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1/2,
(1)
where S„,(x) is the sphere of radius in with centre i e the set of sites at graph distance exactly in n0111 17. ChMn a site and integers a > m > let ft 1,1 be the event that there is an open path P from x to some site in S„ (:c), and let Ix111“!}  be the event that there is an open path P star ting at visiting some site m, y E S,,,(.e). and ending at a site c E 8,,(y) Men Ix C if P is an open path how to S„, + „(x), we may take p to be the first site of P in .9„,(x). As P ends at least a distance n from p, there is a site z E P. corning alter p, at distance exactly /1 from yr see Figure 1 If {:r 1 } holds, then for some y E 5,,, (:c) there are disjoint witnesses for the events {x y} and {y Hence,
pe(f.)?±C') 5 Pp ( 0 ''2")
' In
{
yEs„,00 P P
11)1Pe(l
ES,,,
E IP„(0
y E
y)P„ (0 Z.)
(0)
E„(ro n S„,(0)0PI, (0 2+) S P,,(0 24)/2, where the last step is horn (1) It follows that P p (0 < 2l" / "`J, so 111„ (0 2 ) decays exponentially as n This result was first proved by Hammersley [1957b] in the much Ettore
SO
Exponential decay and el The& probabilities
11
Figure 1 An open path P from :r to Sin H i(r) The portion of P from
:17
shores that
general setting of it 'lemon/ 9
below, well before,e eauden B erg_ Kest en
incqualitN WilS prot ect
4.2 Oriented site percolation lo state the main results of this chapter we shall consider oriented site percolation, where the sites of au oriented graph A are taken to be open independently %kith probability p The reason for considering oriented site percolation is that results for this model extend immediately to oriented bond percolation : and to tuna iented site and bond percolation, by considering suitable transformations of the underlying graph Before we expand on this observation. let us recall some definitions As in Chapter 4 b N a percolation measure P on a (possibly oriented) graph A \Ne mean a probability measure on the subgraphs of A A random element of this probability space is the open sulupaph of A: for bond percolation. the open subgtaph is formed by taking all sites of A and only the open bonds: fin site percolation. the open subgr aph consists of all open sites of A mid all bonds joining open sites POr a site :r of A,. the open cluster C„. is defined from the open subgraph
2 Oriented site percolation
81.
Turning to mientecl site pet total ion. we write = P— = p 4) for Ap the probability measure in which each site of out underlying oriented (multi)graph A is  open independently with probability p As in Chaptei 1, we mite A lot the outsabgraph of A tooted at x i.e., the subgraph of A containing all sites and bonds that may be reached by (oriented) paths [tom x Natmally, we shall consider A as a tooted oriented graph, with toot x We write = Cri",t  for the open outcluster of x. i e . the set of all sites y teachable from I, by open paths in A Note that a path in A is always oriented As usual in site percolation, a path is open if all of its sites ate open. We write O r (p) = 8;i:( A; p) for the Ppprobability that Cy is infinite. and t i (y) = ti:( A p) for the expectation of ICA For each site x we have two critical probabilities, pin ( A: x) suptp : and 0,r(p) = As O r (p) > 0 im:37) suP{P < plies tr (p) = trivially p ( A; < psii ( A xi lu getwial, the critical probabilities pi' l ( A ; ,e) and Pi( A ; ,r) do depend on the site .e However as noted in Chaplet 1, if A is strongly connected, i e if theia is an (oriented) path from x to y ro t am two sites x rind y. then the critishall see later that cal imobabilities are independent of the sites a weaker condition is enough V\ banthe (l aical probabilities tie independent of the site x. we write /4 1 ( A) and ( A ) for then common yr/hies In cartel to spell out in detail the connections between ( )limited and Imo/ iented site and bond percolation, we shall need one mote definition fps on (possibly oriented) Let us say that two percolation measures e is an absolute constant A > U such graphs A, ate equivalent if thei that, whenever {i. j} = {1,2} and r is a site of A i , there is a site y of Aj and a set S of at most A sites of Aj , such that J OC„I
< PACs ]
APj( U
u/.4)
(2)
for all n > 1 Tins definition is toughly analogous to that of equivalence for metric spaces The reason fa consideting a set 8 of sites in the tippet hound is that, in bond percolation. we consider the open cluster containing a given site, which is the union of the open clusters containing all bonds incident. with that site, or in the oriented case. directed front that site < x lot all sites 3: of mid Fr ate equivalent. Ind Note that if cotuse i \ ;„ is defined < x lot all sites y of A, (Here A i , then
82
Exponential decay and critical probabilities
using tire expectation corresponding to P I , and x y that corresponding to P2 ) Roughly speaking, it follows that equivalence preserves the critical probabilities p H and pT For an liner Wnted graph A. let y,(A) be the oriented graph on the same vettex set, where each bond xy of A is replaced by two oriented bonds, 5 and /77; Site percolation on A and oriented site percolation on yos (A) are equivalent: Indeed, (2) holds with A = 1: the distributions in the two measures PsA,p and Pt . A = ys (A), are identical of A ,p •
apt', let v ) ,( A) be the oriented linegraph If A is an oriented (multi)gr, of A. which Las a site for each (oriented) bond of A. and an oriented edge from C to f whenever e and f are bonds of A with tie head of 7 equal to the tail of f The subscript b in the notation y t, reminds us that it is or iented bond percolation on A that we shall model by oriented site percolation on ;::, 1,( A ) For an tutor iented graph A. let L(A) be the usual linegraph of A, with a site for each bond e of A, and a bond joining e and ,f if c and f share an endvertex in A. and set v i ,(A) = y,(L(A)) It is not hard to show that, provided the degrees of the under lying graphs are bounded, oriented bond percolation on A is equivalent to oriented site percolation on y;1 1,( A ), and that bond percolation' on A is equivalent to site percolation on L(A), and hence to oriented site percolation on :,,, b (A) ys (L(A)) We give a formal proof of the first of these statements Them em 1. Let A be an oriented (inulti)graph in which emery site Pm each p E (0, 1), the has ontdegree at least 1 and at most 0 < pi)(1) percolation meaSillt.9 PIL. and P. arc equivalent. where Cl A
p
AI p
Proof For a bond 7 E E( A ). let us write v,(7) for the corresponding P2 = ryip by taking We couple the measures P i =and site of Ap 11 to be open if and only if the corresponding bond of A is each site of Iopen For a site 3, E A , let E(Cs) denote the edgeset of the open cluster Cr of the bond percolation on A Inn other words, E(CA is the set of bonds c of A such that there is a path P in A with initial site :a and final bond . all of whose bonds ar e open Since A has maximum outdegree at most Li. we have
IC„.1— 1 IC 1E(CJI
Ard
4.2
Oriented site it:notation
83
for any open cluster Cr in A We shall show that (2) holds with A = max{A, 2,1/p}. If :e is a site of A, let E + (z) denote the set of bonds of A having :I, as initial site As an open path P in A corresponds to a path Pj of open sites in we have =
U C..„.(7) eE (:ra
Hence, PI
(Kidd
tr)
P i (1E(C.J1
It —
P2
U
(7.:
1r1EE.1(1')
As 1.6+ (x)I < A, this gives the upper bound in (2) with i = 1, j = 2. for any rt > 2 The condition for yr = 1 is trivial, as P I (ICA > = 1. K, and we have chosen while F,OC (7) 1 > = p for any bond e of — A > 1/p. For the lower bound, pick any 7 E (;r) Then (1C„I ?_ 11.1
P I (1E(C,,)1
P2OC„,(7) >
It remains to prove (2) with i = 2, / 1 Let y be any site of Ai, so y = y(ab) for some bond ab of A Then (!s, C and, whenever y is open, E(Cb ) C Cy. It follows that pPi
^
II + 1)
completing the proof
P2 (1Cy1
^. it) 5.. P I (IC„I7 n/A).
❑
The proof of the Imo/ iented analogue of Theorem 1 is similar Theorem 1 implies that 1.1'1 ( A) = whenever one of these critical probabilities is well defined. Similarly, (X) A.(yr,( A )). Also, for an unoriented graph A. we have pl!i (A) = (y b (A)), and so on Thus, when proving results of the type 'yr = p H ', it suffices to consider oriented site percolation There is a reason for the maximum degree restriction' in Theorem. 1. If each bond of a complete graph K,, on It sites is open independently with probability p, and Cr is the open cluster containing a given site x E K„, then C,.1) = e(n) as n 4x, with p fixed, while 1E(IE(C„,)1) = e(//2)• Let A be the graph formed by attaching a sequence of complete graphs with sizes nr, 1/ 2 , to an infinite path, as in Figure 2 Taking's; = 2i,
Exponential decay and
Sd
tie& pi obabilities
to
complete graphs attached to an infinite path
Figure 2 A series say,we have y„,„(,A;p) = so 1/1 =
e
pi21. while.
A 1 (L(A);
p) = e (E1pi41).
(cn,(A)) A (L(A)) < dt(A)= 1/2
The construction mat he adapted to the oriented case by orienting ever y bond 'away front :1'0 ' , ienting each complete graph transitively 01
Returning to the general study of or Witted site percolation on a glapl/ A. let us say that two sites fr and y ale outlike if the out  subgtaplis A and A ut rue isomorphic as tooted oriented graphs (Broadbent and ilannneisles [1957] used this tent' in fr slightly different way ) The distribution of = Cit. depends only on A it so fin outlike sites r and q we have OAP) = 0a(P) a nd
= \u(P)
By the outclass of a site we mean the equivalence class mulct the outlike relation that contains :if In other i\ ds, [r] is the set of sites q such that J. and y ale outlike lAe write CT lot the outclass graph of A. whose ettices are the outclasses, in which thew is au talented edge with t' how [r] to fly] ii and oni if there are sites fc 1 E [x] and tj E 1,
a bond of A We allow [a] = WI, so the outclass graph may contain loops Since the outclasses appearing among the sites in A: rt. depend Duly on the isomorphism class of Al and hence only on kb there is art — — edge Front Ix] to [lr] whenever xi? E E( A) lot some y' E (q] For example, if A is E then there is a single outclass.. and the outclass graph has one vet lex with a loop If A is a rooted tree in which sites at even levels have 2 children and sites at odd levels have 3., as in Figure 3, then there ale two outclasses, one corresponding to all sites at even levels and one to all sites at odd levels. and in Ci T there is an oriented edge from each outclass to the other Much of the time. we shall he interested in oriented graphs with CT finite In the unoliented case, the analogous condition is somewhat
4
$5
2 Oriented N ile percolation
A NA
r1\ T\
LVJ 1;.
r''
6‘k
ti'AV4 VAV4
4
F t„ are 3 A rooted oriented I t ee in which ontdegrees2 and 3 alternate along any directed path.
simpler: recall from Chapter I that two sites .1: and y of a graph A are equivalent if there is an automorphism of A mapping x into y The graph A is of /mite type if there are finitely Malt equivalence classes of sites uncle/ this relation. In particular, vertextransitive graphs are of finite type, with only one equivalence class. In the most interesting examples, A will be vertextransitive ha example, the Archimedean lattices studied in the next chapter, shown in Figure 18 of that chapter, have lids proper H Occasionally, luwever, we shall stud y mote general finitetype graphs. Mr example, the duals of the Archimedean lattices other than the square, hexagonal and triangular lattices For oriented percolation, A will almost always he such that any two sites are outlike, so ICIT I = 1 However the greater generality allowed by assuming only that C:v, is finite does not complicate the proofs, so we shall always work with this assumption, rather than requiring that 1C ,T 1 = The next lemma gives a condition on an oriented graph A that will play the same role as the condition than an in/oriented graph A be connected Recall that an or iented graph is .41rongly connected if. tor any two vertices x and y, there is an or iented path from .t; to y Lemrna Let A be an Infinite. locally finite or ienled multigraph with C v shrrngtroamOrd Then there are real :umber :N (A ) and A( A ) such that
, 1(
A ix)
for all sites:( of A
=141(7C)
and itt
;
(A)
86
Exponential decay and critical probabilities
Proof. Let and y be any two sites of A As there is an or iented path from [al to [y] in CHc, there is an oriented path P in A nom x to a site it' E [y]. If P has length 1, then 0„,(p) > p 1 0 /Ay) = Oy(p), and :y r (p) > pcx,,,(p) =p/Up). As :c and y are ar bitr y, we find that lt,r (p) > 0 for some site x if and only if 0,,,(p) > 0 for all sites a!, and that x x (p) < oc for some site a if and only if y x (p) oc for all sites x. Hence, the critical probabilities pli ( ; x) and 10F ( A ; r) do not depend on the site :c, as claimed ❑
Lemma 2 should be contrasted with the observation in Chapter 1 that in the unoriented case the critical probability pn m fir does not depend on the initial site for any connected underlying graph A Note that there are many interesting cases in which (.77s. is strongly connected while A itself is not, for example, Z and the tree in Figure 3 For sites x and 'y of an oriented graph A, we say that y is at distance n from and write (kr, y) = II, if there is an oriented path from 1, to y, and the slim test such path has length II We write S.,+, (i) fm the outsphere of radius tl centred at i e for the set of sites y at distance from x Note that y E St+, (x) does not imply c E S,±, (y); indeed, there may he no oriented path from y to :r at all, in which case d(y,x) is undefined When A is obtained from an unoriented graph A by replacing each bond with two bonds, one oriented in each direction, then d(x, y) is just the usual graph distance on A, and ,51,1(x) is just S„ (A; x), the set of sites of A at graph distance n from x in A. In [act, the results and proofs in this chapter may be read for unoriented percolation simply by ignoring all references to the orientation of edges Although it so happens that such complications will not be important, let us note that the behaviour of the distance [Unction d(r, y) on oriented gr aphs tarty be somewhat counterintuitive. For example, let A be the oriented graph shown in Figure 4. Although this is perhaps not obvious from the figure, all sites fir A are outlike. Taking X = (0, 0), y = (0, b) and z (2 b 0, b), we have d(x, y) = b, d(y, z) = 2 b 11, but d(t,z) = b+ n Thus, a site z may he a very long way from a site y close to x, but still z is not far from x. Let us write r (C„, ) for the radius of the open outcluster C„„ i r(C„)= sup{ : C, fl 8,1" (c) 54
Our main anti in this chapter is to study the tail of the distribution of (C), i.e . to study the probability of the event r(C„) > a, which is exactly the event {:c 41 that there is an oriented path from a given site
4 .2 Oriented site percolation.
87
Figure I The or halted graph X on 71, 1 in which two bonds leave each site (a, b) one going to (a + 1, b), and the other to (2a. b = 1) ibis example was given by Paul Balistel
to sonic site y E ,9,11Jit) Note that, as A is locally finite, (Q) is finite if and only if C, is finite. Note also that the length of the shortest open path from :r to 8,T, (w) may he much longer titan a X
In many of the arguments, it will be convenient to wor k with a closer related event that does not depend on the state of X. Let i?„(,r) he the event that there is an or iented path P front r to some site y E S,t(r), with every site of P other than :r open. This event is illustrated in Figure 5 Equivalently, R„(:r) is the event that there is an open path P' from some (out)neighbom of X to a site in 8,;(3). (It makes no difference whether or not we constrain .P' not to use the site x: if there is such a path P' visiting ,c, and y is the site after x along P' then the portion P" of P' star ting at y is a path horn an outneighbour of 'a, to S,t(a) not visiting :r ) In the light of the latter definition, we will sometimes write {x + 4} for R.„(x). We shall study the quantity Pc ( 1, , P) = 111)s)(Bc(39) As the event R„ (a) does not depend on the state ofand if and only if :r is open and R„(x) holds, we have IF;(1(C„.) > a) =
(3, 224) = piR;,(R „CO) = pp „Cr p)
241 holds
Exponential decay and ruling, probabilities
88
Figure 5 An Must radon of the event U t eri Solid circles represent open sites As a) — as it — oc.. Suppose flint CT is strongl y connected Then, IwJ Lemma 2, throe is a si tgle ethical ptobability p L1'( A) independent of t he p„ (:r,. p) = initial site a For a fixed p < p i ( A ). we thus have lot twin site 1 7. In the i ugurnents below. w(:` shall need bounds on ( p ) 811 1)
(abli)
J7E7C
obtain such a bound, we shall assume that CT. is finite. Lemma 3, ltd A be an infinite locally finite ()nettled multigraph with C7v finite and strongly connected. and let p E (0 I) Then there is a constant a > 0 sack that
/Pa( Cji,
"Pit °Cul
for all sites .c and p and integers ti > p <14 1 ( A) then
(P) — 0 US lr —
where IF,, denotes
(3) If
4 :2 ()hoard ate percolation
89
Proaf Since Cc is finite and sli t ()ugly connected. them is a constant t such that fin atIV two outclasses [r] and [y]. there is a path of length at most t horn [d] to [y] in CT Hence, lo t any sites .r and y. there is a path P of length t' < f from r to a site V E Let E be the event that all sites of P other than (pet haps) y' are open Then
1P,,(1c,„1
H)
P,,(En flew]
P„Oc„ii
=
(i) =
pl IP' pac mi I
I,
F)
IP (I C
^
and (3) follows with a = For (I), note first that pi l ( A ) is well defined b y Lemma 2. For p < ( A ). we have fir, (t. It) — 0 tut each t` B u t Pik (3 . , p ) depends only on the out: class fri of .1, Thus p„(p) is the point wise maximum of finitely many functions each of which tends to zero so p„(p) —
41
Let us remark that (3) implies 0,, > a0), and > (1\ y . with a = the constant given by the proof of Lemma 3 It would he tempting to think that (3) holds with IC! replaced by )(C r ) This is. in fact. < true in the case of man Wilted percolation. Indeed. if ( 1(.r y) and (Cy ) > n then there is an open path P horn ir to a site z with > a. so r (C„) > d(y. > n  It follows that p (C, ,,) = ri) = p e1 P(/ (C„)
(')
1
1 P(/ (Cy ) )))):, a)
In the oriented case, however, we may have (1(1 ) no t ch snuffle' than as in the example in Figure I. and thole does not seem to he an ohvions reason win P(, (C,„)> n) should decay at toughly the same rate tot different sites 3 . Alensi t ikov. Molchanoy and Sidorenho (1986] state in their Lemma 6 1 that this does hold for t he oriented case. but, their proof is for the num iented case. FOr innatel y. it nuns out that then Lemma 6 1 is not needed in tlw subsequent arguments. since it may be replaced b y (.1) (In [1986]. Alenshikov outlined his proof for the tutor hutted case. where the corresponding problem does not arise ) Alt hough or iented percolation is sometimes hinder to troll: with that untalented. it so happens that the proofs in this chapter ate cqualb simple for the oriented and tunniented models We state the results fin the (nitr ified case. as this implies the unoriented ease In fact. the [cadet interested onl y in the hate) case nun simphv ignore all relercuces to orientation. and so tend the proofs as if the y had been given for unor Wilted percolation d(y. .11  d(r. q)
CJ
PO
Exponential decay and critical probabilities
4,3 Almost exponential decay of the radius — Menshikov's Theorem Our main aim in this section is to present Menshikov's fundamental result that, under mild assumptions, below the critical probability the open cluster containing the origin is very unlikely to be large A consequence of this result is that the critical probabilities pr and p H are equal Independently of Nfenshikov, Zell/M11/ and Barsky proved similar results in a very different way: we shall outline their approach at the end of this section Recall that R„(.1.) is the event that there is an open path from an out: neighbour of to some site in St(x), so { 24 holds if and only if is open and R„ (3! ) holds We shall star t with a study of p„(x.p) ,1?„(R.„(x)), and how this changes with p As we shall use the Mar gullsRusso formula the first step is to understand what it means for a site to be pivotal for the event l?,, (x) As before, the context is site percolation on an oriented graph A. so P p = A ,p Suppose that the increasing event R„ (a) holds For convenience, suppose also that a is open, although this will riot he relevant. As R.„(x) is an increasing event, and we are assuming that R.„(x) holds, a site y is pivotal fm R„(x) if and onh if R.„(x) would not hold if the state of y were changed from open to closed, i e., if all open paths from 37 to Op pass through y. Let P be any open path from :r to 8",;(r) Then all pivots (pivotal sites) appear on P Let 1, 1 ,b,. r > 0, be the pivots in the order in which they appear along P (Our notation bt is chosen as we think of the pivots as In idges that any open path horn :r to .9;,P (a) must cross.) Note that if P' is any other open path front X to 8,i(x). then not only must P' pass through each of the sites In. but these sites appear along P' in the same order b 1 . Ix,. Other wise, there is an s > 0 such that the order of the pivots along P' starts with . where s 1 But then the MUGU of the initial segnrenb of P' up to lit and the final segment of P starting at bt contains an open path from at to S,t(x) avoiding 14 4. 1 , which is impossible; see Figure 6 In fact, the more detailec.1 picture is as follows: the set of sites on open paths from to 87,(v) forms a graph in which br, i br are cutvertices; see Figure 7 Taking ho x, for 1 < i < let 711 denote the set of sites on open paths horn 1)1 _ 1 to 14, excluding 14_, and b, Also, let Tr+ , denote the set of sites other than b, on open paths horn b, to the set T.; U .5, ,f "(x) Then, for 1 < contains two paths
.3 Alenshileov's Theorem
91 .5;ti (r)
Figure 6 An impossible configuration: the pivots(); appear in different orders along two open paths P. P' from 3. to Stec). The path P is the sit algid line segment from to y: the section of the path P' from to 1, 1 is drawn with thick lines The path P' cannot 'jump' front I), to I), t > s+ since P UP' would then contain a path from t to p avoiding
(a)
Figure 7 The set of sites on open paths from 1 to 9;4; ( r), shown with a possible artangement of the pivots ha for the event. R„ (:r) Note that bib.. LT2 63 mid br,b6 are edges of A er
front 1)1 _ 1 to h 1 that are vertexdisjoint apart front their endpoints; in the case whole E E( A ), this holds trivially: both paths consist of this single bond Also, T,. + , contains two paths front 1.4 to St (3,) that are vertexdisjoint except at b r . For i there is no (oriented) bond front a site in T1 to a site in Ti Let ho a Whenever .11.„(x) holds, let D i , = 1,2, . denote the distance horn hi_ i to h i , where we set a = x if there are fewer than + pivots. Let I < k < v. Since D i is the distance from a to the first pivot, if D I > k then either there is no pivot, i.e., there are (at least) two disjoint open paths front neighbours of x to (4 or the first pivot h i is at distance at least k + 1 horn a In the latter case, there are two
92
Eepoio
duo q and c i it Ica! probabdlLies
and au open path how disjoint open paths fron t neighbouts of a' to b i to 5 11` (r) disjoint flow T 1 and hence how these paths: see Figure S
figure
litst to
then it there are any pivots for the event 1? ” (1)• the > . is at disun i ty mot e than k frpm r Thus thine ate open paths P it owl ) and P ' how S i±r) that are disjoint except at .r S II DI
l i t china case. there ate pat hs P and P' front .1 . to S,+, (r) and to tespecti eI N with P 111(1 P' disjoint except at .1 and all sites of P and > k} is contained in I' tit het titan x open illus. the event R „(x)fi { I lie erect .1 „( Pk(.r) Hence. by t he ran den BEngRester inequality. Theorem 5 of Chaplet 2. tic lime Pip (I?„ (3).
>
< Pp (R„(x))Ei,(111, Tr)),
ie.
P p ( D I > k I Rs (l i )) 5_ 14.(1', The wain tool in Nlenshiltov's proof is a genet alizatiou of tins observation When 1?„(r) holds and there are at least t pivots, we WI Re 11 lot the tilt into1o, el u.stel meaning the set of sites z such that there is a path P born to c with I) P and all sites of .P other than ./ open: see Figure 9 Thus, if c is open, then II is the set of sites teachable fron t Jr by open paths not passing through b, We use bold font fin the random variable I/ because it will be partic u larly impottant to distinguish It we horn its possible values It ot h er random variables, such as do not bother.) When lo = y and II = It , then every site in I 1 U 0/1
4 :t itIen gbikov's Theorem
93
Figure 'the 2nd interior cluster Io consists of the thick hues including all of their endpoints except b 2 Note that z 6 1 1 , oven though z is not on an open path Irian i t neighbour of 3: to 51(x)
open, and all sites in d1 1 other than y ate closed. where 01, is the set of outneighbours of . U 1:0 With this pteparation, we me ready to present the key lemma for denotes the probability mealenshikov's main theorem As usual. sure in which each site of A is open with of the sites are independent
ptobabilitv p. and the states
A and let 11. I nod f Lemma 4. Let x be a Ole of an unfilled !pupil di. eve hare < i < r. be positive inteye l For 1 < k < /I 
FE, (D, > lc[
E) < supl i n(y.p): C.r.
A}.
(5)
where E is the event E and as
R„(.r)
n {D i =
D2 =
(12.
Dr1 = 1}
pk(y. p)=F;,(R),(y))
Proof Throughout this proof, we work within the finite subgr aph of A most n hour a We write IP„ For Fis, We shall pat tition the event E according to the location y of the For and according to the ulterior cluster I= (r— 1) 5 ` pivot b, let any possible value I of I consistent W ith E
consisting of sites at distance at
Ely
= R.„(e)
n
=
n =
so the events E„ r Iona = Feu 1 < i < then I = pal tit ion of E If E„ I holds them as noted above, so does the event /7„ e that all sites in ./u yl rue open. and all sites in RI\ {y} ale closed In tact, Ey e holds
ErponenHal decay and critical probabilities
91
it and onl y if Fr, , holds and there is an open path P horn a neighbour of y to .9,1(x) disjoint from I U DI (Note that y E DI.) Let us write G y.i for the event that there is such a path P Let X = (/u0/) c . and let P IA, denote the product probability measure in which each site of X is open independently with probability p We may regard the event D ili as an event in this probability space. Note that, in the measure P„, if we condition on Ft, , ,. then every site of x is open independently with probability p (The event Flo' is defined 'without looking at' sites in X.) Hence, P„(E mi
F Fp .1 )
= WI;
al I ts 1Pyt.E„ 1 )
=
P y (p,
1 )1? (Gm!)
