, or when it ends. Denote by Q k t he first squa re of Q i, Qj to be hi t (or Qi if none was) and by Q/ the ot her onc, so t hat (k, I) is a super position of (i, j) . Let W be t he percolation configuration in t he immediate neighborhood of t he interface so far , and in Q k. Then W determines }'i, 0 , Ck, while being independent of C/ . T hus by the ident ity in (8.3),
By t he total expectation form ula we conclude t ha t, for i
E [CiYi C, Y;]
~
t- j,
O.
Thus we can cont inue (B.9), leaving only t he diagona l t erms, and use (B.G) and (B.3 ) to wri te
1244
ON T HE SCA LI NG LIMITS OF PLANAR PERCOLATIO N
53
Combining this with (8.2 ), (B.8), (B.9) we conclude
P'(r, R):S
<
~
R Lj P [QJ pivotal for X[ (3r)' (;)'(;f (~r',
o
proving the lemma.
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SECTiON DE J\"ATH~MATJQUES , UNlVER$ITE DE GENEVE 2- 4 RUE DU L IEvRE, C P 64 1211 G£NEVE 4 , SWITZ£RLAND E-MAIL: stanislav.smirnov~unige.ch
URL :
http://www.unige.ch/~Slllirnov
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