Let H y denote the event that X contains an open path horn a nrighbout of y to b' (p). Suppose that E l, j holds and that D, > k Then F;, r holds, and, as in the case; = 1 above, X U{y} contains open paths P. P ' . disjoint except at y, with P joining y to 8,;(a.). and P' .joining y to St (y); see Figure 10 Thus, X contains disjoint witnesses fin the f
Figure 10 The shaded region represents the interior cluster I If E y I holds, then I = I and ti is the — 1)“ pivot 14 _ 1 for the event R.,( 1! ) Ift in addition. D r > k with d(r y)+ k < then there are disjoint paths P. P' horn y tr,) (x) and to Si ;F OI L with P. P ' C = uill)'
/
illerishiltov's Theorem
events Cy r and H„ r. so Crl „ i ❑ H„ r holds As E y( subset of Fil l, we have shown that
95
n {D, > k} is a
Py(E, i )P;\,(C / 0 H y 1).
P„(E„ ,1 ri{D, >
Applying the van den Be t gKesten inequality Theorem 5 of Chapter 2, to the product probability measure P IA, WC have
F
(Cy
0 H„ ,1 ) < IA, (6?„,r)Plx,(H„,r)
g the three relations above, Combinin P p (E,, ,1 n{D,. > lc}) G Pp(E„ Now for any possible ti and p k (y,p) Thus
we
have
P;\ (11„ , r) =
<
P y (R 1,(y))
1.7„ (E„
n
> 0 ) 5 Pp 4E. „ 1)pk(p, p),
ie
Pp (D, > k E y 1 )
pk(q.p)
As the events E„ .r par tition E. the bound (5) follows
❑
Lemma , is the key ingredient in the proof of Nlenshikov's main theorem Although this lemma gives detailed information about: the distribution of the distances to successive pivots. all we shall need is the simple consequence that, on average, there ate many pivots if 11 is large.. To state this lemma, let N(E) denote the number of sites that are pivotal for an event E
Let .r be a site if alt leafed graph A. and let a and P be positive integers Then Lemma 5.
E„
HMO  sup ppOi ,p)) LikRj
(R.„ GO) 1?„(x))
Proof Let D,, D2,
be as in Lemma 1 As the event
D, _
is a disjoint union of events of the form {D, =d i ,
,D,_1=61.,_,}nft.„(d),
Lemma I implies that Pp (D,
< k
. ,D,
G
R.„(.0)
1  sup 14(y, p),
(6)
Erponential decay and critical probabilities
In; whene\
tk < ti In pinticulin Pp <
/?„
— stIP fik (
"Faking t = 2 in (6) we obtain
i
iP( DI.D25 no(.0) r,,(D2 A. ( D I 5_ k, 17,0))
(D i 15
n„(.0)
— suPfik(il• Continuing in this uav, we see that li p
D, < k T?„(r))
(1 —suppk(y, p))
lot ;Inv r < Link_ II D, < k. then tinn y me at least t piv lid sites fin the event !?„(x). so N(R„(0) ^ r Thus. (N(R„ (tin R „CO)
LitpidiPp
,D6t/tij
R tt(1))
L o / k i —slIPPkOrpninfid ❑
as claimed
\Vie e BONN leafIV to puisent the tut) lundantent al tumults of 1\ letishikin [1086] (see also Nienshikm. Molchanoi and Sidotenko [1986]) The fitsi shows that. undo ' ■ wild conditions, we have almost exponential deca y ()I the laditts of the open cluster below p it . As we shall see an innuediate consequence of this result is that. lot a huge class of wind's. the t it it ical ()liabilities p i m i d pH coincide The /est/Its below tue st at ed lo/ site percolation on tit iented graphs. Using the equivalences between iiat ions percolation models described in Sect ion 2. co/ esponding iesults lot bond petcubitkm tund for unmiented aphs follow easik Ilene I hen. is Menshikov is main Heinen ' In this tesu. //i i ( A ) denotes the common value of the ethical probabilities A:1). whose existence is gnat ant eed by Lemma 2 Theorem 6. Let A be an locally finite ° nettled multigraph with (I— finite and strongly connected and let p < p1 pim 1 ( A) Then Then the Pi an o > 0 such that EP;(.,
< exp(—on/(logn) 2 )
(7)
4 3 Alen sh ikon's Thew cm
Jot all N iles J . a ml integer s
97
n>
The proof we present Ibllows that given be Menshikov. Molchanov and po < pin( A it Sidorenko [19861 vent/ closely The bask idea is to fix and to use Lemma 5 to show that at probability pa. if rt is large then the expected number of pivots for the event 1?„(x) is large: in doing this. we use 0(po) = 0 to show that sup p p„(y. Rn) is small lot k large Then, 6 om the MargolisRusso for main, it will follow that p„(x.p') = P„,(1?„(J.)) &et eases rapidly as p decreases Finall , we feed the new bounds on Ore function p back into Lemma 5 Amazingly. even though we have no a pi iot i inhumation on the rate at which sup, / pk (g, po) tends to 0 as k —x appl y ing Lemma 5 again and again with carefully chosen parameters. (7) can be deduced p <
Proof. Since limn/ghoul ow argument 3te wort: with, site percolat ion. we shall write for 1137, and p it for pit As before. for p E (Bl). let p„(p) = suptrE  p„(a . p) Note that by Lemma :3 lot any p < Rn ( A ) we have p„ (h) as
0
(8)
n—x BA Lenuna 5, we have Ei (A; (R„ (a)) l?„ (J.
[le / k
 pk(p))111/bil
fm any site J . and positive integers n and k B y the Afrugulis Russo formula (Lemma 9 of Chapter 2), for any increasing event E we Lace dp
T f,(E) = Ef,(Ar (E))
E (Ar (E)1 ft) =
(N(E) I E)113„(6),
so
dl
log 1111 .(L)
=
I
d E ,(N(E) —P „(E) > 
„(E) (E) flp
In par t ieulat taking E= ? „(r). (1 log p„(x. clp
lei I k (1  p„(p))L”
Fix p_ < p + As ph (p) is an increasing function of p, for at the righthand side above is at least the value at p i.. so Pe( 1 , P)
Pa Cr , P4)
< xp ((p± 
p_)Lit / Li (I
 p„(p+))1"Th)
1.4]
a:pont:Mad decay and al ilia& probabilities
98
As the bound on the ratio is independent of 3', it follows that P„ ( p—)5 p„ (p +) exP — P—)[(// ki — Pb(PH))1•"/Id) (9) Menshikov's proof of (7) involves no further combinahnies: 'all' we need
to do is apply (9) repeatedly However, it is fat horn easy to find the right way to do this.. Continuing with tire e proof of (7),—let p < N IA) be fixed horn now 0 as on. Pick pa satisfying p < pa < pa( A).) By (8) we have p„(pa) n Writing pa for p„„(po), it follows that if no is sufficiently large, then pa < 1/100 and pa log(l/p0 ) < (pa — p)16 Let us fix such an no horn now On Writing pi for p,,, ( p r), we inductively define two sequences no < n t 5 > > • by < • • • and pa > and p 14. 1 = Pr — p i log(l/pd,
=
as long as p i n > 0; in fact, as we shall see later, p i > p for all p Let and p_ = p1+1 Thus = p+ = us apply (9) with k = pk p„,(pi)== p i , so (9) gives pd, and (p+ ) = = 1/1 /k) = p ix:(m+ ) 5 P,,,
/ ex l) HP/ — lb+1)Ll iPd ( 1 — Pi)ri
The sequence p i is decreasing., so p i < pa < 1/100 for every i from which it follows that (I — pni 1 /P , 1 > 1/3 Thus we have p„ , (m +1 ) 5 pi exp Hp; log(1 /p i )11 /9 1 1 /3) pi exp(— log(1/ 9 0/3) = p:+113 As n • •
'u, the event II,,,„(x) is a subset of H,, , (:c) for ally (Pift) < Pa.(Pt+. 1) < Pi1/3 9111 =
so
(
we have pi < pr r so, very udely, < pal for every As x log(I./T) is increasing on (0,1/e),
Fr om (10)
Pa — Pi
og(i/pj) 5
c —J pa log(eilp0)
pa(7 + log( 1 /p0 ) ) 5 2p0 log(1/90 ) + pa .< (pa — p)12,
where the second last inequality uses E i>0 )c < 1 and the final inequality holds by out choice of no. it follows that the construction of the pi continues indefinitely, and p i > (Th 1 p)/2 ha ever y sequences
l.3
Theorem
99
Let p' = (m p)I2 > p At this point we have constructed an increasing sequence n it and a decreasing sequence p i , such that < Pu,(Pi)= for i = 0,1,2, To understand the significance of this bound, we should compare r1a and pa Let s; = 11/Thl, so = no {L c se, and 5o = 11/pol > 100. Since s; > so > 100, the rounding in the definition of Si makes little difference Iu particular, (10) implies that sa± i > say. Hence, for 1 < j < we have < 4 /5) , which gives 5i; 5
S—,:>,(1/9)J Yt
< st.ino Let n > no be 111bitutiv Then them is an
SO /11+1 =
Iii < 11 <
i
with
n i+1 = SW; < 81/10.
But
N(P)
Pu t (It ) 5_ Pt:,(Pi) =
< 2 / S < 2(010)115
AS /to is fixed, it follows that there is a constant e such that p„(p' )< ca 1 /5
(11)
for all a > 1 This weak polynomial bound is a Iar my horn (7), but in tact the proof is almost: complete! All that teamin g is to use (9).. this time in a straightforward way Fix p" with p < p" < > 1, let k t = k i (n) = (T/5/11) Then [n/1:::] = e(n 1/6 ), and, from (11), pk ,(p1 ) 0(11 =1/6 ) It follows (1 pc,(P1))1"0''1 is bounded away front zero Hence, applying (9) with = (a), p' and p_ = ,r7 p„(p") < exp((p '  p")0.(nI/6)) = exp( as a oc. Fix p'" with p < p'" < p" a nd let kg = k 2 (0) Front (12), we have p h...,
=
ri I /6 ))
((log r1) '
(lot
(12) 2).
(lin = exp(Q ((log n) 7 /6 )) = o(n1),
so (1  p1,„,(p"))1"Thi , 1 as a :: z Applying (9) with k follows that p„(p '" ) = exp(4(r1/(logn)7)).
/co (n) it
100
Exponential decay and ci Thin! probabilitiwi
One final iterat ion now gives t he result: taking l':n = log rt(log log O s it follows that pfa (p'") = o(n l ). and appl ying (9) once more we find that p„(p) = exp(—[2(///(log n(log log n i ) s ))).
(13) ❑
and (7) follows
The method above gives a slightly stronger bound than the inequality (7) claimed in Theorem 6 Indeed, the unappealing final estimate (13) above is stronger than (7) Iterating finthel, one can push the bound almost to exp(n/ log n), lint this is as far as the method seems to go. As noted earlier, under a very mild additional assumption. Theorem 6 immediately implies another theorem of Menshilrov, that > p i n, and solr7= Pat Theorem 7, Let A
be on infinite. locally finite oriented with C t finite and stiongly connected If there is a constant C' such that
exp(C l /r/(log 0 3 )
1.5,4(01 fm fret
i ti ligi
t hen M(
(IA)
= Pi1(7)
Prof By Lemma 2, the critical probabilities pi i (X :3) and I/ ( A: ") are independent of the site Let p < ICH ( A ). and let .r he any site of
A Thiel' (A :
=
E
= ,r
11/n;,(// E C's)
I
H ( 3 11firi (.1.
S ;':(.1 I
The final stun converges In Theorem 6, so p < pq( A ; :r) = p (A) As P < Pi t / " as at hitral A it 1°Ikms that (//t ( A ) Pit( A ): so 11/41 ( A ❑ p i ( A ) Using the equivalence between percolation models discussed in Section 2, Theorems 6 and 7 immediately imply corresponding statements for bond percolation on A. and for 1 u/oriented site and bond percolation. Mann, of the most interesting percolation models satisfy the assumptions of Theorem 7 (alter the appropriate translation to oriented site percolation) [ indeed, in many cases (Ion example site or bond percolation on Ed ), the classgraph has only a single vertex and the sizes of the neighbourhoods .51,1"(x) in A grow polk nomially iu n It is tempting to think that for any finitetype graph, cipher the
)
s Tluore tti
101
'growth function' 2 (u) = Sup , , : ‘ 15„(.01 will be bounded by a polymania'. or 2 (0> exp(ur) fin sonic n > 0 Indeed. Milma 119681 conjectured that this assertion holds in tilt' special case of Cayle y graphs of finitel y generated groups. Surprisingly. even this is not true: Gtigorchnk 11983: 1980 gave a counterexample Fut the/mo t e. solving in the negative a problem of Gramm' 119811. AFilson [2004] proved that a glom of exponential growth need not be of unifianth exponential growth (See also Arachnile rind Pak [2001]. Pvber 120011 and Eskin. Mozes and Oh [2005]) The condition that C–c be finite is essential fin Theorem 7. We illustrate this lie untalented bond percolation Let G i he it 2' •gt. id i c . a square subgt twit of r with 2 2 ' vertices. and let A be the graph obtained b y stringing together tire gtaphs G; as in Figure II
corners Figure II. A series of gridgraphs strung toget he t by I heir opposite lot the example, t he sizes of t he grids grow superexponentially.
Note that A may be embedded into E 2 in a way that preserves graph distance, so 1.5„(A; < 1.5„(72; = 4n for n > 1, and the growth condition (IA) is satisfied If p> 1/2. then it follows easily horn the results in Chapter 3 that the probability that there is an open path horn one collier of Cr to the opposite comet is bounded below by a constant (This may be shown by using exponential decay for p < 1/2 and considering the dual lattice ) Also, the expected flambe! ' of sites of that YilaV he reached by open paths horn a given cot net is 0(16l 1) Since Pi] , grows super exponentially. it follows that pi; (A) < 1/2, so. as A C V. we have I); (A) = 1/2 On the other hand. Fir (A) = 1. since. for p C 1, the probability that each of the infinitely many cut vertices is incident with at least two open edges is (I A slight variant of this construction
102
Exponeatial decay and critical probabilities
works for oriented site percolation: take the line graph, and then replace e each bond by two oriented bonds. In the unmiented case, we do require the underlying graph A to be of finite type. However, there is little reason to consider the graph structure of CA, defined analogously to Cdr : the graphs A we study are always connected, so CA is also connected. Note that if A = ips(A) is obtained front A by replacing each bond by two oriented bonds, as in Section 2, then two sites a; and y are outlike in A if and only if they are equivalent; in A Thus, if A is a connected finitetype graph, then C1 is finite and strongly connected For locally finite graphs, the transformation mapping an oriented or unmiented graph to its line graph also preserves the finiteness of the classgraph Finally, these transformations also preserve the growth rate of the neighbourhoods, so Theorem 7 does indeed imply cm responding results for bond percolation, and for unmiented percolation In the statement and proofs of Theorems 6 and 7, we took e ye' y site to have the same pr obability p of being open These results extend easily to a sornew bat more general setting, in which different sites :r have different probabilities pd. of being open, with the states of the sites independent Recall that the definition of the classgraph Cv. or the equivalence of sites in the man iented case, involves isomolphisms between subgraphs of A or amount ' phisms of A Naturally, we now requite any such isomorphism to preser'e the 'weights', i e., the probabilities that the sites are open. Thus, outlike sites still behave in the same way for percolation As we require CTc. to be finite, there is one probability pi for each outlike class, so we obtain a per colation model parametrized by o a vector p = (p i , . pk ) For example, one could take A = with alternate sites having pr obabilities p i and pi, of being open, resulting in two oillike classes Theorem 6 carries over to this 'finitetype weighted context in a natural way: using selfexplanatory notation, the result is that if for some . hr; We have . .pk )= 0 then, whenever pi for each an assertion equivalent to (7) holds in the probability measure Pp, Hew Pp , is the measure in which sites of class i are open with probability flit with the states of all sites independent. There are two ways to see this One is to note that the proof given above carries over mutabs unitantlis In particular. the Margolis–Russo fornmla (Lemma 9 of Chapter 2) states that for tar increasing event E. the stun of the partial derivatives of IPp (E) is exactly the expected number 01(N(E)) of sites
/3 Alenskikov's Theorem
103
that are pivotal for E If we decrease each p i at the same rate, then the calculations in the proofs above are exactly the same as for the uniform case. Alternatively, one can realize an arbitrarily good approximation to the weighted model as an unweighted model, by choosing p close to 1 and replacing each site open with probability p r by an oriented path of length C chosen so that pd is within a factor p of pa,. Using this idea it is easy to deduce the weighted version of Menshikov's Theorem from the umveighted one. Note that we require g p i for all i: in general, it is not enough to require p < p i for all i and pie < p i for some i. This may be seen by considering any graph in which sites of one type are ' useless ' , for example, the graph A obtained from Z  by adding a directed cycle to each site, where the sites of Z 2 are open with probability p h and tine new sites with probability Po Marry of the other results we shall present also have natural weighted versions Most of the time, we shall present only the unweighted version, and shall not discuss the simple modifications or deductions needed for the weighted versions Aizemnan and Barsky [1087] gave a result closely related to Theorem 7. Their proof uses yen v differeM methods to those of Menshikov, although it also relies on fire Vail den BergKesten inequality and the MargulisRusso for mula A key idea of the proof is the introduction of a 2variable generalization of the percolation probability 0(p) Considering bond percolation on a graph A in which all sites are equivalent, this may be written as
moi,
= 1—
E
p( 7;
II),
rt=I
where p = 1 — 6 0 The reason for the reparamettization is that: Aizennmn and Barsky consider a more general model of percolation on Ed, where 'longrange' bonds are allowed: each edge xy of the complete graph on 27,(1 is open with probability 1 — exp(0./(x — y)), where .1 is a nonnegative symmetric function on Z d . Taking It = 0, the sum above gives exactly Ilt,OC„. < = 0(p) Writing VI for .1(X), Alain/Mill arid Barsky prove two differential inequalities for AI = Al(13,11), namely 8111
08
< 1.111
0.111
104
Exponential dung and cIilical pi ° hybrid ies
and <
,r + :11" •fli Ur i
Oh
The latter inegt tlitv genet alizes an eat lie/ tesult of Chaves and Chaves 1986a] that (() < 0(i()2
0(p)i)(e)11))
lot hood pet colation on Z il (For site pet colation. the Lena 0(p)2 replaced by li t 0(p) 2 ) Using these differential inequalities Aizentnan and Barsky show that 11/(el l > ch i/2 for some c > 0. where ;IT corresponds to p i Math the) deduce that p i /in Note that the It) has a simple combinatorial description: if we extend quantity the pet colation model la adding a single /KM vertex G (the ghost' vertex). and join each site of E d to G independentl y probability — then 111(13.11) is the to obabilit v that thew is no open path (tour 0 to G
4.4 Exponential decay of the radius Out aim in this section is to show t hat. undo mild conditions < I he distribution of t he 'whits of the open dust et containing a given site has all exponential tail Floiment 6 does not quite give this. although it conies close Flowevet exponential decav follows from Theo/tin t 7 by tesnit of Elatumetslev )19574 a special case of this result was described at the start ()I the chaplet The statement and pool that follow appear considerabl y tome complicated than those lin the special case of bond petcolation on D I considered at the stint of the chapter Then) a l e two 'fiasc 's: first. thew is a noun ' technical complication that.vises in the case of site petcolation Second we shall separate out the heart of the result as a lemma, and we wish to state a teasonalth silo/1g form of this lemma ha hame 'detente If out aim we t° only to move flanuocuslev's result. I Iwo/ em 9 below. then a weaker km to of the lemma would suffice In tact. the much greater genetaffix consideted here int t oduces no essential complications Given a site t of a directed graph A. let
13'1 ( 3 ')
U
if ( ,I )
be the notboll of r.adius r centred at
e 71 • (I(:r. y) :)); t Let A ,• (.r) denote the numbe t of
Exponential decay
105
of the Indira;
sites in5';r(r) that ma y lie leached In an open path P iu B7( t ) starting at all outneighbour y of c Lemma 8. Let A be ml oriented multigraph r > 1 an integer. mai < 1 a real nymber. If Elis.,(N,)+T.r)) < for every site .r of A. then d fat every site ;I
Of
A um/ CM, y 11> 1
Proof As usual. we write tr,, for IP); Let 1,n > 1 Recall that I?„, = tr. + 14. 1 denotes the event that tin g e is an open path P From an outneighbour of x to a site y e S,t(:r), and that a„,(r.p) = Pp(R„,(r)). By splitting the path P at the point it first reaches Sit ()), we see that Rrr.„(x) is the union of the events E„, y E .9;(,r), where E y is the event: that there is an open path P in a,t,..„(r) horn an outneighbour of r to 87: „(: y ) that first meets .9)± (x) at q (This is not, in general. a disjoint union. since there m ay be many such paths P ) Splitting such a path P at y. the initial segment P' Rom a neighbour of 1 to y lies in 8,5(r). and is thus a witness For the event E,C that theme is a path in 13)/(t) flow an outneighbour of x to y; see Figure 12. The remainder, .P", of P is a path star ting at an ontmeighbout of y and ending in S it i. „(x). Thus P" must contain a site of 5,1)(q). so P'' (ot an initial segment of this path) is a witness for 17„(y) As P' and P'' are disjoint, we have /7„(y)
C
Writing p„(p) for sup y p„(y, p), as before. by the van den 3eig inecpuditv it follows that
P,±n(x,
PpuLT ;,FP,,(R„ on)
Lii )„(E„)< (J)
itCb;!(x)
E IFIY( E/y) p a ye sr))
Err(N;3(I))aa(a) 5 IM(t)
)
we have p, ÷ „(p) < p„(p), As this holds fin every lot every a As Py (x 22+) = pp„(r), the result Follows
so p„(p)
< siln/r1
Note that we work with paths star ting at a neighbour of a given site to avoid reusing' a site when we concatenate paths This complication does not arise for bond percolation, where the corresponding result is that, if the expected number of sites of 5 + (t) that may be reached
1(16
Exponential decay and critical probabilities
Figure 12 The (Wick lines show an open path P witnessing the event R.,E.n(:c)• The subpaths P' from it ' to y and P" front y ' to z are disjoint witnesses For the events E,/, attd R.,,(y)= fy + 22 1 respectively
nom x b y open Paths within B,+ (x) is at most i for P i ,(3, 11) < 1 1" h I holds for ever y kite's and integer a Mom Lemma 8, it: is very easy to deduce Hammeislev's result
every 7 then
Theorem 9. Let A be an oriented multigraph with C hi finite and SilDlIgly
connected. and let p < p4( A ). Bum there is an a
> 0
such
that 11F7,(3: fin
exp
Im)
(15)
every s ite 3, and integer a >
Proof Let t' be a site of A By Lemma 2 we have so ivj, ( p) < x Now
E ()))
PE
E
IIESt (x)
p4
a;

=
4.5 Exponential decay of the volume
107
where Ix + yl is the event that them is an open path flour an out(x)al Also, as neiglthour of x to y (Thus, I?, = Uaes;tcofxi(x) to y before, if {x + y} holds then there is an open path fiom are independent; ) ± r is open' and {x not visiting:, so the events Let "o(c)=
E
P r+
acis;•(3,) Then E r 7, (x) is convergent for each x, so 7,.(r) a As 7, (a:) depends only on the outclass [x] of x, and there are only finitely many outclasses, there is an r such that 7, (x) < 1/2 for every site x Since ❑ 4,(N,+ (x)) < (x), the result then follows how Lem a n 8. Let us remark that, under the assumptions of Theorem 9, if the expected untidier of open paths star ting at each site x is finite, then (l5) can be deduced without using the van den Bet gKesten inequality. Flowever, it is pet reedy possible for this expectation to be infinite even when x =i,(p) is finite, as shown by the graph in Figure 13 Indeed, for this
7N 7N 7N N7 N/ = I in V.111(711 there are exponentially many paths Figure 1$ A graph With of length C from a given site.
graph Pit = = 1, but taking each site to he open independently with probability p, the expected number of open paths of length C starting at xo is ,ri which tends to infinity as C. for any p > lb,F5 Performing the saute construction star ting with a binary tree rather than a path gives a similiu example with pit = pis! < 1
4,5 Exponential decay of the volume  the AizenmanNewman Theorem Om aim in this section is to strengthen Theorem 9: under suitable conditions, not only does the radius of the open cluster . C„, containing a given site a: decay exponentially, but so does its ' volume ' , IC1I Aizenman and Newman (19811 gave a very general result of this form, winch we shall come to later However, a special case of their theorem may
108
Exponential decay and critical probabilitic5
he proved in a ver dilleren t way, using 2independence: we present this Hest shall for initiate the next result for until iented site percolation on graph A Recall that, in this context, two sites are equivalent if there an autorumphisur of A mapping one into the other (This corresponds to being outlike if we replace each edge by two oriented edges ) A graph is of finite type if there are finitely many equivalence classes of sites under this relation As usual, we write S, (c) For the set of sites at graph distance a from a \VC write B„(x) for U7u S i(/1 ) , e for the hall o radius a centred at a' The conditions in the result below are generous enough to ensure that it applies to the 1/10tit hale'fir g tiltieS, including site and bond percolation on D I and on Atchimedean lattices The proof is based on Theorem 6 or Theorem 9, and the concept: of 2independent site percolation Although it is ratlwr easy, and strongly ierniniscent of the pool . of Theorem 12 of Chapter '3, hem we have to soil( hauler to get a covering corresponding to the cover of 2 2 with large squares Theorem 10. Let A la a connected, infinite locollyfruityfinitetype wan tented graph S l ippage that sup 12,011
< t (hig t Vino
(16)
■
fa all soffit amity tangy t o = u(11 p) > 0 such that
net fat meet// p < /)11(‘)
;MCH
Lot all sites . t and all
ill( I
> a) < exp(—on)
>0
Proof The (nem!l plan of the proof is as follows: NW shall cover 11(A) In a set of balls Bo, OIL u E II' that are reasonabl y \\ ell spread out': binning a graph (F) on II" by joining a'. a' ' if they ale at distance at most tit . say we shall show that the maximum degree of D(11) is not too large We then construct a 2independent site percolation measure on D(II ) as follows: a site la E II will be active if some E gi,(w) is joined In an open path to a : distant' site i e a site at distance at least r limn .r: these events are independent if d(W. a') > 6/ Also. an y site in a large enough open utast:el is ,joined to a distant site. so a large open cluster in A implies a l a rge active cluster in D(IJ ) From Theorem 6 (\lenshiky . 's rliemein) and inequalit y (16). each a' is yen in /likel y to
4
5 Exponential decay qf the volume
109
active if t is chosen huge enough: it will then Follow lion Lemma I Chapter 3 that open clusters in this 2independent measure are small, of giving c,  the result To carry out this plan, we shall first show that balls of /minis Tr in A ale not too much huger titan those of radius r Let A < c be the maximum degree of A As A is connected.. (di each pair ([4. [y]} of E Ix] to some equivalence classes of sites thew is a path P hour some E [y]. Thus there is an integer L such that for am .r and (1 , there are E and ij E (y] with d(3: ( (/) < L. It follows that. crudely. )e
i p rODi = i 13,C ri )i
(I +A+
i r3 ,+1.0a =
±AL)18,,(y)l
: :.r E A} and bi = for every t > I Let left' = ntaxilB, where C' depends A}. Then we have I < < C' for every only on A. \\'e claim that <
(18)
holds Mt infinitel y many t Otherwise. there is an r n s uch that
1.11
> t 0 we hate let 11/:"
) > \/7: IC'
Tin t s. lin r large enough, adding log? to = log y increases log/; by at least (1/3)logt = (/3, which implies that log ft > 1 2 /100 lot huge t . contradicting (lay and proving the claim F l om now on we consider onl y r for which (IS) holds Let C 1'(A) he a maximal (infinite) set (}1 sites subject to the halls (w) : w E 111 being disjoint (Such a set 11' exists in am graph Lv Zonfs Lemma In fact. since A is connected and locally finite, A is countable. so IV may be constructed step b y step ) Note that the balls {Tin, (w) : w E It } covet V(A): if sonic site y (lid riot belong to Wc it 8.), ( 11 ') . then q could have been added to IF. We define an auxiliar y graph D(11') with vertex set 11  as follows: two sites W. a E H ate adjacent in D(11 7 ) if d(w. n) < . where t1(.t.. te) denotes the graph distance in A If r„ denotes the set of neighbours of 0 , in D(11"), then the balls (13,06 : u ate disjoint subsets of 137 ,(w): see Figure Hence lb, 5_
Li T3, (u) < [37, (o . )1 < Er„
Therefore. using (18). we have degree A i of D(11") is at most
bt, \AT
<
vT.
t he maximum
110
Exponential decay and critical probabilities
Figure H. Balls of radius i centred at a' and halls are disjoint as (w} U F t , C
variot ites a c Hie
Let us say that a site w E IF is active if there is at least one site y e B0 1 ( IV) for which {y L} holds. Otherwise, w is passive, Note that the disposition (active passive) of w E IF depends only on the states (open (A closed) of ri t e sites of A within distance fir of w Let U and U' be subsets of IF such that no edge of D(1I) joins U to U'. Then no site of A is y distance 4, of both U and U', so the dispositions of the sites in U ale independent of the dispositions of the sites in U' Let P be the site percolation measure on D(OT) defined by taking the active sites to he open What we have just shown is that P is a 2independent site per colation measure.. The probability that a given site in IF is active is at most it, sup s 1F'7) (y 1' ) if r is huge enough then, from (16) and 'Theorem 6, we have or
A
ithin
pt
To apply
(21 )log(2, yam exp(—r Nog/ )2)
Lemma 11
of Chapter 3, let us define
[(I) = Since
exp(—r2/2)
= A(D(11 1 )) <
imAtto
whenever (18) holds, we have f (r) < exp(1)
/640 (
)
4. 5 Exponential decay of the volume
111
In particular, j(r) < 1 if t is large enough, and, as (1$) holds for infinitely many I , there is some r for which (18) holds and f ) < 1 From now on, we fix ;Men an . Applying Lemma 11 of Chapter 3 to the 2independent site percolation measure IP on the graph D(W), we see that there is a constant c > 0 such that, for any z E IV and any nr > 0, i13 0C.(D(11 7 ))]
in) 5 exp(—cm),
where Cz (D(IF)) is the open (active) cluster of DO V) containing z To complete the proof, note that if x is a site of A and I CA > then for every site y E C„, the event {y =} holds. (There is no room for all of Cr inside B, _ i (y)) Let IV' = fw E FP : (w) n 01 Then every site w E IV' is active Also, as the balls Bo, (w), w E TV, cover 1 7 (11), the balls Bo, (w), E IV', cover Ci„ We claim that 11 7 ' induces a connected subgraph of D(11) Indeed, E then there are sites y, y' E with y E .80 1 (w) and if w, As C'„, is connected, there is a path (al) y = !pry° • • yt = E yj Iv( E IV' with Eh E Bo, (wi), with every y i in C . We nilw choose and to, = w, air = i v'. For 1 < i < t — 1 we have death < witvi+, is an = 4r ± 1 < 6r, so either mi = edge of D(IV) He/we, w and iv' are joined by a path in the subgraph of D(11") induced by IV' Fixing E A, let E II be any site such that :r E B2r (2) We have shown that II' is a set of active sites that is connected in D(II), so TV' is a subset of Cz (D(W)).. Also. as C . is covered by balls of radius 2t, we have > Hence, for n > > n) <
HA ) Ill'OC',;(.0(1V))1 ^ n/q)
As r is constant, the proof is complete
exp(—cnnib ) ❑
In the p oof of Theorem 10, we used Menshikov's Theorem. Theorern 6, to obtain a.) good bound for on ner 4 p < Mi . Instead, we could have used Hanunersley's much simpler result, Theorem 9. Modifying the proof in this way would give the satire conclusion, (17), but only for p < . (Of course, under the assumptions of Theorem 10, by Menshilcov's 'Theorem ) =p Next we shall present the general result of Aizernnau and Newman [1984] mentioned at the beginning of the section.
Exponential decay and critical probabilities
I 12
Let A he infinite lacallyfinilc oriented multi graph with and strongly connected. and le! p < pq ( A ) Then th in is art o > 0 such that Theorem 11.
r71111(
E l ;(1Cli I
ir)
fat all sites re and integ e rs n >
exp(—on)
(19)
1
Pox"' As p < pl.(71.) is fixed vve suppress the dependence On pin of
notation Suppose first that Csc has only a single vertex, i e that all sites arc outlike this is the wain par t of the proof: the extension to [had many outclasses is a minor variation plr denote the event that there is an open path As berme let fr y+ from an outneighbour of x to y interpret {.r + — .r} as the event which always holds. Thus — It holds if amid only if .1' ± — II and :1' is open. Note that El
E P(.1. !t)/P= yr(r)/P
PLr li pc.,
1r
ye:\
\\ le shall estimate the moments of ICH in terms of using the van den start with the second moment Berg rkesten inequalit \ We mk, not assume that z. fir and yo ate Suppose that . yr. E distinct. although the Heinle is clearer if we do. As um E there is an open (oriented, as always) path Pr horn x to yr Let Pr, he m shortest open path how a site a on P i to /1 2 : such a path exists as /12 Then Pr and Pr are vertex disjoint except at a (otherwise. Pr could be shortened) Let us split PI into two paths. a path P; horn a: to and a path Pi/ horn a to yr: see Figure 15 ()witting the initial vertices horn the paths P(. P(' and P, gives disjoint witnesses for the events IT + , — I/1 {  !P I }* respectively.Tills ' if 1./2 E Cr. Olen the event 
0 {a + IL}
Ica some tt E A (Recall that 0 is an associative operation) Using the can den Bet g IKest en inequality, it follows that holds
FOP fi g ) <
E
 it)P(u + —
111)P(a +
r12)
4 5 Exponential decay al 11u; voleuntc
Tigre 15. Open paths Pr = Pi U felering the initial sites fl ow 12;, Pi"
Stunninig wai l ,
and P2 showing that In t E and P2. these paths are disjoint
113
Alter
E A, it follows that
Eacid 2 ) = Ey; gt
P O/I, /12 E
112
E
IPI (r: 4) — f () P O I'" — :01 )Pfli+
(/1 where, hour now on. all summation variables rnu (wet all sites of A a7) = foi any As all vertices are outlike, we have E(L+ A Evaluating the triple stun above In first summing over yo, Own over yr. and theft ()VC/ it, it follows that this sum is Eractig (‘T. so E.(1(7,1 2 ) (‘')" For higher moments the argument is ver y but slightIN battler to write down: the only additional complication is that if in.. E Cif and Pt , Pi and v t = rt are defined as above. then if ya is also in C ad the shortest path P3 from a site E Pr U A to ya may start limn a site it7, on P(, on Piu , or on 12, It will be convenient to write tin fot and to denote the 'branch vertices' by We shall say that an oriented tree TE on the set •E
+ 1
{0, 1, 2.
. k, —1, —2.
.—(k 1) }
is a k4ettiplate if it satisfies the following recursive definition: for k = 1, the only 1template is the tree consisting of the oriented edge 01. For > 2, the tree T is a ktemplate if it way be obtained flout some (k'  1)template r by first inserting the vertex —(k — 1) to subdivide an edge of T. and then adding an oriented edge nom —(k — 1) to k Note that the number of k templates is exactly A't; = 1x3x5x x (2k3) = (2k —3)!!, as there are c(T') = 2k — 3 choices for the edge of to subdivide..
111
Erponential decay and critical pinbabilities
= I/o throughout, a realization 1? of a template T Fixing the site . y_ k + i of (not necessarily distinct) sites ,:yk, is a sequence E E(T), with of A such that there are disjoint open paths P u yj; see Figure 16 We call yr,. line the a (minimal) witness for: yt
SI
Fignie .16 A realization of a Ttemplate The directed open path to MI is a witness for Olt.,
from
2
leaves of the realization Note that we Mal' have y; = yj ford 0 j, in which case Pr, is the 'empty path' with no sites or bonds sic E Clz , there Recalling that yo = we claim that, whenever yr, has leaves yr, 1? such that T is a realization 1? of some template The proof is by induction on k. For k = 1 the claim is immediate: as yr}, which is all that Y./ E Ca, and x yo, them is a witness for {fil" is required For the induction step, given a realization 1?! with leaves , yk _r, let P be a shortest open path from a site y_ k+ 1 appear ing in RI (i.e from a: OF from a site in some Pr, in /V) to yk Splitting the witness (path) on which y_ k+i appears as in the case k = 2 above, and taking P (without its initial vertex) as the witness P (k+l)k for
fy k+1
yk l, we find a realization I? of some template T
By the van den BergKesten inequality, the probability that T has a
4.5 .E.eponential decoy of the volume realization 17. corresponding to a
sequence
is 9.0
11)})
{Y;
P
particu lar
115
E
yk E Cr is at most
Hence, the probability that
EE
P
elli
1/k ),
f.
77E1
where
denotes summation over ktemplates, and ,
over sites y_r,
tnunation yk.
A Thus, summing over yr,
E
E(Crik)
(20)
(T),
T Wile/
e ;r(T) =
with the sum
II 75E/
— Yr). ,;(p„
r.r running over sites AT
,y_kr1.1
The definition of 7r(T) may be extended to define 7r(U) tot any or iented abelled tree U with 0 as a vertex: we sum over a variable r ki for: each l vertex i r 0 of U. It is easy to see that if every edge is oriented away from 0, as is the case it U is a template, then
.R(u)=(x/)`gn irked there is a leaf j
0 of
U.
so
(21)
ij E E(U) for some i.
(22)
P(tI;  ) r er
for any we have 7(U) =x 1 )r(U1 ) where U' =Uij. and (21) follows by induction Rom (20) and (21), we have EaCzik)
(.0r2(7) = Nkm2k = (2k  3) ( ))2k1
for every k Hence, for any t > 0 we have L (exp(tIC, I)) = T
E (t. I k
Pr:!)
(2k  3)H
, ( (A )) A k
i
116
Erponentiol decay and el Then/ pinbalnlihes
As (2k a) /1,7! < 2 k . this expectation is finite lot 0 < t < (‘') 2 /2 As E(exp(ird)) 
exp(tn)PaC r i = n),
it follows that 1.11 (r i d = 11) < exp(/o) fin all sufficiently huge 11, and the result follows,. SO fat. we assumed that all vertices ate outlike As plot/Used, extending the result from this case to the general case is sttaightfonvatd Indeed, the pool is the same, except that we 'enlace by \" = SUP
P UP . —
In= SU P
\if (v)I p
In place of 22), we then have 11) (lif
(b) <
!j
so rr(U) < ( \i ")'“)) . As shown in Lemma 2.. when El i is strongly connected, then \ „,(p) is finite Mt one site w if and only if it is finite fin all sites. i e if p < Emilie/ mote, as ‘„,(p) depends only on the outclass of W and (7.c is finite, the supremum above /WO' he taken mer a finite set. so \i " is finite The test of the proof is unchanged. ❑ Let us recap very hiiefly the main themenis of this section; all these exults matt n finitet ' pe graphs with st rough connected classg t aphs. We have moved results of Hammersley and of Aizenman and Newman that.. nuclei mild conditions, for p < p T < p it , both the radius and the VOIIIMP of the open cluster containing a given site decay exponentially. \\M have also moved Menshikov's Theounn that, under a very weak condition on the growth of the neighbourhoods, the critical probabilities pi and pu t coincide. Thus.: fin a very wide class of pound graphs, pu t pun. and if p is less than this common value p, then the size of the open clustet of the origin decays exponentially. A very interesting problem that we have not touched upon is the speed of this decay. Mo t e pecisely, what me the best constants ca in (1.5) and in (10), and how do these constants depend on pas i t approaches flow below. The arguments above give some bounds on the constants in terms of other quantities, butt then beha y iuur as p pr is a \ ti difficult question. about which rather little is known We shall retain to this Welly in Chapter
5
Uniqueness of the infinite open cluster and critical probabilities
Ou t hist aim ill this chaplet is to plesent a result of Aizemnan, Kristen and Newman [19871 that, under mild conditions, above the critical ptobability is a unique infinite open cluster; Button and Keane 119891 have given a very simple and elegant proof of this result Together with Menshikov's Theorem, this uniqueness result gives an alternative pool of the Harr isKesten Theorem; this proof is easily adapted to deter mine the critical probabilities of certain other lattices. 'The key consequence] of uniqueness is that, under a symmetry assumption, the critical abilities for bond percolation on a planar lattice and on its dual must stun to I \\re shall prove this, along with a corresponding result for site percolation, assuming only ordei two symmetin y Finally, we discuss the stardelta transformation, which HMV be used to find the critical probabilities for certain lattices that ale not selfdual
5.1 Uniqueness of the infinite open cluster – the Aizenman–Kesten–Newman Theorem 'Throughout this chapter the uncle t lying graph A will be Imo/ iented„ and of finitetype In CAIRN' words, there ate finitely many equivalence classes of sites undo the relation in which two sites :r and y are equivalent if thew is an auton t orphism of A mapping y, in which case we wine 3' 1') j As usual, A will be infinite, locally finite, and connected \Ye stint this chapter with the result of Aizemm t , Kesten and NOWWWI 119871 that, in this setting, with probability 1, there is at most one infinite open cluster For the special case of bond percolation on ..,•,,. was proved by Harris [1960]; Fisher 11961] noted that Harris's argument carries CVO/ to site percolation' on Z 2 A little later, Candolfi, (trimmed and Russo 119881 simplified the original proof of Ali/TIM/MI.
118 Uniqueness of the infinite open du tiful and critical probabilities Eesten and Newman. A very different and extremely simple proof was then given by Bur ton and Keane [1989]; we shall present their elegant argument below Related results for dependent measures were proved by Gandolfi. Keane and Russo [1988] and Gandolfi [1989]. This section is the only place where we shall use proper ties of infinite product spaces in a nontrivial way (although, of course, we could rewrite even these arguments in terms of limits of probabilities in finite spaces if we wanted to) Recall that the probability measure 111 / , P;\ 4, is a product measure on the infinite product space Q = {0, 1} F(A) . Thus, the afield E of measurable subsets of Q is generated by the cylindrical sets. C(17,0)= ful E Q : rdf
o f for f E F1,
where F is a finite subset of i t (A) and a E {0, 1}F As usual, an event is just a measurable subset of Q Thus, c.ny property of the open subgraph C) that depends on the states of only finitely many sites is an event, and countable unions and intersections of events are events In fact, any property of 0 that one would ever want to consider in percolation is an event. For example, recall that {x 1..} denotes I he property that there is an open path flour the site 3, to a site at graph distance I/ from 3: For a given site x and integer ft, tins property depends on the states of finitely many sites, and so is measurable It follows that {:e x} is an event Similarly, 'there is an infinite open cluster' and her e are exactly two infinite open clusters' rue events An event E is autamarphism invariant if, for every antomorphism y of the underlying graph A. the induced autornorphism y'' : 12 Q maps E into itself. hr particular. = there are exactly k infinite open clusters
1
is an automorphisminvariant event for k = 0,1, oc.,. The only property of automorphisminvariant events we shall use is that any antommphisminvar hint event has probability 0 or 1 We state this as a lemma for site percolation The corresponding result for bond percolation follows by consider ing the line graph, noting that if A is of finite type, then so is L(A) Lemma 1. Let A he a locally finite, finitetype infinite graph, and let E C Q = 0 1'0) he an antonunphisminvariant event Then In p (E) E }0,
5 1 The AlzenmanKestenNenynan Theorem
119
Proof We shall write F for FA 1, Let xo be a site of A Note that, as A is infinite, locally finite, and of finite type, there are infinitely many sites x equivalent to to Let > 0 be given Since E is measurable, there is a finite set F of sites of A and a cylindrical event Er depending only on the states of the sites in F. such that P(ELEF) < E
(1)
Let AI = max{d(xo, q) : y E F} Since A is locally finite, the ball = {z : d(x0 ,z) < 2A1} is finite Thus there is a site x with xo and d(x,,r0 ) > 2.111 Let (p be an autonnuphism of A mapping xo to a. For y E F we have Bom(xo)
d(te,y)(0)
d(au, Y0'0) d(V( to), (PM = der° , 37)  (1.070 ,:y) > 251  AI =
so cp(y) F. Thus the sets of sites F and y(F) are disjoint It follows that the event y(EF ), defined in the natural way, is independent of Er SO
NEL
n v(EI )) = F ( Er) F (y (E r)) =INET )2
Thus, 1P(E) F(Er)2I = 1.P(E n E) P(Er n ))I < P((E n E)A(Ef, n y(EF ))) For any sets A, B, C. D we have (Anr)A(c Thus, as E is automm phisrninwniant,
I P ( E )  P(Er )2
1
<
C (Aa,C)U(BLD).
F(EL1EF)+NEOV(Ep)) P(ELEF)+PP(E)6,y(EF))
= F(EL& ) P(EAEF) = 2P(ELEF)< where the final inequality is from (1) Since 1P(Er)  F(E)1 < IP(ELEF)1 a we have I F( E )  P( E) 2 1
< 1P(E)  P(EF . ) 2 < 2 ±
Since s > 0 is arbitr ^
9E
1 + i
r (EF) 2— P(E)21
45.
it follows that P(E) = P(E) 2 , so F(E) is 0 or 1 0
ii tji
120 Uniqueness of the infinite open cluster and critical pivhabilities In the proof above, we did not need E to be invariant uncle/ all antomot phisms of A. just under a set of autornmphisms large enough that any finite set can be separated horn itself by such an autornmphisin In the context of lattices in R d . for example, invariance under a single automm phism y of A conesponding to a translation of IF'' thiough any vector (a l , .0) is enough. In particular. Lemma 1 ati) (0,0, is often stated for translation invariant events In the terminology of ergodic theory, we have shown that the induced automorphism : R Q is et godic. NAre now turn to the particular event lk that there are exactly k open clusters, where (1 < k < x We star t with a lemma of Newman and Schulman 119811, showing that, with probability 1, there 1 or infinitely many open clusters are
0,
Lemma 2, Let. A be run infinite locally finite. finitetype graph. and le E (0.0 Then
= ft
PA , \:
Ilene( Why IF p (10 )= 1 Ps.%
)
=
)=1
1. tit
By LC/111/18, 1. it suffices to prove the first statement: As before, we write for Suppose for a contradiction that P(/(,.) > 0 fon some 2 < k < c Let To be any fixed site of A, and let be the event that lk holds, and each infinite cluster contains a site in B„(x0 ) As the balls B„(ry) cover A. we have Lk = u n T, k, ,/ P (11,), and Oleic is air 11 such P(Ty that P(T„ , r) > 0 Changing the state of every site in B„(,ry) to open, we see that P(./ 1 ) > 0 Indeed, spelling everything out in great detail, the event T„ A is the disjoint union of the events P100.1
so
=T„
.:n{5' = s}.
where s (s„);,,Bu(,,„,) E (13 " 1 " ;) , and 5' = (5,.):,..EBu(,,„) is the (to) vector giving the states of all sites in Thus there is an s which 0. Now if w E T„ k s and w' is the configuration obtained P (rs k s) horn w by changing the state of each of the closed sites in f3„(ro) h om closed to open, then w' E In Thus k
for
>
P(1 1 )
La ( {if
: E T„
= (p1(1.
p))1(TT„ s ) > O.
5 I The AiriemnanAestenNeuunav Theorem
121
where c is the number of sites in .13„(:ro) that are closed when S =s We have shown that if P(//,,) > 0 for some 2 < k < cc. , then P(I,) > 0 But then P(4) = P(1 1 ) = 1 by Lentela 1 As 1k fl it = 0, this is ❑ impossible To prove the main result of this section.we shall need a simple deterinistic lemma concerning finite graphs L emma 3, Let C; he a finite graph with k components and let L and , es } he disjoint sets of ITT bees of C. with at least one et C = in each coinponent of Let . be integers each at least 3 Suppose that fin each i deleting the reflex m disconnects the component containin f« . 1 info mullet components. in; of which contain mudd i es of L Then
ILI > 2k 4 E(In i — 2) Proof By considering each component: of G separately, we 11/111/ assume that k = 1, i e that C is connected Flemming an edge from Ce can only increase tire number of components of — e i containing elements of L.. so we may assume that C is minimal subject to CI being connected and containing C'U L Thus C is a tree all of whose leaxes are in CUL As > 3, no vertex E C' can be a leaf of C. so all leases are in L (there ma ■ also he internal vertices in 14 Since C — has in; components, the vertex has degree in; But for any tree with at least one edge, the number of leaves is exactly 2} (d(e)— 2), where the sum HMS over internal vertices. As d(v)> 2 for each internal vet tex, the sum is at least. 57= ,(nc —2), and the result follows. ❑
5
Let us say that an infinite graph A is untenable if 1.5' ,,(r)iABflOpi —' as n for each site 3, i e . if large balls contain man y more sites than their boundary spheres. There me several notions of amenability for graphs; in the present context this \In hint seems to be the most useful. In fact, the concept of amenability originated in group theory, where it is defined somewhat differently. Note that. if A is amenable am( of finite type, then the limit above is automatically uniform in r. The following result is dire to Aizemnan, Nester' and Newman [1984 the proof we shall present is that of Bruton and Keane 119891 Theorem 4. Let A he a connoted. locally finite. finite type. amenable
122 Uniqueness of the infinite open cluster and critical probabilities infinite graph, and let p E (0,1) Then either 1FX p (I0 ) = 1 or rA p(B) 1, where Ik i s the event that there are exactly k infinite open clusters in the site percolation WI A Proof As usual, let us write P for P;\ p In the light of Lemma 2, this result is equivalent to the assertion that P(/,) = 0. In fact, we shall show that the probability that there are at least three (possibly infinitely many) infinite open clusters is zero Suppose for a contradiction that this is not the case.. Let to be any site of A, and let X 0 be the set of sites equivalent to zo As the balls Br (ra), r > 1, cover A, there is an p such that, with positive probability, Br (x0 ) contains sites From (at least) three infinite open clusters. For the rest of the argument, we fix such an I Let Tr (_y) be the event that every site in B, (x) is open, and there is au infinite open cluster C) such that when the states of all the sites in B, (a) arc changed horn open to closed, 0 is disconnected into at least three infinite open clusters; see Figure 1 If w is a configuration in which
Figure I A site a for which Tr (c) holds, Ever y site in B, (r) is open; the rest of the open subgraph is shown by solid lines If the sites in Br (x) ale deleted, or their states changed to closed, then the infinite open cluster C) falls into four pieces, three of which are infinite
B, (to) meets at least three infinite open clusters, and w' is obtained from w by changing the states of all the sites in B, (to) to open, then E Ti (a0 ). Hence, IP(Tr (to)) > Thus, for all sites x E X0 we have
P(T, (a)) = a. for some constant a > 0 Our next aim is to sinus that, if
/I
(2)
is much larger than 1, then we can
5 I The kicennionICestenArcoonon Theorem
123
find marry disjoint balls B, (:r), :r 6 X0 , inside the ball B„(:to) In fact, to make the picture clearer, we shall find balls B, (.r) that are far from each other. To do this, let; If c X 0 n B„_, (x0 ) be maximal subject to the ba lls (c), w E W. being disjoint If lot E Xo n B„_, (x0), then (1.04/, w) < 4t for some w E otherwise. w' could have been added to If . As A is connected and of finite type, there is a constant C such that ever y site is within distance C of a site in X 0 Thus, every y e B„_,—a:ro) is within distance t of some w' E X0 n B„_,.(3,0), and hence within distance 4t+t of some w E W In other words, for a > +11, the balls BID4w), w E H. cover B„_, _ ( Gro) Thus, IH1
18121(edi/P1,H:(34
ver y crudely. 1B„,n(x0 )1 ,5_ IB„_,_ 1 0.0 )10 + A + A 2 +
A'±t+/),
where A < x is t he maximum degree of A Thus. since is a c > 0 such that
I
is fixed, there
(:co)i for all n > t
C. As A is amenable, it follows that 111 7 1 > a 1 1,9„ + ("di
if n is large enough, where a is the constant in (2) Let us fix such an a Let us call a ball B, (w) a cutboll if w E 117 C B,,, (::0 ) and T, (w) holds. Note that if Br (a.,) is a cutball. then B, (iv) C B„ (co), and every site in B,(w) is open Since w ti ,r0 for every w E by linearity of expectation ' the expected number of cutballs is
E P(T,(w)) = 011171
IS„.H(370)1
„'Em Hence, as P(Z > IE(Z)) > 0 for any random variable Z, there is a configuration w such that, in this configuration, we have s 18nm(tie)1,
(3)
where s is the number of cutballs As we shall soon see, iris contradicts Lemma 3 For the rest of the argument, we consider one particular configuration w for which (3) holds: in the rest of the argument Uncle is no randomness. Let denote the union of all infinite open clusters of the configuration meeting B„ (4), considered as a subgraph of A. In the configuration
EL)
124 Uniqueness of the infinite open cluster rind ci Weal probabilities w' obtained from w b y changing the states of all sites in cutballs to closed. the (perhaps already discotmected) clustet 0 is disconnected into several open clustels, some infinite and some finite Let the infinite ones Fu Each L i contains 1 be L 1 , L 2 . ,L t , and the finite ones site in 5'„+1(:ro)„ so TIM/]
(I)
Let the cutballs be CIL , C. We define a graph H from 0 by coneach F4 to a single vet tex trading each cutball Ci to a single vertex and each L i to a single vet tex en see Figure 2 In the graph H. there is an edge how Ci to ci , for example, if and only if some site of fq is adjacent to sonic site of CI; Infinite components of (9 cot espond to components of H containing at least one vertex in L = Tints, the condition that Ci is a cutball says exactly that deleting c i Flom H disconnects a component into at least (Mee components containing vet tices of L. Thus we /nay „ s, to conclude that apply Lemma 3 with in; > 3, i = 1.2, t=
2±
E ( 3 – 2) = s + 2.
This cont /adios (3) and ( ). completing the wool
❑
Simpl y put, Theorem 4 tells us that, above the critical probability /4 1 . almost smelt theme is a unique infinite open clustet TO conclude this section we remark that, by a simple argument of van den Beta and Keane (198. 1], Theorem 4 implies that 0(1 .,p) is a continuous function of p, except possibly at p =14[(A)
5.2 The Harris–Kesten Theorem revisited Combined with Menshilrov's Theotem, Theorem t leads to vet another proof of the thurisKesten result that pl'i (Z2 ) = (22 ) = 1/2 This moot will adapt easily to give the exact values of the critical ptobabilities tot certain tithe/ planar lattices We start by improving the 'ease' inequality that p li ),(Z2 ) > 1/2 Mote precisely, we shall deduce Mattis's 'McGloin, restated below, nom Theorem 1 The mg:tin/eat: we give is due to Zhang; see Ctimmett, (1999, p 2891 Theorem 5. Fm bone! percolation cm E 2 l i am 0(1/2) = 0
5 :2 The Harris taster Theorem revisited
125
Figure 2 The upper figure shows the union 0 of all infinite open clusters meeting B„dro) The shaded halls, in which all sites are O pen, are the cutballs. The lower figure shows the corresponding graph H The filled circles ar e the vertices ei corresponding to the cut: balls, the hollow dines the vertices t i corresponding to the infinite clusters L t , and the crosses the vertices di cm esponding to t he finite clusters
Hive( Suppose not. Them applying Theorem 1 to site percolation on the line graph of Z2 we see that iP itq l ) = 1 . where I I is the event that
126 Uniqueness of the infinite open eluslel and critical pmbalatities
there is exactly one infinite open cluster, mid It follows that there is an no such that, if n > no, then the probability that an infinite open cluster meets a given by a square is at least 1  101 Let n = no + 1, and let 5' be an by square in 22 . Suppose that some site x in S is in an infinite open cluster Then there is an infinite open path P starting at Let y be the last site on .P that is in S, and let P' be the subpath of P starting at in then P' is an open path from S to infinity, using only bonds outside 5'. Let L i be the event that there is an infinite open path P' as above leaving S upwards, i.e with the initial site y on the upper side of 5, the initial bond vertical, and all bonds outside S. as in Figure 3 Let LO, L 3 and be defined analogously,
Figure 3 An infinite open path P starting at a site a in a square S subpath P' [torn the last site y of P in S leaves S upwards.
rotating 8 though 90 degrees each time Thus P 1/2 (L 1 ) = P i/2 (L i ) kn all i. We lime Pip (LI U L9 UL3UL 4) > 1  101
(5)
As the L i are increasing events, it follows from H a rris's Lemma by the 'nthloot trick' (equation (8) of Chapter 2) that P(L 1 ) > 1  1/10 for each i: other wise, INV!) > 1/10 for each and, as the L ate decreasing and hence positively correlated, L?) 101, contradicting (5) Recall that the planar dual of the square lattice 2 2 is the lattice Z2 + (1/2,1/2), and that we take a dual bond e* crossing a bond c of 22 to be open if and only if e is closed Let 5" be an 11  1 by 11 1
5.2 The HarrisKesten Theorem revisited
127
Figure 4 Infinite open paths P, and P4 in the lattice V, leaving a equate S to the right and to the left The infinite open paths P. PI in the dual lattice leave 8 1 upwards and downwards. If P( and may be connected by open dual bonds in S' t hen there ate at least two infinite open clusters, one containing A, and one containing Pi
square in the dual lattice inside 8, as in Figure As the bonds of the dual lattice ate also open independently with probability 1/2, and as n — 1 = ne > no, the argument above shows that P i r(L;) > 9/10, where is the event that an infinite open dual path leaves the ith side of Si . Thus the event E = fl L2 fl L43 11 LI illustrated in Figure I has probability at least 1 x (1 — 9/10), 6/10 > 0. The event E depends only on the states of bonds outside 8'. Thus, with positive probability, E holds and every dual bond in 5' is open But then the paths P. and .P:13 may be joined to form a doubly infinite path P' that separates the plane into two pieces. As P' consists of open dual edges, and an open edge of Z2 cannot cross an open dual edge, the open paths ./?, and lie on opposite sides of P', and thus in separate open clusters. Hence, there are at least two infinite open clusters
In short, starting horn the assumption IP, /2(1 1 ) = 1, we have shown that with positive probability there me at least two infinite open clusters: a contradiction It follows that P i pe(11 ) = 0, and hence that 0(1/2) = 0
El
128 Uniqueness of the infinite open cluster, and critical plobabilitics Using Menshikov S l /WU/ it is veil, eas y to complete l et another proof of the fla t r is [Kristen Theorem. Them cm 13 of Chapter 3. restated below Theorem 6. Pot bond p rcolation on the square lattice pm ! = pr = 1/2 e
Proof As misted in Chapter 4. Theorems 7 and 9 of that chapter. stated
for oriented site percolation. appl y also to tu t or iented bond percolation and in particular. to bond percolation on E 2 (Formalit y one can apply the theorems to the graph obtained front the line gr aph of 2:2 by replacing each bond Ir y two ofmositd y oriented bonds ) In particular, lenshikoyis result, 'Theorem 7 of Chapin 4. gives ',I li C.2;2 1 = p r (7, 2 ) Since 0(1/2) (2V) < 1/2 implies ph (;Y,') > 1/2 it tints suffices to prove that This is immediate flour Theorem 9 of Chapter I and Lemma 1 of Chapter :3 Indeed. suppose that p.11(272 ) > 1/2 Then. In the first of these results. as 1/2 is less than the critical probabilit y, there is an o > (1 such that LP it i(f 21() < exp(—mt) for all sites ,r and integers n. where Tr 21 1 denotes the own that .r is joined by an open path to some site at graph distance n front r Taking n large enough. we have j
JJ) C 1/(10(M) 113 1/2 (ft "H Let S be an n by n square in 2T2 As before. let 11(8) be the event that thane is an open horizontal crossing of S II 11(S) holds. then one of the n sites on the left of 5 is joined by art open path ill S to a site on the right of S. tit distance at least n — I Hence. (H(S)) <
nP 1/2
<1/100
But this contradicts the basic fact that P 1 1 ,01(S)) > 1/2. which v,e know from Corollary 3 of Lemma 1 of Chapter 3 As the Harris Rester/ Them cm is so fundamental. let us briefly summarize the different approaches to its proof that we have presented here. All the pools start hum the basic fact that either a rectangle has an open Ina izontal crossing. or its dual has an open vertical crossing Then, to prose that p ill (Z2 ) > I/2, one ma y prove a RussoSeymourWelsh (RS \V) type theorem as in Chapter 3. 'elating crossings of rectangles o crossings of squares Alternatively. one can deduce the result horn the Aizeurnan Kristen NtiNV/IMIt ' filet)/ C/11, Theorem 4 lb prove that i ii(G 2 ) C 1/2, ha y ing moved an RS NV type theorem. one can apply one of tat ions sharpthreshold results as in Chapter 3 Alternatively
58
Site percolation on the Niangidat and square lattices
129
one can deduce that ph(Z) = (Z2 ) > 1/2 directly Flom Monshikov's Theorem (Fm the last pint: we do not need exponential decay of the radius as used above: the almost exponential decay given directly by Theorem 6 of Chapter 4 is more than enough ) The approach used in Chapter 3 is perhaps mote downtoearth, and simpler it/ any one given case The advantage of the Afenshikow Aizennnt Kest:enNewman approach illustrated in this chapter is that the tools rue very general, so genetalizing the moth to other settings is easier. Of course. one still needs a suitable star ting point, given by sortie kind of selfduality Let us note that, in this latter approach, moving that percolation does occur undo suitable conditions. which was historically much the l u odei part of the Flattisliesten result, is vent easy: the deduction hour Alenshikov's Them em is very simple. and can be applied in a peat: variet y of settings. In contrast, showing that percolation does not occur. which was histoticallt the easier part of the result, is more difficult: the deduction Man the Aizen n ian Kesten Newman Them ent is not quite so simple, and requi t es write assumptions. We shall see this phenomenon again when we consider percolation on (whet lattices
5.3 Site percolation on the triangular and square lattices We next to p side! the (equilateral) tit iangulat lattice T C IF:2 Pot definiteness, let US imam and scaler so that (0.0) and (1, 0) ale sites of T. and all bowls hate length 1 Portions of T me illustrated in Figures 5 and 7 below Out aim is to show that psil (1) = = 1/2. As a stinting point, we need a suitable selldualit y property In bond percolation on V. the outer boundary of a finite open cluster can be viewed as tin open cycle in the dual lattice V ± (1/2,1/2) For site percolation Oil T. it is east to see that any finite open cluster is bounded by a closed cycle in the 5anie lattice T. Also, an open path in T cannot stair inside and end outside a closed cycle in T: indeed, the latter statement holds for site percolation on any plane graph, as a cycle in a plane graph separates the plane into two components These observations give a sufficient still ting point to enable us to prove that Mr(T) = p)(T) = 1/2, using the results of Menshiko y and of Aizenman, Kesten mid Newman. In fact, as in Chapter 3, it is easy to prove a striking ; hugescale' consequence of the selfduality As usual, we write t he percolation nwastue undet considelat ion, in this case, for P)1/ P„ Lemma 7. Let 1?„ he the rhombus in
until 11 sites mi a ,side s"loam
130 Uniqueness of the infinite open cluster and critical probabilities
in Figure 5, and let fl(R,,) be the event that there is an open path in T consisting of sites in R.„, starting at a site on the lefthand side of R„, and ending at a. site on the righthand side. 'Then P 1/2 (11(1?„)) = 1/2 for every rr >
• • • •• .•A. • • ••
Figure 5 closed sit
V
to rhombus nu: solid citcles represent open sites, and hollow circles
Plug. Let IiI (R„) be the event that there is a closed path in R„ joining the top of R,, to the bottom. Reflecting R„ in its long diagonal, and exchanging closed and open, we see drat
1E„(11(R„)) = Pr_glit'(/?„)) for any p and any n. In particular, Pit,(H(R„)) = 1P it (17 *(R„)). It thus suffices to prove that 17[72(H(R„)) +FP1/2(r(R„)) = 1 As in Chapter 3, we shall prove the much more detailed result that, whatever the states of t h e sites in R„ exactly one of the events 11(1?„) and 17 *(1?„) holds.. The proof is essentially the same as that of Lemma 1 of Chapter 3, although the picture is somewhat simpler One can replace each site of T with a regular hexagon to obtain a tiling of the plane. Thus, what we have to show is that in the game of Hex, no draw is possible: if all t i re hexagons corresponding to R„ ate coloured black m white, then either there is a black path from left to right, of a white path from top to bottom, but not both (On a symmetric board, it follows easily that the first player has a winning strategy. )
5 .9 Site percolation on the triangular and square lattices
131
To see this, we shall consider face percolation on the hexagonal lattice, which, as noted in Chapter 1, is equivalent: to site percolation on T More precisely, let; us replace each open site of R„ by a black hexagon, and each closed site of R„ by a white hexagon, and consider additional black hexagons to the left and right of R„, and white hexagons above and below R„, as in Figure 6
Figure 6. A partial tiling of the plane corresponding to Figure 5, obtained by replacing each open site in Ro by a black hexagon and each closed site by a white hexagon, with additional black and white hexagons around the outside The thick line is a path separating black and white hexagons, starting at :r, with black hexagons on the right This path must end at y (as shown) or at re The rest of the proof is exactly as for Lennart 1 of Chapter 3: let I be the interface graph formed by those edges of hexagons that separate a black region horn a white region, with the endpoints of these edges as the vertices. Then every vertex of I has degree exactly 2, except for the four vertices ,r, y, z and w of degree 1 shown in Figure 6 The component of I containing a is thus a path. Following the path star ting at there is always a black hexagon on the right and a white one on the left, so the path ends either at y or at ro In the for men case, the black hexagons on the right contain a path iu T witnessing H(R„), in the latter case, the white hexagons on the left witness V"(.11.,,) As before, 11 (1?„) and V"(R.„) cannot both hold as other wise Kr, could be drawn in the plane Using the results of Menslrikov and of Aizeinn, Kesten and Newman, it is easy to deduce that the critical probability for site percolation on T is 1/2, a result due to Kesten [19821 Menshikov's Theorem (Theorem 7
132 Unpteney; of the infinite open Haslet and et Hirai ptvbabilitics of Chapter 1) cells us that m = p H in this context: horn now on, we wine Th. for their common yable Theorem 8. Let I be the equilateral trianottla, lattice in the pla ns Then ms.(T) = 1/2 Proof By Theorem 7 of Chapter 4 we have M i (T) = Suppose first that /VT). it; (T) > 1/2 Then, by Theorem 9 of Chapter 1, we have exponential decay of the radius of an open cluster at p = 1/2, i e., there is an a > 0 stud/ that 111 1/ 9(0 < exp(—or/).. DefinR„ as in Lemma 7, any of the sites on the righthand side of is ing at distance at least it — t hour any of the a sites on the lefthand side. so Pit2
(11(R„)) < Itifi l/2 (0
:)
n exp(0(n — 1 ))
As n — oc the thud humid tends to zero, contradicting Le11/1118 7 Suppose next that p;'(T) = M i (T) < 1/2, so 8(1/2) > 0 Then. MI heorem 4, in the p = 1/2 site percolation on T there is with probability I a unique infinite open cluster Let /1„ be the hexagon centred at the or igin h " sit es on " eh side dm)" iu Figure 7 As U„ = T, if n is large enough then the ihrp t obability that some site in H„ is in an infinite open cluster is at least 1 — sap. Numbering the six sides of 11„ in cyclic order let L 1 be the event that an infinite open path leaves horn side i Mote precisely, L i is the event that there is an infinite open path in T with initial site on the nth side of fl„ (we may include both cot nets), and all other sites outside ll„. Then j i L 1 is exact') the event that there is an infinite open cluster meeting > 1 — 10<6 ks the events 1, 1 me increasing. and, H „ by symmetry. each has the sauce probability, it follows hum Iiattis's Lemma (Lemma 3 of Chapter 2) that P i /,(L) = > 1 — 1/10 for each I Let L is be the event that an infinite closed path leaves ./I„ from the itir side. Then F„ (L;) Ifil_.„(Li), so E")/0(Li)
= > 1 — 1/10
Hence, with probability at. least 1 — 4/10 = 6/10 > 0, the event E = fl n fl L .; holds: this event is illustrated in Figure 7. Now E is independent of the states of the sites in 11„_ i Thus, with positive E holds and every site in /1„_ 1 is closed But then the closed paths R„`„ R guaranteed by the events L and L may be joined
u.8 .5Vic percolation on the Niangalar nil squat( lattices
[33
Figure 7. The hexagon Ho with the initial segments of infinite open paths /41 and P4 leaving its 1st and 4th sides and of infinite closed paths P; Pr; leaving the 2nd and 5th sides \Miaowr the states of the gte t sites (and the fl L; holds n andrawn sites), the event. E = L, n to forth a doubl y infinit e closed patil sepal. at ing t he open paths PI and Pi guaranteed by L I and L 4 It follows that. with positive inobabilih.
Untie ace at least two infinite open clusters, cuntIndicting Thement An alternative proof of Theme/a 8 is given in Bollohas and Montan 12006H, based on an RSW type theorem and the sharpthreshold tesults in Chapter 2 The method used to prove Therrien' 8 above may be applied to site percolation in the square lattice This time, the claim} probability cannot be obtained in this way, as the lattice is not selfdual. Indeed, let A D =Z 2 be the plaint] square lattice, and let Az be the graph with vertex set Lt. in which any two yet Hefts at Euclidean distance 1 or J/I?7 are adjacent Thus Az is obtained from Ao by adding both diagonals to each face of A D It is easy to see that a finite open cluster in the site percolation on Ao is bounded by a set of closed sites that form a path in A. and vice versa Also, a path in Ao cannot cross a path in Az without the two sliming a vertex 'These observations ate the starting
134 Ull'iglieliGSS of the infinite open cluster and critical probabilities point for tire proof of the 'duality' result for Ao and AE, Theorem 10 below. Let PI, be the product probability measure in which each site of open with probability p. and let I3 be a rectangle in 7L2 For A A D or A = A D , let HA(11) be the event that there is an open Apath crossing 11 horizontally, i e a set of open sites of I? that forrn a path in the graph A crossing R horizontally Similarly, let VA (R) be the event that there is an open Apath crossing R vertically. The following result corresponds to Lemma 1 of Chapter 3 Lemma 9. Let A be one of A D and Az, let A* be the Whet. and let R be a rectangle in Z 2 11 7hatettei the states of the sites in R. there is either an open Apath crossing I? from left toright, at a closed A"path. Glossing R from top to bottom. but not both In particular.
Pp (H.\ (R)) +P 1 4,,(14A . (R)) = 1.
(6)
Lemma 9 says that, if we colour the squares of an a by n i chess board black and white in an arbitrary manner, then either a rook can move hour the left side to the right passing only over black squares. or a king can move from top to bottom using only white squares, but not both. Bollobas and Riordan [2006b] gave a very simple proof of this result; this proof is eery similar to the corresponding arguments for bond percolation on Z2 and for site percolation on the triangular lattice presented here Figure 8 below, reproduced from Bollobas and Riordan [20061d, is essentially the complete proof Although this fact is not needed for the proof, let us note that the tiling in the picture is a finite part of the lattice (I,8 2 ) shown in Figure 18. The same lattice was used in a different way in the proof of Lemma 1 of Chapter 3. Theorem 10. The critical probabilities for site percolation on the lattices A D and A D obey the relation (A 0 ) +K(A D ) = 1. This result was first proved by Russo [1981] (see also Russo [1982]), by adapting the original arguments for bond percolation on Z 2 , in particular, the RSW Theorem An alternative presentation of this approach is giver/ in Bollobiis and Riorclan 12006bj Once again, Theorem 10 is easy to deduce from the general results of Alenshikov and of Aizenman, Nester' and Newman; we shall describe briefly the steps in such a deduction.
5.3 Site percolation o the triangular and square lattices
135
/ I/
Figure 8 A rectangle R iuE 2 with each site drawn as au octagon, with an additional row/column of sites on each side 'Black' (shaded) octagons are open. Either there is a black path from left to right, or a white path (which may use the squares) from top to bottom Following the interface between black and white regions starting at re, one emerges either at y or at w In the first. case (shown) t he event. K ALI (B) holds Otherwise V(1?) holds
Proof Once again, by kfenshikov's Theorem.. we have MI =p1. for 10 and for A9, so it is legitimate to write p for their connnon value Given an assignment of states to the sites of by a Aclusler we mean a maximal connected open subgraph of A, where A = A 0 or Az. Suppose first that ms (A iD ) pr,(11. 0 ) > 1 Then we um„y choose a p E (0,1) with 1 — pis,(A9 ) < p < p.,1(A 0 ) Note that 1 — p < ps,(110) Taking each site of Z 2 to be open independently with probability p, by Theorem 0 of Chapter 4 we have exponential decay of the radius of the open A 0 cluster containing the origin, and exponential decay of the radius of the closed /1 0cluster containing the origin For large enough a, this contradicts Lemma 9 applied to an rr by n square. Suppose next that pr.(A0 ) p;1(A9 ) < 1 Then there is a p with p > p,(A 0 ) and 1 — p > Ths,(Az) Taking each site open independently with this probability, by Theorem 4 there is, with probability 1, a unique infinite open /1 0 cluster, and a unique infinite closed Azcluster. It follows as before that with positive probability there are infinite open ADpaths leaving a large square S from two opposite sides, and infinite closed Aspaths leaving S from the remaining sides. Hence, with
130 Unaptencss o/ the infinile opal rin g let awl
(chiral
probabilitics
positive mobability them ate at least two infinite open Anyclusters a
❑
cont //diction
As we shall see in the next section. Theorems S and 10 ate special cases of a mote general result (Themern 13) concerning symmettic lattices
5A Bond percolation on a lattice and its dual The results of Nlenshikov and of Aizemnan. Kesten and Newman imply that. under a mild symmetry assumption. the e t hical probabilities lot bond pe t colation CM a planar lattice A //rid on its platun dual A' satisfy 11? (A) + 11: (A') = 1
(7)
When A Z2 , the /elation above is exactly the Hattie Resten heotent Later, we shall prove (7) in sonic generalit y (Theorem 13); first, we illustrate it with another simple example As before. let I be the (equilatet al) triangular lattice in the plane. Let /I be the planin dual of defined in the usual wa y Taking the sites of H to be the centres of the laces of T, then H is the (tegulat ) hexagonal lattice, or honeycomb: see Figure 9..
Elgin e 9 Portions of the iangulat lattice T and its dual 11, the hexagonal
of hancyromb lattice
As we shall see later. the critical probabilities for bond pet colati
on
4 Band percolation an a lattice and its dual
137
71 and on H have been deto mined exacth by NVienuan (1981d confirming a conjectme of Sykes and Essatn 11963; 1961] For the moment:, we show only that these critical probabilities stun to 1 Theorem 11, The triangalat lattice T and honeycomb lattice H satisfy pc.1 ( T )± v!)(if) = 1
(8)
Proof The result follows easily front the general results of Menshikov and of Aizetunan. Is:esten and NeWinall. Suppose first that p"1.(11 ) 11 4!(//) > 1 Then we may chose p E ((l,1) with p < .1)!!(T) and 1 — p < p!)(11) Let us take the bonds c E E (T) to be open independently with probability p, and each dual bond c' E E(H) to be open if and onl y if c is closed. Then la Menshikov's Theorem we have exponential decay of the t adius of open clusters both itr T and hi H.. Hence, taking a huge enough 'rectangle' R as in Figure 9, with probability 99% t here is neither au open path in 71 massing R how left to tight, not au open path in H c t ossing I? limn top to bottom But by planar duality, them is always a path of one of these two types: this is a special case of Le11/111i1 2 of Chaplet 3, w hose proof is the same as that of 1,0111111i1 t 01 that chaplet In this case. the figure obtained la replacing each degree d site of H ot its dual by a 2dgon, and each bonddual bond pan by a squate, is the (3,12') lattice shown irr Figure 1$ To complete the moor of (8), it suffices to show that lot any p, at most one of the petcolation probabilities 0(T; p) and 0(11; 1 — is strictly positive 'This follows from Theorem I as above: if both 0(1;p) and — p) ate strictly positive then, taking bonds of T open independently with probability p, with probability 1 them is a unique infinite open cluster in T, and a unique infinite open cluster in II Considering a huge enough hexagon in T. it follows as berate that with positive probability thew ate infinite open paths in 71 leaving the hexagon flout the 1st and Std sides, and infinite open paths in H leaving from the 2nd and dth sides. If the bonds of H inside the hexagon me also open, we find a doubly infinite open path iu H separating two infinite open components in It a cmaradict ion ❑
Having proved (7) in two special cases. fin A = V, and rot A = 21, we bun to a considerably more genet al result The arguments we have given so lar used the fact, that A had a suitable rotational symmetry, of onto 4 in the M i st case. and order 6 in the second In fact, the weaker assumption of onto 2 rotational symmetry is enough, although
138 Uniqueness of the infinile open (larder and critical probabilities one has to work a little harder to obtain (7) in this case Also, there is a natural generalization of (T) to certain settings in which bonds of different 'types' may be open with different probabilities. In this context, it is convenient to wad; with a weighted graph (A, p), i.e , a graph A together with an assignment of a weight p c E (0,1] to every bond e of A Fin each weighted graph there is a corresponding independent bond percolation model NI ARA, p), in which the bonds of A are open independently, and each bond e is open with probability pc To state a formal result, by a planar lattice we mean a connected, locally finite plane graph A (i e a planar graph with a given drawing' in the plane), with V(A) a discrete subset of 1R 2 , such that therm are translations 21,, and of IR 2 through two independent vectors tt i and rto each of which acts on A as a graph isomorphism In particular. all the Archimedean lattices are planar lattices Recall that two sites and a', or two bonds c and e', are equivalent ill a graph A if there is an automorphism y of A mapping r' to c to c' Note that any lattice is a finitetype graph, in the sense that there ale finitely many equivalence classes of sites and of bonds under this 'elation To allow for models in which edges have different probabilities of being open.. we define a weighted planar lattice (A, p) as above: A is a planar lattice, and there are two translations T„, and T. as above acting as antonnophisnis of (A,p) as a weighted graph, i e preserving the edge weights Perhaps the simplest nontrivial example is the square lattice, with = p for every horizontal bond and p r = q lot ever y ye t Heal bond, where 0 < p, q < 1 kesten [1982] showed that in this case, percolation occurs it and only it p+ q > 1: see Themern 15 below Another simple example is shown in Figure 10. Note that in a weighted planar lattice, there can only be finitely many distinct edge weights We sus that a graph A drawn in the plane is centrally symmetric, or simple symmeta lc, if the map a tY from P. 2 to itself acts on A as a graph isomorphism For a weighted graph, this map should also preserve the weights For example, the (weighted) planar lattice shown in in Figure 10 is str ut/nettle if one takes the origin to be the centre of an appropriate face If A is a planar lattice then, taking the vertices of the planar dual A* to be the centroids of the faces of A, sav, one can draw A' as a planar lattice, as in Figure 10 We assume 1:111°110/out that the bonds of both A and A* are drawn with piecewise linear craves in the plane If A is symmetric, then we may draw A* so that it is also symmetric The dual of a weighted planar lattice (A, p) is the weighted planar lattice (A*,q) in which the dual e' of a bond e
5.4 Bond percolation on a lattice and its dual
139
Figure 10 The planar lattice A (solid lines and filled circles) obtained by adding diagonals to every fourth face of Z 2 If the horizontal. vertical and diagonal bonds are assigned weights p, q, and r respectively, then A becomes a weighted planar lattice. The dual A' of A is drawn with hollow circles, at the centroids of the faces of A, and dashed lines
has weight qc . = 1 — As shown by Bollobtis and Riordan (2006d1, percolation cannot occur simultaneously on a symmetr is planar weighted lattice and on its dual Theorem 12. Let (A. p) be a squattete ic weighted planar lattice. with 0 < p„ < 1 fat every bond c Then either 0(A: p) = 0 al OW: where (A", q) is the dual weighted lattice Proof Suppose lin a court adiction that 0(A; p) > 0 and 0(„V;q) > 0. In the proof of Theorem 4 it was not relevant that all bonds were open with equal probability. Thus. writing Al and AP for the independent bond percolation models associated to (A, p) and to (A*, q), we see that in each of Al and AI* there is a unique infinite open cluster with probability 1 As usual, we realize the bond percolation models 11I and Al* simultaneously on the same probability space, by taking the dual e* of a bond e E A to be open if turd only if e is closed Throughout the proof we write I? for the probability measure on this probability space The basic idea of the proof is as follows: as before, any large square S is very likely to meet the unique infinite open clusters in A and in A". If we had fourfold symmetry then, using the 'nthroot trick', we could deduce that for each side of S. with high probability there are infinite open paths in A and ili A* leaving S from that side With only central symmetry, all we can conclude immediately is that there is some pair
110 Uniqucee s of the nrfiurlu open cluster and cr rhea/ probabililics of opposite sides of 5 how which infinite open pat hs in A leave S with high pr obalitlik■ The key idea is to move the 'corners' of S while keeping S the Sante Mote precisely. iliStetuf of a square, We take S to he a circle whose 24,/, and cot/sides infinite boundary is divided into four arcs A l , open paths leaving S hunt each A; If we move the dividing point; between two arcs, then pa t hs leaving one become more likely, and paths leaving tin other less Mi ch If we move the dividing point grachially, then the ptobabilities will change in a roughly continuous umnner, so at some point they will be roughly equal 13v moving two opposite division points while preserving symmetry. we can find a symmetr ic decomposition of the boundar y of 5' into four ales so that open paths of A leaving the tom arcs a t e roughly equall y likely Now, using the lout throot for even/ ate A 1 it is sett likel y that tittle is an infinite open path in A leaving S front this ate We cannot say that the same applies to AL as we have chosen the ales for A and not for A' We observe. however, that among out lour arcs there is sonic pair of opposite at cs of S horn which infinite open paths of A' leave with high probability Indeed. this Follows from symmetr y and I he squawroot hick This gives us infinite pat hs in A. AL. A and A' leaving the aces of 5' in miler and, as before, we can deduce a contradiction lit showing that the two paths in A' may be joined within .5. giving two infinite open clusters in A. We shall now make this aigument precise. Let .5 = S r be the dicky centred at t he or igin with nadirs r Let E(S) bet he event that some site of A inside S is in an initiate open clustet iu A, and let P(S) he t he men( that some site of A' inside .5' lies in an infinite open cluster in A" \\ T ilting D, for the disc bounded In 5, , we have IR', so the union U , E(S, ) is simply the event that there is U, D, infinite open cluster somewhere in A. and we have lin t ,— NE(5,.)) = S(A. p) he ;r positive constant that we and similar Iv for E' (5', ) Let shall specify later Choosing r large enough, we have NE(S,)) l —s 00d P(E' (S,))? I—
(9)
For simplicit y, we shall assume throughout this ptoof that no site of A ot A' lies on 5, and that tic bond of A ot A' is tangent to ,S, (Mote pletiselv, recalling that bonds of A and A' me drawn as sequences of line segments, we asSinne that none of these segments is tangent to S, ) This assumption is satisfied ha all but a countable set of values of r. Lin the test of the wool we fix such an r large enough that (9) holds, and mite tot S,
5 4 130nd petcolaijon on 0 lattice and its dual
141
Let e 1 1 < i < 4, he four distinct points on the boundfu y of S. 'nunbered in anticlockwise order We write c for the quad/ uple (c,,/!,,c3 We shall always choose these points so that no c; is on a bond of A of. A* We write .4 1 = A t (c) lur the boundary 01 r of S front cr to ciir t taking e tr r If MC is a bond of A with o inside S and to outside. then ate leaves S front the arc A; iL travelling along the (piecewise linear) bond ca! from V to 0', the last point of S lies on the ate .4 1 Let E 1 = E 1 (c) be the event that there is an infinite open path in A leaping 51 from the arc outside S with co inside S and i.e an open path P = roo t co. for all j > such that non leaves S from the arc .4 1 ; see Figure 11
Figure I I Possible open paths Pt and Pa witnessing the events E t = (c) and Ea = E:dc) Usuallt, the bonds toe straight line segments, as in Pi but they need not be
The precise details of the definition are not that important: the soft' arguments we shall present go through with many minor variants For example, we could consider the last time the whole path leaves S. even if this is MI a hood soul with vo S Fot 'nice' drawings of 'nice' lattices, a bond typically crosses S at most once, so the condition is essentially that vo vf crosses Ai Set fdc) = INEi(c)) and odc)
— f dc) = P(Ei(c)"),
y find define f and 91 similarly, using the dual lattice As aninfinite
142 Uniqueness of' the infinite open dusted' and critical probabilities open path starting inside S must leave S somewhere, U 1 E1 (5') is exactly the event E(S) The events Ea(S) are increasing, so then complements .E1 (S)" are decreasing Thus. by Harris's Lemma (Lennna 3 of Chap. ter 2), for any c we have P(E(8)1
P (Ei (c) c )=
E;(c)c) i=i
H
(c)
er_r
Front (9) it follows that gi(e)
1.17( c )
<
64
EA
i=1
The key observation is that, as we move one point, (», say,the probabilities fi (c) changc, in a 'continuous' mannet . For a precise statement, it is mote convenient to work with m(c) The only properties of c that the event Ei (c) depends on are which bonds of A leave 5' from which arcs A i .. Tints, as we move C. the probabilities Mc) and m(c) can only change when moves across a bond of A Of course, m(c) does jump at these points Our claim is that there is a constant C = C(A,p) such that, at any such . jump, no m(c) increases or decreases by mole than a factor of C Let c and c' = (e t , c!,.c3 , n i ) be such that exactly one bond c leaves 5' front the ate c,cf, Without loss of generality we may suppose that o)c1,,, ea lie in this order mound S; see Figtue 12 Thus, defining arcs .4 1 using the division points c, the bond e leaves S across the arc A I = eme2 while, defining arcs using the points c', the bond c leaves across A{:, All other bonds leaving S do so across corresponding arcs Thus, for i = 3, 41 the events E1 (c) and Ei (e) coincide, so R e ) = fe(ci Let E„ be the event that tire bond e is open The event E, (c) is defined in terms of open paths leaving 5' across the arc ..4; If e is closed, then Ito open path leaves S along the bond e, so which arc e crosses is in (Amami: Thus, the symineti ic difference of E i (c) and Ei (c') is contained in the event E, In other words. Ei (c) c fl
= E;( c' )` n
5
4 Bond percolation oma a lattice and its dual
es
CI
143
Figure 12. The effect of moving co slightl y to a new point c!): various bonds of A ate shown as dashed lines. We choose ct; so that a unique bond c leaves S between c 2 and gt The ales .4; and AC are determined by A I = c i co. itt; = A2 = coca and Xi= etc:,
holds for each i Lemma,
Now
E) (e)" and .6,1: are decreasing events, so, by Elan s
P(Ei(c)c n Etc,)
P(E)(c)c)r(E))
Tints me) P(E) (c')`) P(Ei (ct n = iP(Ei (c)" n
P(Ei(c)c)r(Ej
cfn(c),
where c = c(A,p) = int IP(Ect ) = int — Pc) > 0, as A has finite type and each Pc < 1. Similarly, gi (c) > cfn(c'), establishing the claim Set C = 11c > 1 Let us fix e l and e5 as opposite points of S Consider moving c2 from very close to c i to very close to c3 . At the star t of this process, no bonds cross A t , so El (c) cannot hold, and (Li (c) 1 > go(c) Similarly, at the end, [Mc) = 1 > g i (c). Each time c2 crosses a bond, g i (c) decreases by at most a factor C. and 02 (c) increases by at most a factor C It follows that we may choose c,) so that 1/C G g i (c)/g2 (c) C Let c.1 be the opposite point to c2 Then by central symmetry we have gi (c) = gi +2 (c) and g;. (c), 94 0 (c).
(12)
111 Unicl acnctsof Un
open cluslcl and el Meal prababililks
Thus.
arh(c) = gi(C) 2 g2(e
gi(C)4 /C2
Using (10). it follows that 9 1 (c) < C' 2E. and hence that gi(e) < C:3/26
13)
lot cow y i Front (11). there is some j with g(c) < 5 As (13) holds for even/ we Min' assume without loss of geneialitv that ) = L Thus, using (12) again. q:1(c) = f/i(c) 5 s and
Th(c)
=512 (c) < 0 /2 E
(
El)
It is now easy to complete the pool of Theo/ern 12, although, to avoid the need to consider exactly how the bonds of A and A" leave S, especially Ilea/ the division points we shall introduce one more technicality Let d be a constant (much) laigei than the length of airy bowl in A or in A' Let F1 = li (c) he the event that there is an infinite open path P in A leaving S across the ale such that no point of P lies within distance d of any ci This event is illustrated in Figure 1:3 Let D(c) he the event that all bonds passing within distance d of any c; are closed Thew is a IllaXi1/1111.11 munhet of bonds of A that any disc of radius d can meet:, so there is a constant c i (A, p) > 0 such that F(D(c)) > for any c Clear l y . if D(c) holds. then Ei (c) holds if and only if Fi(c) holds Using Hartis s Lemma as above, it follows that h i (c) = N./2) (cl')
Co i (c),
(15)
where C I = 1/c 1 Replacing C I by 1/ minfc l (A. p), c t (A t . q)). then both (15) and the cot lesponding equation 11(c)= P(Fr(c) e ) 5 C f/7(c)
(16)
hold fin any c y Let E = 1/(1.0C 31'CI ), noting that this quantit quantity depends on A and p only, not 011 I 01 c From (14) and (15) we have 11
while fron t (
2 (0 (c) <
(7 3/2 E
<
1/10.
and (16) we haw h;(c)
IK:1 (c)
< C 1 5 < 1/10
rid peicolaiiorilattice awl its dual
115
Figure 13. Open pat hs P, and P i witnessing the events F2(c) and F t (e) The open dual paths P( and PT witness i7(c) and ET (c) No site or bond of any or these. paths wets the shaded clacks It follows that Pi, for example, leaves the larger circle .5" through the arc col responding to As
Hence, with with probability at least I —
4/10 > ft.t h e
event
F = Fi (c) n F2 (e) fl Ps (c) fl P1(c)
holds, i.e., them are infinite open paths R and P i in A leaving 5' front the arcs and .4 4 , and infinite () mut paths F,* and P.‘; in A* leaving S from the arcs .4 1 and .4 3 , with no P1 or Rt passing with distance d of the endpoint of any arc 4 This event is illustrated in Figure 13. When F holds, there ate subpaths P.; and P; of P, and P 1 leaving the larger circle S' with radius] +d from the arcs corresponding to A, and .4 4 We may connect: .P1* and P3k by changing the states of all churl bonds e" that meet 5' to open The corresponding bonds c lie entirely within 5'. so after this change the paths p.!, and P.; are still open But then we have, with positive probability, two infinite open paths separated by a doubly infinite open dual path This implies that there are two infinite open clusters in A, contradicting Theorem As an immediate cotollat y of irheormn 12 we obtain the desired relationship between the critical probabilities km bond percolation on a planar lattice A and on its dual, assuming only central svutineti
146 Uniqueness of the thfinitc open clustc; and critical probabilities Them em 13. Let A he a symmetric piano] lattice, and let A" be its Amu dual Then p c1.)(A)+ p cll(A") = 1 Proof As before, the inequality p lc,)(A) + plAA*) < 1 follows easily horn Menshikov's Theorem, by considering a large region in the plane which must be crossed one way by an open path in A, or the other way by an open path in A* In the other direction, pir !(A) +111. (A*) > 1 is immediate from Theorem 12: if this inequality does not hold, then there (Al But then 0(A; p) and is a p E (0, 1) with p > p ck.'(A) and 1  p > O(A'; 1  p) are strictly positive, contradicting Theorem 12 ❑ Theorem 13 includes the HarrisKesten Theorem, Theorem 13 of Chapter 3, and Theorem 11 as special cases. It also applies to many other lattices, for example, all the Archimedean lattices shown in Figure 18 Turning to site percolation, Kesten [1982] pointed out that the site percolation models on certain pairs of graphs are related in a way that is analogous to the connection between bond percolation on a planar graph and its dual; he called such graphs matching pairs, and noted that any planar lattice matches some graph To see this, let. A be a planar lattice, and let A x be the graph on the same vertex set obtained from A by adding all diagonals to all faces. For example, if A = A 0 , then A'' = Ao The wool of Lemma 9 extends immediately to show that, for a suitably chosen 'rectangle' in A, whatever the states of the sites of 11 (A) = 1,7 (A" ), either there is an open Acrossing from the left to the t ight. or a closed A x crossing front the top to the bottom: to obtain the picture corresponding to Figure 8, replace each site v of A with degree d by a 2dgon that is black if v is open mid white if v is closed, and each fsided face of A by a white fgon If A is symmetric, then trivial modifications of the proof of Them ern 12 show that 01,(A) and 0 1 _ 0 (A X ) cannot both be strictly positive, while p,5,(A) + p s,„(A ) < 1 is again immediate from 'A lenshiltov's Theorem, giving the following analogue of Theorem 13. Let A be a symmetric planar lattice. and let A' he the graph obtained from A by adding all diagonals to all faces of A. Then
Theorem 14.
K.(A) + p,s,(A x ) = 1
❑
As noted above, = A E , so Theorem 14 implies Theorem 10 Since every face of the tr hingular lattice T is a triangle, T X = T. so Theorem 1.1 implies Them ern 8 as well
5.4 Bond percolation
On
a lattice and ids dual
147
We conclude this section with an application of Theorem 12 to a weighted graph. Let (2 2 , p,., pr ) be the graph 22 , in which each hot izontal bond has weight p, and each vet deal bond weight pm Kesten [1982, R 82] showed that the 'cr itical line' for this model is given by px ±pg = 1. Theorem 15. Let 0(Th„ p„) denote llw probability that the origin is in an infinite open cluster in the independent bond percolation on (2 2 ,p,, p„) Fm 0 < p,,, pr < 1, we have 0(p„,,py )> 0 if and only if >1
Proof The planar dual of t he weighted graph A = (2 2 , p„, p„) is A' = (22 + (1/2,1/2), 16, t p„), the usual dual of 2 2 with weight 1 py on each horizontal bond, and weight 1 on each vertical bond Rotating and translating, A' is isomorphic to (Z 2 ,1  p,, 1  pil). Suppose first that O(p,,,p„) = 0 for sonic p„, p,, with p„, > 1, and fix 0 < < p„. and 0 < jig < pr with 11, > 1. By the weighted version of Menshiko y 's Theorem, the radius of the open cluster containing a given site of A' = (22,[4,p'9) decays exponentially. As 1  p'y < Tit and I  < p'„, the same is tr ne in the dual, (22 + (1/2,1/2), 1  1 1/J..) But then the probability that a large square has either an open horizontal crossing or an open thud vertical crossing tends to zero, contradicting Lemma 1 of Chapter :3 We have shown that the condition > 1 is sufficient for 0(p„,,p„) to be lionzero To show that it is necessary, it suffices to show that p, 1  p) = 0 for every 0 < p < 1 Since (22 , p. 1  p) is symmetric and selfdual, this follows from Theorem 12. ❑
As it happens, one does not need Them ern 12 to prove Theorem 13; the proof of the HarrisKesten Theorem given in this chapter adapts immediately Indeed, suppose that O(p, 1  p) > 0 Considering a large square S, it follows from the 'four throot' trick that there is some side of S from which an infinite open path leaves with high probability Of course, the same holds for the opposite side The dual weighted lattice is isomorphic to the original lattice rotated through 90 degrees, so infinite open dual paths leave the remaining two sides of 5' with high probability, and one can complete the proof as before. In the next section we shall apply Theo/ em 12 to prove a mot e difficult result, that a cer talc analogue of Theorem 15 holds for the triangular lattice
118 Uniqueness of the infinite! open (A ide, and e l itical prohabiltlics 5.5 The stardelta transformation Sykes and Essam 119631 noticed a second connection between bond petcolation on the hexagonal and ttiangulat lattices H and T. other than that given by duality This connection involves the starthavilln formation 01 stardelta transformation, a basic transformation in the theory O f electrical networks To describe this, let G I and Gi be the two graphs shown in Figure lt Suppose that the bonds of G I ate open eit
y
Figure Id A triangle, G I and a star, and
02,
with the same :at lac:lune ' sites
independently with probability p i , and those of GO with p t obabilit■ ya. In either glaph, there rue five possibilities fot which sites among 1,r, y, :1 me connected to each other by open paths: all thtee may be connected, none 111M , he connected and some pair may be connected to each other but not to the third In other wo t ds, the par tition of fr. y, cl induced {{,r}, f lb ca one In tety subgiaph 01 0 1 Of of 02 is {{:r. {c}} These cases have the of the three pat titions isomorphic to {{,r, probabilities shown below: pairs connected probability in C I ;ill none
+ 3/4(1 — pi) (l — / 03 Pr ( 1 — Pt )2
probability in at
— p2/ 3 ± 3/12( 1 — P2 p4(1 — Ai)
Serendipitously, there is a solution to the duce equations suggested by the table above, i e., to Pi
+ 3/4(1 — p i ) = ( 1 —
/ 3 =
Pi — P0 2 =
P2,
— p2Fi IIp2 (1
p ) 2 . and
(17)
P_( 1— P2)
i i Substituting p = 1—p 1Indeed, the last equation is satisfied whenever into either of the first two equations gives t he stare equation. iii
—$pi=1=(l.
5 5 The stardelta tionsjhrination
119
This e quation has a unique solution in (0.1 ), nameh pp = 2 sin(7/18) = 0 3172 Let a, i 1,2, be the random (open) subigraph of 65 obtained by selecting each bond independently with probability pi . whew p i = pc, and pc = 1 – pa As all Hum equations in (17) me satisfied, the random graphs 0 1 and a, are equivalent with respect to the sites 11, and z: We may couple 0 1 and a so that exactly the same pairs of sites front { Ti m ate connected in 0 as in 0 In the context of independent bond percolation, each bond of a (usually infinite) graph is open independently with a certain probability, which we may think of as a weight The obser ations above mean that it a (finite ca infinite) weighted graph A has G as a subgraph, with bond weights pu, then We may replace G I by 6'0, with weights 1–pp. Ignoring the Internal' site of this operation does not change the distribution of open clusters This is a simple example of the 'substitution method' that we shall return to in the next chapter Using the startriangle transformation, it is easy to deduce bow TheOFC111 11 that p(1'(T) = pp This result was derived by Sykes and Essan [1963; 1961] without rigorous moof. \Vicuna' [1981] gave the first rigorous moot, based on the startriangle transformation and a RussoSeyniourAVelsh type theorem
Theorem 16. Let T be the triangular lattice in the plane.and FI the hexagonal oi honeycomb lattice Then p(1 .'(T)= 2 sin(–/18) and
p{I VI) = 1 – 2sin(rr/18) Proof As before. it follows nom Nlenshikov's Themem that the two critical probabilities associated to each lattice are equal, so it is legitimate to write pr:' for their common value. Let H' he the graph obtained by replacing every second triangle T by a star with the same attachment sites; see Figure 15 Then H' is isomorphic to /1; we shall keep the notation separate to indicate the different relationships to 71, reserving H lot the planar dual of T. Infonnidly, bond percolation on I with parameter po is equivalent to bond percolation on H' with parameter 1 – po, so both ate supercritical. both me
CSN
150 Unigaeness of Ike infinite open. cluslet and critical probabilities
WAWA WA res WA' a A SS WA TA Tata Figure 15 The triangular lattice 7, and the graph H' 01) each downward pointing face of 7' by a star. The sites o (solid and hollow); its bonds ate the dashed lines.
red by replacing are the circles
01 both are critical By Theorem 11, it Follows that both are critical, giving the result More formally, by a (10M(1411 in or in H', we shall mean one of the triangles to which we applied the startriangle transformation, or the resulting star in .11 ' Consider the probability measures 111 ,1 mi in which bonds of T me open independently with probability pa, and nu, in which bonds of H' are open independentl y with probability 1 — Po. We have shown above that the restrictions of these measures to a single domain D may be coupled so that the same attachment sites of D are joined by open paths within D in the two measures We may extend the coupling to all domains simultaneously by independence. Any path in T . or, in H' between two sites of T may be split into a sequence of paths P1 within domains D i , with the ends of Pi being attachment sites of Di. It follows that, under our coupling, two sites y of T are joined by an open path in T if and only if they are joined by an open path in H'. Recalling that 0 is a site of T, let Co be the open cluster of T containing 0, and let C') be the open cluster of H' containing 0 Under our coupling, we have (2() n V(T) = Co. As Cf, is a connected subgraph of H' and sites of V(H') \ V(T) me joined only to sites in V(T), which have degree three, we have 'Cid < 41C1, n 1 ,1 (T)I Thus
subevitic
C 10) 1
I
411C01
holds always in the given coupling, so 111. 0co
l
n) POC(d
POCH ^
151
5.5 The situdelta leeinsfortnatimi
for every a Letting n — we see that 0(H';1  po) = 0(T; po) In other words, as H' is isomor phic to the hexagonal lattice H. we have 0(11; 1 po) = 0(T; po) In proving Theorem 11, we showed that, for any p, at most one of 0(11; 1  p) and 0(Thp) can be strictly positive Thus, = 0,
0 ( 11 ; 1 — Po) = O CT ;
which gives p,i (T)> po and pcl.'(1.)> 1  po Since p (1 )(T)+ plc)(11) = 1 ❑ by Theorem 11, it follows that; Al' (T) = po and 144.11) = 1  po. Let us summarize what the results above, the Harr isKesten Theo/ern. Theorem 8 and Theorem 16, tell us about the critical probabilities associated to the three regular planar lattices. Theorem 17. Poi the aware lattice V. art have mi ./ (E 2) = 1 /2. fin the trianyalai lattice T. K(T)
= 1/2 wad 1 . (T) = 9 n(d/18),
and lot Mc he:rayon& of honeycomb lattice TT ple'(H) = 1  2 sine,T/18)
In a sense, the summary above is a little misleading: for these three lattices, four of the six critical probabilities are known exactly. but there are very few other natural lattices km which even one critical prohabilit\ is known exactly Tire observation of Sykes and Essar [1903 .1 concerning the stardelta transformation is a little more general: let G I be a triangle in which the bonds have probabilities p„, ph and pc, of being open, and G, a star in which the corresponding (i.e., opposite) bonds have probabilities r,,, and ra of being open Then G i and Go are equivalent if and only if 1'1)0
 r n
=
k
(1.8)
lot fi. j.1.1 = {a. 0, el Pa
pap,: + (1  P)PbPc ±
pa ( 1 POP
p„ph(i — )
/ a/ /
and (1  p)(1 pb )(1 pc ) = (1  /„)(1 10(1 le)* l„(1ts)(1 t„)+(11„),/,(1/J ±(1 t„)(1
152 Unignenciis of the infinite open elimle, and critical probabilities The equations (18) ate satisfied by taking then froth the remaining equations reduce to
= 1 —
p„pem — pa — ye — pc + I = 0
for each i and (19)
This more general ,taitriangle transformation was used by Sykes and Essarn 119041 to study percolation on a ti iangular lattice in which the states of the bonds ate independent., but the probability that a bond is Open depends on its orientation They derive (normigoirmsl y ) equation (19) lot the critical surface' in this threeparameter model Using Theorem 13 in place of [Theorem 11, the proof Of Theorem In given above adapts immediately to this weighted model, to give the Following result = fir . p„. y z ) be the weighted le iangulal lattice in which bonds in the three directions have weights Th. IL, and fi, sprelively when.: 0 < pr . p„. fi, < I Let 0(),„, Th) be the mailability that the oe igin is in an infinite open citadel in the independent bond maculation model corresponding /o A Then 0(p,. y y . p„)> 0 if and otilg
Theorem 18. Let
+ P y P P ;PO) z > 1
❑
This result g as Lamed by Grim/nett [1999[.. using ideas of Kristen [1982; 1988] In the light of VIenslnkm i s Thement, the hind pint is to show that percolation does not ()Celli when ys p„ T p, — pr y„p, < Kristen [1982] deduced this result Flout a theorem that he stated without proof A N'elSiOn of this liniment that is in litany was !note general was later proved by Gandolfi, Keane and Russo L19881, but then result assumes synunetiv untie/ reflections in the coordinate axes, which this model does not have. A stardelta tt ansfot mat ion 'elated to that discussed above is important in the them, of electrical networks. where it has a much longer p iston The operation on the graph is the same. but the weights (resistances) t i anslo t aiireleutiv. to satisfy the different notion of equivalence (that the ' espouse i . net cadent at each attachment vertex. to each input. i e. set of potentials at the attachment vertices. is the same) 6n electrical networks. it hums out that even sla t is etpthalent to kt triangle. and vice versa: see Bollobris [1998. pp .13 H] Rental kablv. Al t man 119811 w as able to use the star—triangle tianslotmat ion with unequal edge weights to obtain t he exact critical probability
5 5 111P, si tudella Emusformation
153
Cot a certain lattice, where each bond is open with the same probability Let ,5 1e be the square lattice with one diagonal added to every second lace, shown (rotated) on the left of Figure 16, together with its dual D
Figure lb 1 he lattice S obtained bow the swami Ire t ice by ridding a diagonal to ever} other face, shown on the left (solid circles and lines) toget hen wit its dual D (dashed lines and hollow cir cies) Fru clarity is draw n sepalateIN on the light Let S' he the lattice shown on the Hi in Figure IT below, obtained Inuit S by replacing each of the diagonal bonds b t a double bond Then SI. with tinily bond open with probabilit y p. is equivalent to 5' ' . with the otiginal bonds of S t open with probnbilitq p, and the new bonds open with probability =1 — (I — p)' /.1 (As usual. the states of different bonds are independent ) Appl y ing the star—triangle t t anstot motion. one obtains t he lattice I)' bu nted hour D by subdividing ce t lain howls: see Figure 17
Figure IT certain bonds of 5'' arc replaced by double bonds, and the tlianglestar t t ansibi mat ion is applied as shown. the resulting lattice is I) %vit h certain bonds subdivided Taking the undivided bonds to be open with probability q = I — p. and the divided bonds with probability q = q = I — p'. we see
151 Uniqueness of the infinite open cluster and critical probabilities that D' is equivalent to D, with every bond open with probability q. The conditions for equivalence in the startriangle transformation are satisfied provided (19) holds with p„ = = p, P c= p' This condition reduces to — p — 6p2 + 6p3 — ps (20) Using these transformations and arguing as for the hexagonal and triangular lattices above,\Vierman deduces that p!),(S ÷ ) = 0.101518 a loot of (20). An Architnedean Ionia is a tiling of the plane by regular polygons which all vertices are equivalent, i e the autornorphism group of the tiling acts transitively on the vertices. The square, triangular and hexagonal lattices are all Archimedean, the lattice 5' and its dual are not The complete set of Archimedean lattices is shown in Figure 18, The notation, which is that of Griinbaurn and Shep pard [19871, is selfexplanatory: it gives US the orders of the faces when we go round a vertex At this point we have essentially exhausted the list of Archimedean haft:CS Pi exact critical probability is known; there are two further examples that may be easily derived from those above. Let K be the kagona'r lattice. shown in Figure 18 Then K is the licegraph of the hone y comb II, so we have iu
gdIC), p(1,(H),
1 — 2 sin( 7/18)
Also, let K' be the (3, 12 2 ) or extended Kagoind lattice shown in Figure 18. Then IC + is the line graph of the lattice H2 obtained by subdividing each bond of II exactly once As noted in Chapter 1, the relation fki (11(i), p!(H) I12 is immediate: an open bond in the subdivided graph is only 'useful' if its partner bond is also open. Thus, as noted by Sliding and Ziff [19991, among others, fise (K), pcIVI2 ) = p A l H) 1/2
(1 — 2 sin(rT/18)) L/2
(21)
In the next chapter we shall review some of the upper and lower bounds for the critical probabilities of Archimedean lattices.
5.5 The stundelta hamsformahmt
•• •• •• •• •• •• •■■ ■ • ■ • SUOMI
•■■ ■ •■■ •
•■■ ■ •■■ • Sq nu ):
tome: (3,
(a,6, 12)
4•1■•■•• a • •^ • • • • a • • • • 0 11 0 (3' .1
•A 114 •A ............A .41 y
Triangulm: (36)
3,6)
155
extended Kagon
Hexagonal: (63)
(I. 82)
(3, 1, 6, 1)
1
ally A A w IS • • A•sffa Alwarar A SAY AT • A
A A 111a • FaVa •r a•Al • rA A Va• A ValrAwA VANS • ATAYAT VA
,6)
1)
(3 6)
Figure 18 The 11 Archimedeatt lattices, i e filings of the plane with regular convex polygons in which all vertices ate equivalent The notation for the unnamed lattices is that of Griinbainn and Shep pard (1987) 10 of the lattices me equivalent under 'oration and translation to then 'Milo' images The final lattice. (31 ,6), is not, and is shown in hot h forms
6 Estimating critical probabilities
In genet al. there is no hope of dele l mining the exact critical probabilities /OA) and pl,;(A) for a general graph A. even if A is a planar Iant ice Neverthel ess, there a l e mane ways of proving rigorous bounds on these critical probabilities In this chapter we shall describe sever al of these. soil ing with the substitution inclhod, a special case of which we saw in the previous chaPlel
6.1 The substitution method Tu describe the substitution method, we shall use the fenninolug i, of wcightcd graphs: all our graphs will haw a weight p, associated to each bond e. with 0 < < t We shall consider independent bond percolation Olt a weighted graph (A. p), whew each bond e of A is open with probability p, independent Iv of the ot het bonds. We often suppress t he weights in the notation. A weighted graph (G. p) with a specified set A of attachment sites generates a 'widow partition n of A: two sites in A are in the save class of El if they rue joined ln an open path in G As in the precious chapter. we SUN" that ( WO weighted graphs G I . C, with the saute set .1 of attachment sites ate equivalent if the associated I andoin partitions Il i II, have the same distribution. i e if the col lespondiug pet colat ion measures can be coupled so that IT = 11 2 always holds In general. exact equivalence is too much to hope for Let us saw that a partition 7ir. , of A is counsel than "Th. and mite 71 . ) > if anNr two sites of A that ale in the saute class in 7 1 ar e also in the same class in79 In other words. r, is (ionise' than r i if and in this context. a coarser partition is onl y if Ti is a 'ohne/new of 'better'. as it will correspond to more connections in the percolation model: this is the reason for our notation 'We say that a weighted
G.1 The substitution method
157
gr aph CI, is shott ipo than Cr, and WI > il rte DIM couple the
co l I esponding percolation measures so that II is alwa y s coarser than H I In this case Flo stochastically dominates FT,. Note that C I and Ga t ue equivalent it and onl ■ if each is stronger than the other Let A, and Aa he two infinite weighted graphs Suppose that A l may be decomposed into edgedisjoint domains Dr j . i = 1.2, . each having a specified set A j of attachment sites We assume that each D I is a suligmpli of Ac that (l i cit union is A t and that t. WO dOtliltillti 1118V meet onl y in sites that are at tachment sites of both Typically the graphs Du me all isomorphic Suppose that Aa has a decomposition into domains hew each Du has the same attachment sites as D i \\: e have seen an example of such a decomposition ;heady. ill connection with the sumtriangle transformation Indeed wi le/ i ing to Figure 15 of Chaplet 5. we IWIV take A l to be the tiitungular lattice. A, the hexagonal lattice fr. the domains D I j to he ever y second triangle in A l and each Du to he the cot tesponding star in A . , Aiguing as in the pool of Theorem 16 of Chaptet 5. since am, path bout one domain to ;mottle:: must pass Huough an attachment site. the onl y inopei It of D I ; that is teleiant tot percolation on A i is the induced partition 01 the set TI; Hence. if is equivalent to 171 j fin eve/ then petcolat ion occurs 011 A l if and (t h if it Deems on Am fin any fixed attachment site the percolation probabilities O(A j : 3 ) ;o ld 0(Aa: .r) ate exactly equal then 0(Aam ) > 0(A 1 : ./.): one Similarly. if Du > tot every call couple the percolation ;leashes on A, and A, so that any pair of at tachment sites that a t e joined in A i ale also joined ill A, Tins tact allows its to let iVe i elat ionsltips between the critical probabilities of two lattices: if the ethical mobahility of one is known then we can hound the critical p l ohabilit ■ of the other This technique is known as Hie substitution method. and is due to Vieunan 119901 At frost sight. it is not cleat hots one can tell whether a given weighted graph is stronger than another, but Hume is a simple algorithm An upset ill the partition lattice on a set A is simpl y a set U of partitions of E U and 71 > 71. then ira E Ll It is not too A such that whenever haul to show that > (7, if and onl y if. fo i even upset. we have E > P ( H, E U): in fact.. this is an eas y consequence of Halls p 77j) hus, the 'Matching Theorem [1935] (see also Bollolnis (1995. condition CD > G' t is equivalent to a finite set of pol nomial inequalities on the weights of the bowls Let its illustrate this with a simple CNillipie l (it ) = = p!"(10 giving hounds On the critical probability p T the whew It is the Anytime lattice shown on the light of Figure 1
Eqimatilly dim( probabilities
158
Figtue 1 A twostep transformation 11'0111 the hexagonal lattice /I to the liagomet latt ice K: first subdivide the bonds of to obtain the lattice ifs shown on the left. Then apply the startriangle transformation to the original sites of /1 (middle figure) The result is the kagotn6 lattice (righthand figure). notation of atiinbaum and Shephatd [19871 for Ai chimedean lattices, is the (3,6,3,6) lattice. Ottavi [19791 noticed that the Kagoin6 lattice may be obtained horn the honeycomb or hexagonal lattice H by fist subdividing every bond, and then applying the statthangle transformation to the (non edgedisjoint) stay s centred at the original sites of H see Figure 1 (A mote inimitite sequence of stair ' equivalent ' graphs was shown in Figure 6 of Chaptet 1.) Let Ss denote the sla t with attachment sites {,r, y, :di in which each bond has height s, and Tr the triangle on {:v,:y,s} in tvhich each bond has weight t We have seen that 8, is equivalent to T, if and only if t po and s 1 — po, where po = 2 sin(x/18) We would like to know lot which pails (.s, t) we have > Th and lot which pans we have Ss <111. Recall that there ate 5 partitions of {x, y, z}: one itt which all three are connected (are in the same part), one in which none ate connected, and three in which exactly one pail is connected We shall 100 to these as the partitions of type 3, 0 and 1 tespectively, so the type of a partition is the number of connected pairs. Repeating the calculation in the last section of the previous chapter, the probability that Ss induces a given partition of type i is Rs), where I3(s) =
li(s) =
—
and fo( s ) = ( 1— 5) 3 + 3 '4 1 iw/2
The co tt espondmg ptobibihities tot T1 ate given h t Mt). with firr(t) =
±
3t2 (11).
yr
= 1(1 — 1) 2 and tto(t) = (I —
159
6 1 The substitution method
There ate 10 upsets in the par tition lattice on a Onceelement set: two are trivial: the empty upset, and the upset consisting of all partitions. Any other upset must contain the type3 partition, cannot contain the type0 partition, and may contain any subset of the three type1 partitions. Let us write for one of the nontrivial upsets containing ) partitions of type 1 In this symmetric setting, we have Ss > Tr if and o i ly if four inequalities hold: each uj must be at least as likely in the partition induced by Ss as in that induced b y Ti In the partition induced by S. we have (NUJ ) = h(s) +1 i(s), while in that induced by T, we have P(U1) = g(t) + !in (0, so Ss > Tt if and only if ni(s)1 lids)
g 3 (1)± Rh (0
holds for ) = Similtuly, Tr > told only if the reverse inequalities hold. 01 course, if (I) holds for ) = 0 and for . j= 3 then it also holds for j = 1 and j = 2, so there ate only two conditions to vet ify We know the critical probability tot H. the lefthand lattice in Figure 1 Indeed, writing p„ for p H or p 1 (which are equal fin any lattice by Menshikov's Theorem), from equation (21) of Chapter 5 we lime p(1 .)(/12 ) = pti !(H) 1 /2 = (1— 2 sin(11.8)) 7 11
=
so
say As shown in Figure 1. this weighted graph 11,, with bond weights s, has a partition into weighted sous S, Replacing each stat with a triangle T, we obtain the Kagontê lattice with bond weights!. It follows that, if 0(119:s) > U and T, > then 0(K:1)> U As o(it,;s) > 0 fin any s > so.. we thus ha‘ e mi,)(K) < int
: T, >
for sonic s > so
inl ft :
Ss
p(1.)(k) > slug/ : 8, 0 > Tel
Solving the simple polynomial inequalities (1) fort with s = so, method gives the bounds U 51822
this
< pcl ."(K) < 0 51128
These inequalities and the proof we have just given are due to Wierman [1990] In the same paper, lie obtains the stronger upper bound pl:(K) < 0.5:335 by considering a larger substitution  replacing the union of two adjacent: stars in HO by the union of the co/responding triangles in IC In this case there are lour attachment sites, so the partition
9
1(i9
Estimating t tlical 1)101)001,1u s
lattice is now complicated 13% using huge, and lilt get substitutions. hot ten and bet k g bounds m i t\ l i e ()With/v(1 1lowc) \ el. the calculations quickly become Wiwi/Oita' ii C111110(1 0111 ill a 11111\ C \la \ Using yxtions methods of simplifying the eill(ititat and it sttl/stitittion \\ it h six attachment sites. \Vic/in/an 120031)] shat wined these 'mitt/cis considrilabl Theorem 1, Let Ii ix the ha q Dm( o, (3.6.3.6) lattice Then 5299
<
❑
phK ) < 0 5291
Pet tuning to the stintitriangle Panstointation. the condition
>
is fin some PluP o s es. anueeessaliin strong. Suppose that we Wee a \\ eight ed aph A l with as a subs' aph (joined out \ at the attachment sites). mud we replace S, bt 1, to Obtain A9 Nle would like condition's
on s and t that allow us 10 deduce that 00/C01/1110/1 is mo l e likel y in Al than in A.1 Mote wriciseh we should like conditions that cosine that the went {0 — that a particular site (the might) is in an infinite open (lusty ' . is at least as likel y in A l as ill A . ) lVe write Ss :7 7 / it this holds Ion all /Wits of weighted graphs A.)) /elated in the win we have described. Oita\ i 119791 round the set of pai l s (s. ) lie which 5, 1– .Ej To present this t esult let us Ws( decide the states of the bowls in E the set of all bonds of A I outside 5. Note that E is also the set of bonds ()I A . ) outside T, Eon each attachment site p E u there 111/0 1/1 Wilt not be 1111 open path limn 0 to 1' in A I \ S s . and there nia \ 01 /100,' not he an infinite open (inst il ()I A i \ meeting ° diet "°° 111 111C ( " 1/18 {{) } and { } 11111% of m i n not hold \ Chen the states of the bonds in E. the conditional probability that — x depends owl \\ Melt of the events and — hold in A i \ 5. Indeed. tl — x in A t if awl only it thew ale sites el. E dial are connected within .5 – with 0 — and x in A i \ Hew. e l it.) is allowed Much of the time, this conditional wobabilit \ is 0 (it 1: it 0— v and u rot some e E z} then 0 — x even if all bonds in S„ me closed Similarl y if 0 f r fia all 1' e f.r C }. on r f x for all then 0 — x cannot hold. whatwei the states of the hoods in 5, This leaves only One() nont thin! cases: we must have an attachment site di sat, with — :0 and a/101w/ site tr. tin) with 11)101 the Hind site, we roan 1000 0 — x. 01 0 71^ c 7 x In the last case. — x in A t if and only it ,r and it me connected in .5, II 0 —c and x. then 0 — x A l if and 0111V if one of and : is connected to
inl
1 The substitution method
y Slunk)/Iv if d sc., then 0 — sic in A i if turd only if one of irf and c is connected to :J . in S, The relevant events in S s have respective (s) ± ( fa( ± 2 f (s) and .f:Ws) + 2 fi(s) Pwilmbililies lane shown that PA, ,X) is a weighted awlage of the quantities 1. 0, 1 3 (s) + h(s), and nr(s) + 2[ 1 (s), where the weights depend sc) is a weighted average of 0, on A t \ S. Fut Him more s P A2 (0 weights, determined by y3 (t) I in (1). and g3 (t)1 2g i (f) with the sonic \IT/ A i \S, Tints S s bolds mecisciv when the two inequalities
E(s) ± [ 1 (s) ^ 00= !EU) and / 3 (s) + 2f ( )
113(0
20 1 (1)
(2)
hold. i . when (I) holds tot = I and j = 2 This is a much tveaket condition than S s > Tt flu i trivetse relation Tr S. holds pun Wed the revel se inequalities to (2) hold so is the (*initial probabilits for [1,, and I = Ottayi showed that il s 0 52803. then E > so the sultglaph S s of run weighted gin/tit A may be teplaced and the probabilit y that a given site is in an infinite cluster ( 0 1 that a given pair of sites ale connected by an open path) will not decrease It might seem that the inequalit y p(1 9 fi) < 0 52893 follows easily: un101 I mutt els. this is not the case As the Wit III al at gement is so close to working let us ex tunine it in detail, to see where it fails One would hope that. if T .5.. then percolation is at least as likely in a graph Bunted In gluing together copies of 1st at tlwit illlath/nem sites. as in the col responding graph obtained from copies of 5, However. the relation 8 S s allows its to replace (me cop%of Tj lw a copy of S, in an 'outside wank w [licit is the same beton . and after the substitution. but it does not allow its to conti nu e and !vitiate a second copy After enlacing the first cop y. the outside gl aphs E l and Eg ate different: in Eg Nu , have ahead ■ replaced a cop y of S s by 7r, One might hope that. as IT, is 'bet lei' than E l them is no teal problem But what does 'better' mean? ft could mean that an y connection in the outside graph E i between an at tachment site of mu second substitution and ft or isc is also present in Es Until we have looked at the second substitution. we do not know which connections we will tric l inic., so we should impose the condition that the first substitution preset yes all such connections (while pet Imps adding new ones) ilhe condition Tr S. does not allow us to do this only to p i esti' NC a connection chosen in advance The arguments above illustrate the power of the notion of stochastic domination: if I) > S s then it is ye t y cans v to prove that we nun replace as main copies of S s by copiris of Tt as we like I In ke y to the application
Estimating critical probabilities
162
of the substitution method is to find suitable weighted graphs an with Gr > ay. \Viet/Ilan 12002] used the substitution method to obtain bounds on pit i(A) for other Archimedean lattices A, obtaining the following result. Theorem 2, Antony the Atchimedean lattices A, the extended I< againá, 0013i70/2CS p!?(A), with 07 (3.12 2 ). lattice 0.7385 < pci Vii+ ) < 0 7119 Although we have described the substitution method only lot bond percolation, essentially the same method can be used to studs site percolation. For example, \Vietnm [1095[ adapted Ids method to obtain an upper bound on pcs,(7.9) Theorem 3. The et itical probability pi,(22 ) for site pet ()lotion on the S01107V lattice satisfies pis.(22 ) < 0 679+192 Po" a list of other rigorous bounds on the ccritical probabilities for lattices. see Wiennan and Pm viainen [2003]
6,2 Comparison with dependent percolation Another method of bounding critical probabilities is implicit in the use of dependent percolation in Chapter 3. Let us Wrist] ate thus by giving an upper bound fill icii(Z 2 ) = M i (22 ) = 14(22 ); we shall mention a much more sophistica.ted version' of this idea later. As usual, we write = P_:. t,for the independent site percolation measure on 2 2 , where each site is open with probability p; we denote this model by M. Let be a parameter to be chosen later, and partition the yet tex set of Z2 into C by ( squares S,„. e E Z2 Thus, ha v = (a b), S„ =
= Ur, y) E 2,72 : at 5 < (a +1)1', Itt < y < (b +
The set of squares St has the structure of Z2 in a natural sense To make use of this, for each bond e = u e of Z2 , let R = S„ US,.. so let is a 2t by (or t by 20 rectangle. If e and are vertexdisjoint, then the rectangles R.„ and R./. are vertexdisjoint Thus, if we define a bond percolation model M on 22 in which the state of a bond e E E(22 ) is determined 1w the states of the sites in Rt . then the associated probability measure on 2 E(:2) will be 1independent. The idea is to define IC/ so that
6.2 Comparison with dependent percolation
163
an infinite path in Al guarantees an infinite path in tare original site percolation Al We have seen a way of doing this in Chapter 3, using 3 by 1 rectangles for clarity in the figures We use the same idea here Recall that 11(R) denotes the events that a given rectangle R is crossed horizontally by an open path, and 17 (R) the event that I? is crossed vertically by an open path. For a horizontal bond e = ((a, b). (a + 1, b)) of Z2 . let (AR() be the event Win b)) illustrated in Figure 2. For a vet tical bond
Figure 2 The figure on the left shows open paths guaranteWng that t he event
0(10 holds, where c = ((a. b), (a + l.b)) is a horizontal bond of 2: 2 ; the sites shown are open 'The corresponding event lot a vet deal bond c is shown schematically on the right.
(0, b + 1)) O r 22 , let 0(11%) = 1 .7 (R,.) n H(SR„ 0 ) in either case, let e be open in Al d and only if 0(RJ holds Since hor izontal and vertical crossings of the same square must meet, if there is an infinite open path in A/ then there is an infinite open cluster in Al Note that the probability Pp (G(Rt .)) is the same for all bonds c Let p i = p i (Z2 ) be the infimum of the set of p such that, in any 1independent bond percolation measure on 7G2 in which each bond is open with probability at least p, the origin is in an infinite open cluster with positive probability. As we saw in Chapter 3, it is very easy to show that p i < 1 Here, the value of p i is important; we shall use the result of Balistet Bollobris and \Valte i s [2005] that a t < (1 8639; see Lemma 15 of Chapter 3 Suppose that for some parameters s and p we can show that = ((a, 6),
lt(P)=Pp(Cl(R.,)) > Then lTr
is a 1independent measure on 72 2 in which each bond is open
Estimating .
161
'cal probabilities
so with positive Limitability there is au with some probabilit y p > and hence in II/ 'Thus p> p11(Z11) Al infinite open path in Suppose that p > p11(Z 2 ) is fixed Then, by 'Theorem 10 of Chapter 5, we have 1 — p < t ,11(Am), where A D is the square lattice with both diagonals added to every face Exponential decay of closed clustets A D follows by Menshiliovs Trhecneum so the probability that a 2( by t rectangle R is crossed the shott way by a closed A D path tends to zero Hence, by Lemma 9 of Chapter 5, the probability that R as ( is crossed the long YVilV by au open path in An tends to 1 It follows that i f (p)(• 1, so thew is sonic 1 Stith f hat li (p)> CI 8639 > Thus,
in plinciple. Hie method above gives arbittarilr good upper bounds: for each (', find the minimal rot an almost mini/nap value of p, of p such that heir) > 0 8639. Then each p, is an tippet bound on p11(2, 2 )„ and the sequence p, converges to pi1(Z 2 ) boo, above. ft is easy to check that, 16p6 q2 +8p7 q 1 p s , \\dune q = 1 — p, giving the bourn! ',f p ) = 6p 5 M 1 p11(Z) < 0 8798 With a computer, it is eas y to show that fa(p)
117/tq l ° ±1399p9 q9 1737p lugs 1027p 11 q 7 = 5166p12q°
+ 1527p 1 "q 1.'
2335/ J il t/ I  I  7571, 15 q3 153 /) 16 ,12 F 18p ri + pis
giving p1(21: 2 ) < 0. 815 An exact (ruination of 1,(p) gives /.)`,(II32) < 0 817 Linlin unat el ■ t he st r night il/Wald met hod of evaluating rpr mnd count inglia which of t he 2' 2 configut at ions in a 2l In 1 rectangle 11), the event 0(11, 1 holds, quickh becomes imm act lent. at mound I = 5 \\Ten the subst it Mimi method can be applied. it tends to give beton bounds for the same computational shirt Note, hotveier. Hurt the method described hew is much more robust Fin example. fi l eg1 1 1e/Iiii( IS in die graph Me riot a ploblem plodded we can show hat 12'1,(G(1, )") > i > pr ha every bond c one can perform different substitutions in diffetent pails of the gtaph the substitutions lune to fit together exact) to give the stt ucHne of a graph with known critical probability This will often not be possible Another very imputeant adiant age of the 1independent approach is that, if one is p l epa l ed to accept an ewer probabilit y of 1 in a million. say. or 1 in a billion. then good bounds can easil y he obtained b y this method Indeed. there is a yeti easy wa y to estimate the numerical configtuations in a 2( by value of E(p) very plecisek: generate rectangle I?, at random using the ['wasn't , and count the number .I/ of configurations 1hr which 0(1?,) holds. The random number Al has p) so if A is huge. a binomial dishHa t tion with parameters A . and
n',2 Commi t Cion with dependent percolation
165
then with lel \ high probability 11//A will he close to /r(p) The y IreV point is that one can boned the error probabilit y. ercu without knowing ir (Pr Given any > 0, one Call give a simple procedure for producing upper. bounds on /OE') ry ltich provably has probability' at most E 0[ giving an incorrect bound: foist (by trial and 0110 1 . or guesswork) decide on parameters I Will p for which )r(p) seems likely to he huge enough Then calculate numbers N and Ma such that TdA. > J1,(0 ) < whew Bi(N. 0 8639) Then generate A samples as ;drove. m i d if Al > 1110 o f t hese have the propert y Ci(g). assert pe r r.... 1 p as a bound IL in fact 0,(23 2 ) > p. then h(p) < 0 8639. and t he probability that the sampling procedure generates at least .3/0 successful trials is at most This method works ver y well in 0010 and the dependence ()I the I mining time on is ver y modest For eXa/111A0. using the parameters = 10 000. p = 0 591, N = 1000 and .110 = 935 with a moderate computational efhat we obtain the hound i t,".( 2 ) < 0 591 with all 19 lot plObabilit \ Of E < 1012 Similar Iv, 161 the matching lattice AE with vet I CX set out bonds between each pair of vertices at Euclidean distance I or v' 71. using the pat ;mulcts l = 20 000. p 408. N = 1000 and A' = 935 we obtain K( A 3) < 1(18 w" a c""lidellue of1 1g12 Together these bounds give /):1( ) E 10 592.0 5911 with extremely high confidence. A little 'note computational effort (1= 80.(100. N = 1000. Itltt = 915) gives the result K(Z 2 ) E [(1.592 0593] with 99 9999cA confidence Note that we cannot: sat that the probability that pr(r 2 ) lies in the inlet val (0.592,0 593] is at least 99 9999%: the critical probability is not a random < / mutt:itv. so this statement either holds cm it does not In the language of statistics. we have given a confidence interval for the lillk110111/ deftlillillititie quantity ir,'.(Z2 ) As described, the procedure produces a confidence interval whose upper limit is either the bound 0 593. sm. that we aim for. or infinit y, if Ill < 21.4) Narrow confidence intervals for mars or her critical probabilities have been obtained by .Rionlan and \Valters [2006]; Ica example, p;1(11) E [0 6965..0 6975] with 99 9999 1X confidence. where is the hexagonal lattice As usual in statistics. we must be a little careful in generating confidence intervals: Ito example it is not legitimate to per fornt various runs
166
Ethmaling critical probabilities
of the sampling procedure with various parameters, and only use one result In practice, this is not a problem for two reasons: finitly we can perform as many • durnuty' Inns as we like to get an idea of parameters that are very likely to work, and then one teal run with these parameters Also, it is easy to get a failure probability of 10 9 , say, for each run It is then legitimate to perform 1000 runs with different parameters, and take the best bounds obtained: as long as the probability that each run gives an incorrect bound is at most 10 9 , the final bounds obtained still give a. 99.9999% confidence inter val for the critical probability. There is another pitfall to bear in mind when implementing the probabilistic procedure described above: we have implicitly assumed that a source of random numbers is available.. In practice, for this kind of sinndation one usualli uses a pseudorandom number generator Not all the widel y used ones are sufficiently good lb/ this pm pose Indeed, the current standard generator I t anclourfl used with the programming language C is not For example, using this generator we obtained estimates for 1' 10 (0 731) of 0 8661 ±0.000 2 and 0 8631+0 0002, depending on the order in which the random states of the sites were assigned (The turc:er tainties given ire t wo standar d deviations ) Using the much better I Merserme Twister' generator MT19337 WI ititen by Matsumoto and Nishimura, we obtain li o(0 131) = 0/8630 ± 0 0002 Note that this (presumably) true value is smaller that 0/8639, /i hile one of t he values obtained using I andom() is larger. in generating t he confidence intervals given above, we used the lersenne Twister. Of course, one can rerun the same procedure tr itlr a different generator. or with I n rte' candour numbers obtained horn, for example.. quantum noise in a diode The methods described above can be easily adapted to give good upper and lower bounds on p4!(.(1) and pII(A) for any of the Archimedean lattices A. 1701 lower bounds. to prove that percolation does not occur at a particular value of p one can use 1independence as in the proof of exponential decay of the volume in Chapter 3. For another approach to lower bounds, let S„(0) be the set of sites at distance n 11010 the origin, and let N„ be the number of sites in 5„ (0) that may be reached from 0 by open paths using only sites within distance 11 of 0 Lemma 8 of Chapter 4 (or its equivalent, for bond percolation) states that if, for some n, we have 1/1/ ,(N„) < 1, then there is exponential decay of the radius of the open cluster Co, which certainly implies that p < pc . In am lattice, for any p < Menshikov's 'Theorem implies that: a p (N„) — 0 as n ^ ob , so arbitrarily good bounds may in
6.3 Oriented percolation on 22 167 pr inciple be obtained in this way As before, exact calculations of Ep(Ar„) are impractical except for vet y small but estimates with rigorous error bounds may be obtained for larger re This observation is applicable to any finitetype graph, including any lattice in any dimension For bond percolation on a planar lattice A with inversion symmetry, to find a lower bound on plc' (A) = 14.11 (A) = pi (A), one may find an upper bound on pl„)(A*) and use Them ern 13 of Chapter 5, which states that p ,l (A)tp1JJA*) = I. Similarly, for site percolation, we may find an upper bound on p4(AX) and apply Theorem 14 of Chapter 5 This approach seems to give better results in practice The reason is that it is easier to estimate the probability of an event by sampling than to estimate the expectation of a random variable that, might in principle take quite large values occasionally
6.3 Oriented percolation on Z2 The study of oriented percolation, and, in panic/dar t bond percolation on the or iented graph 23, is a major topic in its own right.; see Durrett [1984] for a stir vey of the early results in this area. iented percolation is in general much harder to work with than unoriented percolation, a fact that is reflected in the difficulty of obtaining good bounds on Z 2 ) Indeed, Durrett described the question' of finding a sequence of rigorous upper bounds on 1.21!1 ( Z 2 ) that decrease to the true value as an important open problem His bound, pi'( Z 2 ) < 0 84 , Was very fat from the tower bound of 0.6298 obtained by Mar [1982]. In contrast, for lower bounds, it is easy to produce hounds that do tend up to the true value Indeed, let N„ be the number of sites on the line x y = that may by reached from the origin. If Eih,(N„) < 1 for some n, then Lemma 8 of Chapter 4 shows that p < pfi ( Z 2 ) < Indeed, Hanumnsley [1957b] deduced that pl I (Z 2 ) > 0 5176 from the fact that Eb (No)  = 4p2 — 11 1 . Of course. by Nlenshikov's Theorem,Z ) = p!iFE 2 ) =1.4,'(72), say
There have been many Monte Car lo estimates of pN Z 2 ): such results are not our main focus, so we shall give only a few examples, rather than attempt a complete list: Kertósz and Vicsek [1980] gave the estimate pcb ( 2.,Y ) = 0 632 ± 0 004. Dinar and Bar ma [1981] reported p[ (G2) 0 6445 ± 0.0005, and Essam, Gultruann and De'Bell [1988] 0 644701 ± 0.000001, for example
(Z2) =
168
Eqinading et Theo! probabilities
Bali:der. Bollolgis and Stace y [1993: 1991] gave Indict tippet bounds
ou p :l ( Z 2 ) using the basic sir ategv of comparison with lindependent percolation. Init ill a much mow complicated WaV than that described iu the previous section. Their approach does give a sequence of rigorous upper bounds tending down to the true value As Hie alguntent is lather involved. NiAi shall give onl y an outline \Alien consideting oriented percolation on
OW aim is to decide
tot which p Ha' percolation probabilit y O(p) 00 (p) is m e nzero In doing so. we was of cows() restrict mu attention to that prat of 2)2 that Mak C011telVabh : LC l eached hoot the (nigh). namel y the positive
(radian (2 The basic idea is to use independent bond percolation on Q to define a leSealed Iindependent bond percolation on Q To (I() this, We choose parameters b and h. and at 'fume rhombi with II+ I sites along the bottomlett and toptight shies as in Figure 3 Mow precisely
Pigmy: 3. Rhombi with sites along the base (bottom left)) and the top (tipper light) II each t hondtns is replaced b y all oriented edge and I he top and Lot too t of each rhombus be a \et tux the tesulting or iented pupil is isomm to 2 In realize the rhombi ate lagged : each contains exact IN :I sites ham each of t la: 9 la% epf of 772 that it meets
f
for : .r. y > ± q = for bagel wlft iug = Ei each site r iu L 1 we choose a set 5,, of consccutke sites in L h(t+. 1) . so that the sets Se ate disjoint Also. fin each bond 7 E Q we choose a (toughl y rhombic) subgtaph of Q in such a wry that
R R
and P7
ate disjoint whenever 7 and f me bonds of Q. that do not sha l e a site \\ : e legnite that it e = UP then the bale of R T , defined in the nattu al 5„ , turd the top of 117 is 5, wr y( is exact
3 Oriented percolation On
V
169
Cit for the open outclunk] of the i e., As before . WI it e open (oriented) the set of sites that may be reached fron t the might ( ) < 1 such that, paths. It is easy to show that there is a pu = for any lindependent bond percolation measure F on  iu which each > bond is open with probability at least p H we have P(rof the ingument is vets similto to the moot of Lemma in Chapter 3 Indeed, if Co is finite, then there is a dual cycle S surrounding the origin.. such that every oriented bond star Ling inside S and ending outside is closed As shown in Chapter 3 there me, ve tt ctuclely, at most :te3212 dual cycles S of length 2( surrounding the or igin For each, there is a set E of exactl y f oriented bonds starting inside S and ending outside Two horizontal bonds in E cannot share a site, and nor can two vertical C E consisting of fl/121vertexdisjoint bonds in E. so there is a set bonds The fillprobability that all bonds in ate closed is at most (1 — ) 1/2 so the expected number. of cy cles with the required property is at most
which is less than I if p i = 997. sa t Note that the expression above is exactl y the same as that appearing i l l the mool of Lemma ll of Chaplet 3, not b y coincidence By counting more car Mont the cycles .5' that ma t actually arise as the boundary of Co. as in Balister , 11n1 Stacey [19991, one can obtain a bet ter bound on pi If we define an event C:( U T ) depending on the states of the bonds in fly so that an infinite or iented path of bonds 7 for which C(R 7 ) holds guar antees an infinite open oriented path in Q. and if. (or some p. we can show that P p (G(R T )) > pr for me t y(T then it Follows easily that TT, i ( ) < p Howevet , it is not eas y to see how to define G(R 7 ): piecing together paths in this context is much harder than in the unot iented case. The actual atgm/rent of Balist et. Bollobfis and Stacey is notch more subtle Choosing the regions fly so that they are isomorphic to one another, the y seek a Wont—trivial upset A C 'P(11/± II) with the Following proper ty: let i = 17 1 t, and consider the random set U of sites in S„ that may he leached nom the otigin b y open oriented paths. This set may be naturall y. identified with a subset of [bf I] Let V be the set or sites in S, that may be reached horn U br open paths in again this set may be identified with a subset of [b+ II The condition [conked is that.
Estimating el die& pmbabilitie5
1 70 for some p p
E (0,
l),
whenevet U E A, then 171,(1 .7
E A U) > p
(3)
There is a further restriction, that A be symmetric under the operation b + 1– I. so that the identifications described above may be made i consistently. Bollobas and Stacey [1993] show that, under these assump–'' lions, one can construct a rescaled oriented bond per colation on Lin which each bond is open with probability at least p, and which is effectively 1independent (Mole precisely, given the states of the bonds below sonic layer, the bonds in that layer ate 1indepertdent..) They show that, if p > (3 – 07) )/2 and p > 1 – (1 – p) 2 , then such a process dominates the original independent process Using Menshikov's Theorem, ). one can then deduce that p> In summar y, if one can find a suitable region R., au upset A, and a p > (3 – V17) )/2 such that (3) holds with p = 1 – (1 – p) 2 I'm all U E A, then p is ' in tippet bound fin p l!( Z 2 ) Note that this does not con espond to a direct construction of a 1independent percolation model as in the unotiented case Such a direcl construction could be achieved by defining 0(1?7 ) so that if G(Rn" ) holds and U E A then V E A; indeed, one could take this as the definition of 0(11 7 ) But then the condition that 0(1 7) ha t e probability at least p i amounts to Pp(U E
A
V E A) >
which is infinitelt stronger than (3) with p In order to apply the method above, it seems that, we have to make a large rininiter of choices: lust, we have to choose a suitable legion P. E4'.fP/t/n and a probability p Then, we must also choose one of the 2 possible upsets to test Fmtunately, there is an ;lige/Ulm' given by Balistet, Bollob 'is and Stacey [19931 that, for a given I? and p, tests whether there is am] upset A with the required property (and finds the maximal one if there is) Using this algorithm, together with the guluent s outlined above, they moved that p:( L 9) < 0 6863. They also showed that. as in the case of man iented percolation, the method gives arbitraril y good bounds with sufficient computational effort This is a much harder result than the (almost trivial) equivalent for fluor iented percolation described in the previous section Using a more sophisticated version of the argument, involving a more complicated 'eduction how the dependent percolation to independent
6 3 at iented percolation Oil Z2 171 percolation. Balistei Bollobas and Stacey 1199II obtained tare following hounds. Theorem 4. The critical piobabilitics for oriented bond and site percolation on the square lattice satisfy ( 72 %; ) pc1;'(27: 2 ) < 0 6735 and pj"(1
<0 7191
❑
As in the case of umaiented percolation, a (simpler) version of the op/molt; just described may be adapted to give bounds with 99.99%) confidence!, and fa/ stronger bounds can be obtained in this way with much less computational effort Indeed, Bollobtis and Stacey [19971 showed 1 , 7 < 0 617 with 99.999967% confidence. This bound is very that pc ( close to the value of appioximately 0.64415 suggested by simulations; Liggett (19951 used a beautiful and totally different approach to bound p,/,'( Z 2 ) and pr.(2), based on au idea of David Williams, who conject i ned that p(1 '( Z 2 ) < 2/3. To describe this, note that ()dented peicolit7/, corresponds to a Mar kov chain in a natural way Recall that. lion on the ith layer is the set L i = {(:r, y) E 1Z 2 ) : t y > 0, a' F y = t.} The be reached from the origin by open paths set TIt of sites in L i that and depends only on and on the states of the bonds between L t Obi oriented bond percolation). 01 the states of the sites in L i (for site percolation) More explicitly. let us code R t by t he rcoordinates of the points it contains, setting A i =
: (r t t — :r) may be reached from 0 by an open path}
Then Ao = Fat bond percolation, given ./14 , the probability that :I! E = 2p—p2 according to whether lA t ofx1, j is 0, = A,+1 is 0, 1 01 2. Furthermore, given A t , the events E A t+ , } are independent for different c;. For site percolation, the jiMarkov chain is the same, except that p i = p2 = p. Thus, it is natural to extend the definition of < 1 and 0 < pa < 1 the Maticov chain (A t ) to any pan/meters 0 < Note that this Markov chain has stationary transition probabilities: the distribution of 1 t t , given that = A is the same for any t. Of course, this Nlarkov chain underlies any analysis of ot hinted percolation; lot the approach described above, we did not need to define it explicitly Liggett j1995] !moved the following result. Theorem 5. If the parameters 1/2 <
< 1 and
mid ha satisfy the inequalities (1 — ) < p 2 <1,
(1)
Estimating critical probabilities
179 Then
the
chain
(A 1 ) defined above satiqics Pec't : A l = (0) > 0
conditions Em bond pcs ("dation. 'slagpm p and Ps = 2p — ti of I ()mew 5 MU satisfied fin p > 2/3 For site percolation. whew p, the relations (I) hold fin art's p > 3/1 Thus Liggett's pu = Ps mediatels implies the following bounds result in t Theorem 6. The (atheist probabilities lot minuted bond and site percolation on the square lattice satisfy
< 2/3 and p.( 2 ) G Given a finite set A C E. let denote the landont set obtained by minting the M iukov chain lot k steps stinting with the set A Thus, if A = {01, then .4" has the clistlibut ion (if A„ The basic idea of Ligget t's pool is as follows: we hope to define a Function on the finite subsets A of Z so that II(0) = I awl M I all A 0 lye have fl(A) < 1 and (5)
EgH( A I )) < H(A)
Of course whether such int I7 exists depends on the parameters p i and of the :\ farkov chain If such an 11 does exist, then from the Nla i km" opei tv awl (5) we have
po
E(B(.
E(H(..1„_1)) <
(AO) = 11({0))< I
lira civets i s This implies that the pelcolat ion incites); is supple' it ical: = 0) — 1 as n — x As 11(0) = I awl otherwise. we would Inge /1(A„)) — I. a contradiction fl is bounded. this would imph Liggett [1995} showed that. it the conditions (I) hold then an H with the I equited mopet ties can be found The weight fitnet ion 11 used by Liggett is Lather complicated To describe Liggett's argument. let T be a minion 's variable taking positive integer values. with E(T) < and let I( be the stationary ' encash' meastue on sequences q E 10 1 t associated to h Thus. the diSt / Haitian of I he landain secnwnce rt is itr y ltiant under tt mishit ion. and the gaps between successive Is a t e independent and have the distribution of IT Lhe weight 11(A) is defined as the icprobabilit y that li (y) = 0 for met v .1‘ E) A. Note that this etamint y satisfies 11(0) = 1 awl 11(A) < t Fm = 0. Of tom w het het the kev equation (5) is satisfied depends on the choice of
(1 (bleated pelfolation on Z
173
\\j i lting F(n) f or ENT > a). Liggett shows that (5) is satisfied in the special case that A is an inter val if and only if
p 2i 110 + I) =
—
.17(k)F(n
— k)
a1)2F(7I)
(6)
fin ever > 1 El he derivation of this condition is relativel y simple: roost of the work lies ahead. First. one must show that. if (I) holds. then the solution to (6) with F(1) = t makes sense. i e . cooesponds to F P(T > n) fin some T I iris amounts to showing that 11(a) is decreasing and that Y[' . n F(n ) L(T) < x Second. OM must show that (5) holds not only for inle t als. but also for atHutto sets .4 Even though the Mat kov chain operates on disjoint Mor t vals sepruatelv. some sense, due to the complex definition of If t his is by no means a simple consequence of t he interval case Indeed. t he tuguments of iggett [1995] for both steps are far horn east Let us rcuuulc that oriented percolation on 2 fray be thought of as a discietct hue version of the contact process on E Indeed. Liggett s proof described above is similar in outline to an argument of Holies and Liggett [1978] ghing an upper hound for the critical parameter of this contact process As noted 1)5 Liggett [1995] it is much harder to tutu the outline into a proof for oriented percolation than tot the contact process. due to the discrete nature of the process. At tire tint(' of WI iting. Liggett's Intim ( ' < 2/3 is the bestrigorous upper hound on p /l (L 2 ) Liggett notes. however. that Iris method cannot he pushed further: the solution C (a) to (6) above has the required properties if and only rf t he conditions (i) are satisfied. In conOast. the method of Batiste/. Bollobtis and Stacey [1993: 1990 gives arbitraril y good bounds with increasing computational effort bus at some point it will become feasible to obtain a better bound in this way . Indeed. this nay have aheadv happened: the amount of computing power available now is much larger than it was when the bound p[[( Z1 2 ) < 0.6735 Willi obtained in 1994 FM site percolation. the bound of Batiste/. Bollobris and Stace y [1994] given ill Ille01 CM already better than 3/4 As Liggett le/narks. Lincoln Chaves pointed out to hint that Theotem 5 gives bounds on the critical probabilities km percolation on oriented lattices ot h er than 72 twitted. let TI he the oriented hexagonal lattice shown in Figure 4 Taking ever v second column of sites as a laver L I as in the figure. a given site in tr.j. j runs be reached only hoo t one
Estimating critical probabilities
t
:3
igme 1 'The portion of the oriented hexagonal lattice that may be flinched loom the origin Considering the sets of points in evert . second column t liat may be reached from the origin by open paths leads to a simple Marlcov chain
of two consecutive sites in 1, 1 Further mote, all routes from L i to a site E L t+1 are disjoint from all routes from L I to any other site to E Hence, both for site and for bond percolation, given the set PI of sites in .L 1 that may be reached horn the might, each site E Lg +1 is in R1_1 independently of the other sites. Thus, the sets Rr may be described by the same Markov chain used in the case of Z 2 with
Pi
p and
P2
= P( 2P
P2)
for bond percolation on H. and Pl = P2 =
for site percolation ou H Consequently. Theorem b implies the bounds 3t3 P I: ( 11 )
mid /PM
9
6 4 Nonz iymotts bounds
175
64 Nonrigorous bounds It seems that the critical probabilities for even very simple graphs can be determined exactly only in a small number of exceptional cases In the light of this, it is not surprising that a huge amount of work has gone in to estimating such critical probabilities. Here we shall describe briefly some of the techniques used, and mention a few of the older and a few of the more recent papers on this topic: a complete survey of this field is beyond the scope of this book ??lost nonrigorous estimates of critical probabilities for lattices ate based on Monte Car lo techniques, i e., on simulating the behaviour of percolation on some finite lattice However, there are many different ways of extrapolating front such finite simulations to predict the critical probabilities For simplicity, we shall consider site percolation on the square lattice Z2 Let A„ be the finite portion of this lattice consisting of a square with n sites on each side If A„ is any event depending on the states of the sites in A.n , then we May estimate Pp (A„) = PE; 24,(A„) in the obvious way: for each of N trials generate the states of the sites in A„ at random, according to the measure P. and count the number M of trials in which A„ holds The problem is this: how do we estimate p„ (Z2) = f(Z2 ) from the probabilities of events depending only on the states of finitely many sites? One possible approach is the following Let H,, = H(A„) be the event that A„ has an open horizontal crossing From Menshikov's Theorem and Lemma 9 of Chapter 5, we know that lim„_, P p (H„) = 0 if p < Pt , while lim„_.„ „(H „) = 1 if p > p„ For each ti re function Pp (Hn ) is increasing in p. Hence, as a gets larger and !anger, this fluteHon increases from close to 0 to close to 1 in a smaller and smaller 'window' ar ind pc In particular, the value p(n) at which Pp(„)(11„). 1/2, say, tends to pc. By estimating Pp (II„) for various values of p, we can estimate p(a). By choosing n as large as is practical, or by considering several different values of n and extrapolating, one can then estimate pc = lim„_,p(n). A problem with this approach is that, even for a single value of a, we must estimate P„(11„) for many values of p A computationally efficient version of this method was developed by Newman and Ziff [2000; 2001], based on the following simple idea.. Let Q„ be some random variable that depends on the states of the sites in A„, whose expectation we wish to estimate For example, C2„ could be p(rin). If there the indicator function of the event LI, so EgQ„)
176
Estimating critical pan/abilities
ate N = IV(P) sites in A„, then. writing X for the random number of op ens i te s in A. we Ittoe E i,(Q„) =
E(Q„
= ittliPp (X = k)
Eb( Ar.p. fit)E(Q„ X = k
k=t)r4)
where lt(N,p, k) = (fiT)pk (1  p) Nk is the probability that a binomial landorn variable with parameters N and p takes the value k 111/IS, to obtain (nonindependent) estimates of Iii i,(Q„) for all values of p. • suffices to estimate E(Q„ X = k) for each value of k. After conditioning on = k. the set of open sites is a iandon, subset 01; consisting of k sites of A. distributed unifor nily o yez all ( tin such sets \Ve call efficiently generate a sequence (0,; )ii of random subsets with the light (marginal, i e . individual) distributions by considering a random set process: start with. = 9, and generate (9t .i. t horn 0, by adding a site of A„ \ 0, chosen at random, with each of the N  l sites (Totally likeli II we generate It such sequences independently, for each k we may estimate E(Q„ X = A') as the me/age of Q„ over the I? samples for Or, we have generated Combining these estimates as above gives estimates of Ei,(2„ ) for all p simultaneously Often, the behaviour of Q„ varies in a simple way as the state of one site is changed how closed to open. in which case this poet:Mire is en fast For example. taking Q„ to be the indicant' function of the event H„ = 11(11„), the dues of Q„ for an entire sequence (Or,) Call he calculated in lincep time (in the number of sites), using a 'union/lied' alp/Hun to keep track of the set of open clusters at each stage This enables Der y accurate estimates of the curve ilii,(99„) to be made even lot (mite huge n, which in turn means that ',Mean be estimated accurately Using a variant of this method, considering the event than some open cluster on a toms 'wraps around' the toms in a certain way, Newman and Ziff report the following estimate fon pc = gt(Z2): li„(E")
= (1 502716'2110 00000013
Flowevm. there ate two problems with the approach just described One is that the estimates of E i,((2„) kit different p ate not independent 'This makes the statistical analysis somewhat Made/. A ouch mote set ions moblent is the effect of Jinn CsiZC scaling': we know that p(p), sai tends to pc , hut not at what rate. It is believed that 17)
6. 1 Aton  1.1gormt, bow, tb
177
o: this would follow hoot l i re existence of a certain critical exponent': see Chapter 7 `the existence of this exponent is known only for site percolation on the triangular lattice, where prin any case known exactl y. Bleu if (7) were known, it would not, help to give rigorous hounds: such a relation still tells us nothing about the relationship between p„ and the finite set of Values p(n) we have estimated, as the o(I) term could he very huge (or small) for small values of n. In 8111111/011 V, Monte Carlo 'nethods as described above produce estimates for pt. that do (70 /1/1/1 ge in pi obabilitt, to the tun e value, as the size the finite lattice studied and the number of simulation 111/IS increases loweyer while the statistical en of can he analyzed. the r ate of convergence to p c is unknown Thus an y HIDt bounds given by these methods amount to an educated guess as to the final uncertainty This contrasts with the rigonms 99 99% confidence intervals obtained by consider ing 1independent: percolation, where the only stance of (trot is statistical, tund the error probability lot a given bound can be determined exact ly To show that this problem of estimating the uncertainty is a real one, note that there is 80111() disagreement about the rate of convergence for the Newman Ziff method described above. Newman and Ziff [20(11] say that the finitesize error decreases as N 11,18 , when) N is the number of sites in the finite lattice studied however, the numerical Jesuits of P/I/ [2005! stiongl ) suggest that the r ate is closer to _y  fir There are also many examples of reported :results contradicted by later estimates, or even by rigorous results An example is the esinnate 6'1 ( 7,, 2 ) = 0 632 ± 0 00+I given In keitesz and Vicsek [19801 we mentioned cattier Nonetheless, these methods do give very atell/ ate estimates oh pc : the trouble is that one cannot be sure how accurate! lb/
501/K/ ("011S/ Hifi
7 invariance  Sn mov's Theorem Conformal
The celebrated 'conformal invariance' conjecture of Aizemnan and Langlands, Pouliot and SaintAubin (199,1] states, roughly, that if A is a planar lattice with suitable symmetry, and we consider percolation on A with probability p MA), then as the lattice spacing tends to zero certain limiting probabilities are invariant under conformal maps of the plane P 2 C This conjecture has been proved for only one standard percolation model, nameh independent site percolation on the triangular lattice The aim of this chapter is to present this remarkable result of Smitum [2001a; 20014 and to discuss briefly some of its consequences In the next section we describe the conformal invariance conjecture, in terms of the limiting behaviour of crossing pr obabilities, and present Cardy's explicit prediction for these conformally invariant limits in Section 2, we present Smimov's Theorem and its proof; as we give full details of the proof, this section is rather lengthy. Finally, we shall very briefly describe some consequences of Str u t nov's Theorem concerning the existence of certain 'critical exponents '
7.1 Crossing probabilities and conformal invariance Throughout this chapter we identify the plane P 2 with the set C of complex numbers in the usual way A domain. D C C is a nonempty connected open subset, of C. If D and D' are domains, therm a conformal : D D' which is analytic on D. crap from D to D' is a hijection l is then analytic on D'. i e., anal y tic at every point of D Note that • Locally, a conformal map preserves angles: the images of two crossing line segments are curves crossing at the same angle; this is why the ter m conformal is used By the Riemann Mapping Theorem( (see, tor example,
7 1 Crossing probabilities and conformal invariance
179
Duren (1983, p 111 or Bear don (1970, p 2061), if D, D' C me simply connected domains, then there is a conformal Map from D to D' Roughly speaking, co/Mutual invariance of critical percolation means that certain (random) limiting objects can be defined whose distribution is unchanged by conformal maps. Here we shall only consider a mole downtoearth statement concerning crossing probabilities For this, we shall need to consider domains whose boundaries are reasonably well behaved We write D for the closure of a domain D Let us say that a simply connected domain D is a Jordan domain if the boundary D \ D of D is a Jordan curve, i e the image I of a continuous injection 2 : T C, where 7 = PlZ is the circle, which we m a y view either as [0,1] with and 1 identified, or as {z : = We shall write F(D) for the boundary of .D, and 7 = 7(D) for a function 7 as above, noting that 7 is unique up to parametrization.. By a kma,11,:cd domain (D; we mean a Jordan domain D together with k points Pi , P), , Pk on the boundary of D We always assume that as the boundary F(D) is traversed anticlockwise, the points 121 appear in this order We shall often suppress the mar king in our notaticm, writing Dr, (Of (.19; Pk) Given a marked domain Dr; , we write A 1 = (D,,.) for the boundm y arc from P1 to Pi+i (in the anticlockwise direction), where we include both endpoints, and the indices are taken modulo k; see Figure 1
Fiume 1 A 4nnuked
domain
= (D; Pi , Pi, Ps, PI)
Let A be a planar lattice Given a real number S > 0, by SA we mean the lattice obtained by scaling A by 6 about the origin Thus, for example, OZ 2 is the graph with vertex set {(So, : b E Z} in which two vertices at distance rS are joined by an edge Suppose that we have an assignment of states, open or closed, to the bonds or sites of SA If = (D; Po, P3, PI) is a 4mar iced domain, then by an open crossing of D from. A i to A3 in SA, we mean an open path non r., t. in OA
Conpain& invaDaaer Smanov s Theorem
180
such that /7 1 , cr_i lie inside D 1 ,0 and id ale outside D. and t he line segments eon and et _ i n meet the arcs = (Dt land A 3 = A30).0, respectively When the context is cheat, we IllitY omit the references to the arcs .1 1 and A3 and to the lattice SA The quintessential example of a 1mat Iced domain is a rectangle with the comers marked Let D = (a,b) x (e..d.), and let be the vertices of the rectangle. D, labelled as in Figure 2 If A is the square lattice P2
P1
J
1
I I m 
' i , ,
, ,
, , 11 1
R  ,,, 11 I II [
Figure 2 A path P in the lattice 0::1 The path P is an open (dossing open ciossing of D. 1 in site (a bond Inn iZOIdal n ossing of t he act angular
DI
1111111 11: I 1
I [ I ,
I r I
, , , ,
I I I I 1: Ill 11111 111111'
II
Ir
!
crossing a reel angular •Imarked domain it all sites or bonds ()I P are open An percolation on 6: 2 is exactly an open sulnpaph l? of <5222 shown in the figure
Z1 , then the subgraph I? of SA induced b y the sites in D is a rectangle — . d — c}. an open in ST2 in the sense of Chapter 3. For n < clossing of = (13; Pl . Pi 8.. P 1 ) in the site or bond percolation on SA is exactly an open horizontal crossing of the rectangle II' obtained front by adding an extra column of points to each vertical side \Ve shall see later that the precise manner in winch we heat the boundary is not Minot tant: we could have defined an open crossing of at as an open path inside .0 starting at a site 'adjacent to' A I awl ending at a site 'adjacent to' A3. for example be a 4marked domain, iind A a pl a nar lattice Considering Let either sit e on bond petcolation on A. for b > (1 mid U < p < 1, let f;) (D ) . A. p)
111'1,(D. ) has an open (dossing in SA),
eher e IF,, is the site or bond percolation measure on SA in which sites or bonds a l e open independently With probability p 01 course. as usual we should indicate in the notation /d ) (D.I . A. p) whether we consider site in howl per (dilation, but this will not be necessin : shot tly. we shall testi ict
7 1 emssing probabilities and conformal invariance our attention to one specific model. site percolation on the
181 iangular
lattice
p < Th.
= p 1 0),
pr (A) is fixed. then it follous from Menshikoy's Theorem (see Theorems 6 and 7 of Chapter I) that Ps(D 4 , A. p) —tt 0 as , 0 Indeed. the arcs Ar = Au (D 4 ) and fla .51:1 (D4 ) ate separated 1w a distance // > 0 Roughly speaking, as a , 0, if D i has an open G lossing then one of the 0(1/6 2 ) sites near A I must he joined b y an open path to a site at graph distance (1)(/./O) = e(1/6). By Mensbikov's Theorem and exponential decay below p t (Theorem 9 of Chapter 1) the pmbability of this event is 0(1/6 2 ) exp(6(1/6)) = of I) as /5 0 (The hound 0(1/52 ) on the number of sites irm D 'adjacent: to' A t is of coutse rather crude: note, howevei, that this munbei need not be 0(1/5), A l has a fractal structure fin example ) Similarly. considering the dual lattice (for hood percolation) or matching lattice (fin site percolation). it follows that Ps(DA . A. p) 1 as 6 0 with p > pc. fixed This begs the question of what happens when p = Flom now on, we take p to lie the critical probability = ThjA), and
mite /),;(D.1 , A) for .Ps(Dj . A. pc ) To be pedantic, we should indicate whether we consider site or bond percolation. but we shall not do so. \1'c have seen in Chapte i 3 that it D. 1 = (a.b) x (c, (/) is a rectangle with the corners marked, then the RussoSeymourWelsh Theorem implies that there is a constant: o(D.1 ) > 0 such that
<1—
o
(I)
holds fior all sufficientl y small 6. (hi fact. a < inin(b — d — c) will do, since this condition on a enmities that D must contain points of OZ 2 ) A corresponding statement for any lattice with suitable symmetry can he proved along the same lines Using Elm ris ls Lemma, it. is not hard to deduce that (1) holds lot any Amarked domain D.i. In the light of (1). it is highly plausible that for any lattice A and any 4marked domain D. 1 . the limit lintszo P,t (D. 1 , A) exists and lies in (0,1). Langlands. Picket. Pouliot and SaintAuhin 119921 studied time behaviour of this limit assuming it exists, by perrot/Mug 1/1/11/CIical expet intents with rectangular domains Dr, for site and bond percolation on the square. triangular and hexagonal lattices These experiments suggested to them that the limiting dossing probabilities .P(D. I . A.) are universal, i e. independent of the lattice A (This is au oversimplification: in general. one must first apply a finest transformation to the lattice A: for the cases listed. this is not necessary) Aizenman then suggested that these crossing p i obabilities should he confin wally inyar
182
Canfonital thou] lance  Strtirnoo's Theorem
ant; supported by additional experimental data, this was stated as a 'hypothesis' by Langlands, Pouliot and SaintAubin [1094] Conjecture 1. Let A be a. 'suitable' lattice in. the plane, and let D ., =_ p3, ) 4marked domain Then the limit. (D; P(D. 1 , A) = lim Po(D.,, A) exists, lies in (0,1), and is independent of the lattice A Ful thermal and ry, nit cmrformally equivalent 4marked domains, then P(D.1 )
P(D'4)
Let us spell out the meaning of conformal equivalence in this context. Car atheodor y's Them ern (see, for example, Duren [1983, p.12], or Beardon [1979, p. 226] for a proof) states that if .D and D' are Jordan domains, then any conformal map f from D to D' may be extended to a continuous map f from D to D' As f
7 I Crossing probabilities Wad conformal inuariance
183
for any twodimensional lattice A, there is a nonsingular matrix AI such that P(D, AI A) is confonnally invariant, and equal to P(I),Z2), say They also state that the same result should hold for many nonlattice percolation models, citing experimental results of Maennel for Gilbert's disc model (defined in Chapter 8) and Yonezawa, Sahamoto, Aoki, Nose and Hori [1988] for per colation on an aperiodic Penrose tiling Conjecture 1 is also believed to hold for random Volonoi percolation in the plane (also defined in the next chapter); see Aizenman [1998] and HeMatnini and Schramm (1998] As there are so many conformal maps, Conjecture 1 is extremely strong. Indeed, a n y simply connected domain D C is conformally equivalent to the open unit disc B, (0) Thus, any 4nnu Iced domain is conformally equivalent to the unit disc with some four points zo, 23, 21 specified on its boundar y The conformal maps from B 1 (0) onto itself are the MObius transformations. Given two triples of distinct points on the boundary of B 1 (0) in the same cyclic order, say (2 1 ,20,23 ) and 4), there is a unique ?bolus transformation mapping z i to 2; Thus, any 4marked domain D., is equivalent to the unit disc with the four marked points 1, VII, —1 and 2, say, where kir:: 1 and Im(z) < 0 Hence, the equivalence classes of 4marked domains under conformal equivalence may be parametrized by a. single 'degree of freedom' Although, throughout this chapter, we work in the complex plane C LPf2 , we shall not often need to write complex numbers explicitly Thus, we reserve the letter i for an integer (as in z i above), and write j7. rather than i A natural parametrization of 4marked domains is in terms of the crossratio: given four distinct points 2 1 , 23 , 2.1 appearing in this cyclic order around the boundary of Br (0), their crossratio )7 is the real number = ,
Z:3)(z2
Z1)
124 — 22)(23 — ZI
E(0.1)
(2)
Mains transformations preserve crossratios; in fact, two markings of the domain B 1 (0) are conformally equivalent if and only if they have the same crossratio Given a 4marked domain ,D4 = (D; P2, P, PI), we may define the crossratio n(D.1 ) as the crossratio of any marking of Br (0) conformally equivalent to D4: this is unambiguous, as any two such Matkings are themselves conformal! ), equivalent, so they are related by a Maius transformation and have the same crossratio With this definition, two 4marked domains are conlonnally equivalent
184
Conformal alma altar Sail! non s Theorem
Tints the conjecture of Aizennum and if and only if 11 (1) 4 ) = Langlands, Pouliot and SaintAnbin [1994] states that P(D 1 .A) is given a(D,1) by some function p(q) of the crossratio Inspired by Aizenmatfs prediction of conformal invariance. Cindy [1992] proposed an exact contormally invariant formula 7 (D4 ) fin the limiting crossing probability P(D I , A). i e . a formula "T (q) for the function p(q). Using methods of conformal field theory, which, as he stated, are not rigorously founded in this context, he obtained the formula p(q)
3F(2/3) 7r(q)
r0/3)
2F, (1/3.2/3; 4/3; a)
Het 2F 1 denotes the standard lopengeomett ic function, defined by a Fi(o,
a(72)1,(H) z" c(")
3 , ( " ) is the rising Pictorial x i " ) = ,r(x +I )(x + 2) • • (x+ —I) (see Abramowitz and Stegun [1966, p 556]) It is not clear whether Candy's derivation can be made rigorous In fact, there is (essentially) only one case where conframal iuvariance has been proved rigorousl y, namely site percolation on the triangular lattice As we shall see in the next section. Smir nov's proof gives conformal invariance and Cand y 's formula at the same time The case where Di is a rectangle with the corners marked is of special interest Let D.,(/ ) be the domain ' (0, ) x (0,1), with the corners marked as in Figure 2 Then the aspect ratio of .D,1 (; ), i e the ratio of the width of the rectangle Di (r) to its height, is I: the crossratio ti(D.1(t)) is given by some function q(r ), which is easily seen to be a decreasing ticular, function flour (0, co) to (0,1) In par, there is one rectangle D4(1) uulted domain is confor wally for every crossratio q E (0,1), so any 4run equivalent to sonic rectangle DI D ) Fin this reason', equivalence classes of 4marked domains w i den conformal invariance rue often known as whe
conformal r ec taargies
As every rectangle has an axis of symmelny, mid any 4marked domain is cordon math, equivalent" to a rectangle.. Conjecture 1 implies the iucru ianee of P(D. 1 ) under reflections of the domain D i ; see Figure 3. the crossratio ti (D.1 ), and hence Candy's formula it(D 4 ), are preserved by reflections, and hence by Mutiliolornor pink... ? maps, i e jections "which preserve the magnitude but not the sign of angles = P4). set Given a ilmanked .Jordan domain Di = (D: PI , tossing of D. i e an open crossing of al An open (" m , ). .p„, P i (D; P. p
/ CIO
ng plababilitias Ind con . .minal Huta/lance
185
f Figure 3 If f is a contramat map, then so is the map y : Hance. a Turarked domain D m i d its minor image may be confornially mapped to the same rectangle
[t om its first tO its third is simply an open crossing of D4 1101/1 A, to A., In the light of results such as Lemma 1 of Chapter 3, it is natural to expect that P(1),1 ) P(D,1) = 1 As q(D , i) = 1 — q(D.O. one can check that Cm dy's for mula does predict this, i e., that 7(1 — q) = 1 — 7r(11)
By constructing an explicit: confirm/al map from the circle to A(l), one can use Cindy's formula to evaluate the value 7(D. 1 (1)) predicted for P(D, I (t)) In fact, Cindy [1992] worked with the upper half plane U = {z : > (l} instead of the open unit disc. We shall not consider unbounded domains hen± but the definitions and results extend easily to such domains under suitable conditions M a rking four points 2 1 , z± 011 the !cal axis to obtain a T. /narked domain U, = (CT; (z i )), the crossratio t(l.i ) is also given by (2) For 0 < k < 1. let U l (k) he the domain U with the foul points —14 1 , —1,1, marked Then (1.1(k) 11/10' be mapped by a Schwartz Christoffel transformation (If \/(1 —1)(1 — k/) to the interi o r of the rectangle I? w it II corners iihk(k 2 ) k2 ) \,/T. whine A
00=
di I V( — U)(1 —
A:(k2)±
186
Cortformal invariance  Spirtt i tre's Theorem
is the complete elliptic integral of the first kind (Our notation here follows Abramowitz and Stegun [1966. p 590) The same notation 1(1,7) is sometimes used for K(u2 ) ) The aspect ratio of r is r 2K(1c2 )//C(1 — k 2 ), so, to find 7r(D.di )), one can invert (minim jenny): 7r(q), where ri= this formula to find k; then 7r(D.I (r)) = 7r((.1.1(/,:)) tkU.0)) = (1 — k) 2 /(1 + k) 2 (In particular, this shows that tkr) lAD4 (1)) is monotone decreasing in r ) Starting from Cardv's formula, Ziff [1995a; 1995b] gave a relatively simple formula not for r(D.1 (1)) itself, but for its derivative with respect to : 20ITI (2/3) (fi k(D, 1 (0) — 1(1/3)2
(n±lit
The calculations outlined above illustrate an unfortunate propert) of conformal invariance: given two families of 4marked domains, each: parametrized by a real parameter, even if both families ate 'nice' the conformal transformation from one to the other may II ansform the N. rameter in a rather ugh. way. Thus, one can n ot expect Cand y 's for muk to take a simple for in for a given Mice' family of domains. There is, however, one family for which Cm dy's formula can he written very simply, an observation made by Lennart Carleson in connection with joint work with Peter Jones Let .D be the equilateral triangle in (0. 0), nnc (1/2, 4/2) and .p, IR2 C with vertices P i = (1,0), P, let Pi = (x, 0), 0 < <1, as in Figure 4
Figure 4 Cat leson s '1marl
do
Car leson showed that for the special 4nueked domain T;',, = (1); Cindy's formula takes the extremely simple form
= .r
(3)
7.2
SmintorC5
Theorem
187
As we shall see in the next section, it is in this form that Smir 1/0V proved Cal(' \ r' S ft/1/ . 11111a for site percolation on the triangular lattice. Note that relation 7r(q) 7(1 — r t) = 1 is easy to verify from (3) Indeed, permuting the labels of the rum ked points to obtain the dual domain Writing ri n(21„) for the Tx* as before, Tjj is the mirror image of crossratio of L. , we have ti(13.*0 = 1— 0(1).1 ) for any 4marked domain, so
r(1 —
= r(Trj) = 7T(7) 3 .) — — X = — 7r(77,) = 1 — rt(q)
(4)
The hypothesis stated by Langlands, Pouliot cord SaintAubin [1991] is a little more general than Conjecture 1 Their conjecture extends to events such as 'there is an open crossing horn „4, to .,4 3 and an open crossing from .40 to 24,/, and corresponding events involving any finite number of crossings of a domain whose boundary is split into a finite number of arcs Conjecture I also has consequences that are at first sight unrelated, namely the existence of various critical exponents We sh a ll return to this briefly in Section 3.
7,2 Smirnov's Theorem For the rest of this chapter we restrict our attention to site percolation on the triangular lattice T. and the rescaled lattices ST Throughout this section we consider only critical per colation, i.e we study the probability measure F in which each site is open independently with probability p = p7.1 (T) = 1/2. As noted earlier . , Smirnov [2001a] (see also [20016]) proved the remarkable result that the conformal irrvarlance suggested by Aizenman does indeed hold in this case. Once its walls have been breached, one might lave that the castle would rapidly fall, i e that a proof for general lattices, or at least for other 'trice' lattices such as V, would follow quickly Although this was widely expected, no proofs of other such results have emerged, except for an adaptation of Sruirnov's argument to a somewhat unnatural model based on bond percolation on the triangular lattice given by Chaves and Lei [2006] It seems that Sruirnov's proof depends essentially on special properties of T In this section we shall present Sinn nov's proof in detail. Let us note that even the expanded version of this proof in Srnirnov [200114] is only an outline For the heart of the proof, showing that ceitaM functions related to crossing probabilities are Innmonk, it is very easy to dot the CS and cross the t's However, one must also deal with certain boundary
186
invariance Smirnov Theorem Conformal invariance
conditions Steil 110V 12001bi gives a suggestion as to how this should be done, but it does not seem to be at all easy to tour this suggestion into a proof. Bellina (20051 and independently Tad/ suggested considering certain symmetric combinations of Sinithov's harmonic functions; Be faro showed that tans greatly simplifies the boundary conditions, eliminating the need to consider partial derivatives at the boundary (2005) has used these ideas to give an expanded account of Smirmov's proof: nevertheless, even this presentation is far from giving all the details \\e shall move Stnimov's Theorem in the precise form below: the original statement is somewhat note general in toms of the domains' considered The proof we shall present here is lather lengthy We follow the strategy of Stith nov. as modified by Beffar a Along the way, however, we prove the many 'obvious' statements that are requited Theorem 2. Let D C C be a simply connected, bounded domain whose boundary is a Milian curve Let P i . 1 < i C I. be distill() boundary points of D, appealing in this cyclic order as I is travelsal, Ai(DA) so D I = (D: P. Pe. P3 PO is a 1lnat hell domain. Let he the arc of F from Pi to PH. I . inhere p, is taken to be Pi Then P,;(D,I,T) exists and is given by Candy's Mtmala where P(.0 1 ) = p,s(D.,. I) is the probability that, there is an open crossing of D front to A R in the ci Hirai site percolation on the lattice ST that genet The basic idea of the moot is to define cm taro functions alize crossing mobabilit MS We regard as a 3marked domain D 3 by temporarily lotgettingthe fourth nuanced point P I , replacing the boundin V arcs :la. A4 1):1 a single arc A3 = A3 U A4 'We then define functions is the probability that 1, 2,3 Roughly speaking on D. i to A1.3 separating z hour An; the then e is an open path in 6T fl from i emaining are defined similarly These functions generalize ctossin probabilities: t ot: tuning to the 4mat Iced domain DAo an open crossing hour An to )13 is the same as a crossing of Da from An to A'3 separat hom ,4,i; see Figure 5 Thus, the crossing probabilit: ing the point T) is just the Mine of n(z) at the point Sinn nov moves a certain 'colour switching lemma' which implies an equality between cert ain discrete derivatives of the functions r i; It will follow that, as b 0, the functions f converge uniformly to harmonic functions I' These functions satisfy bouuchu v conditions that ensure that thec tr anshinin in a cc,intia math inviu iant way as D 3 is transformed,
7 2 Sin g nov Theatem
189
1
There is an open classing of I) loan A l to A 3 it and only if throe urine is an Open path how the ate A t to the arc = .1 3 U Ar that separates 111 from A 2 'The indicated path separates R t ti am An and o how .4 2 , but does not separate a from .1 2 The function .1 f. (z) is the probability that there is an open path in 67. fron t A t to si t:, sepa t ating z Crow .42
and that if D3 is au equilateral triangle with the cornets flnked, then Bch t i is the linear function that takes the value 0 on the are .1; and 1 on the opposite point. Cardy's formula in Carleson's form then follows
7.21 A consequence of an RSWtype theorem. One of the key ingredients of Stith nov's proof is a consequence of the RussoSevmouril relsh (11SW) Theorem. concerning open paths crossing annuli with veiv different il/11C1 and outer radii So fat. we have proved an RSWtype theorem only for bond percolation on the square lattice Of the many /nook of this result:. most (perhaps all'?) can be adapted (ably easily to site percolation on the triangular lattice to educe the following result Theorem 3. .Let p> 1 be constant Them is a COP shin/ e(p) > Il such that, if n > 2 and R pn by n rectangle In IR' with any orientation, then the probability that the critical site percolation On T contogys an open crossing ofR joining the two short sides is at least e(p). Here. the notion of an open crossing is that km tmarlced discrete domains. i e au open path uor,r v i in iL with m i . insicle such that the line segments von t and e t _ i V I meet opposite short sides )1 R. For a detailed proof of Theorem 3 based on the strategy used for bond percolation on 22 in Section 1 of Chapter 3, see BolloNis and Riordan (2006b]
190
Conformal irraarian CC Smintatis Theorem
We shall use the following immediate consequence of Theorem 3 several times Let A be an annulus, i.e the region between two concentric circles C 1 and 0). We say that A has an open crossing in the lattice ST if there is an open path from a site inside the inner circle C 1 to a site outside the outer circle C.)). Lemma 4. Let. A be an annulus with ginner radius r_ and onto, radius ± If 7 +11 > 2 WO r_ > 10008, then the probability that A has an open crossing in ST is at most (r_11 + )", where (11 > 0 is an absolute constant_ Proof. R eplacing a by a/2, we may assume that / 4 . = 2 k r_ for some integer k > 1. In any annulus with inner and outer radii I and 21 . , we can find six rectangles as shown in Figure 6. If 6/7 is small enough (and
Figure 6 Six rectangles inside an ru t /mins A of inner radius r and outer radius 2r If each rectangle is crossed the long way by a closed path in OT, then the annulus A cannot be crossed by an open path
< 1/1000 will certainly do), then the shorter side of each rectangle is at least 26, and the 6neighbourhood of each rectangle is contained in the annulus, so the event that it has a closed crossing depends only on the states of sites in the annulus As K(T) = 1/2, Theorem 3 applies equally well to closed crossings Hence, for each of mu six rectangles, the probability that it contains a closed path joining the two short sides is at least c, where c > 0 is an absolute constant By Harris's Lemma
7 2 Stair non's Theorem
191
(Lemma 3 of Chapter 2), the probability that all six rectangles contain such paths is at least c 6 Hence, with probability at least d i , there is a closed cycle in ST separating the inner and outer circles of the annulus. Recall that = 2 k t Thus, inside the given annulus A, we can log(r+//_)/ log2 disjoint annuli A j C A with inner and outer find k radii and 2/ 7 , respectively Let E1 be the event that A; contains a closed cycle separating its inside from its outside. If J_ > 10008, then 1P(E 1 ) > di for each i. If A has an open crossing, then none of the events c an hold. But the events E1 are independent. so Pfil has an open crossing) < Pfno E i holds) =
(E1) < (1 — ,_r
and the result follows with a = — log(1 — c 6 )/(log 2)
❑
Thinking of open sites as black and closed sites as white, a path P is monochromatic if all sites in P have the same colour. Lemma 4 applies equally well to closed crossings, and hence to monochromatic crossings Roughly speaking. Lemma 4 says that if we have a 'small' legion of the plane then, when our mesh a is fine enough, this region is unlikely to he joined either by an open path or by a closed path to any part of the plane 'far away . To a certain extent, the form of the bound does not matter: any upper bound of the form f(r_/ r +) with 1(a) 0 as 0 suffices for the proof of Smitnov's Theorem. Note that even this weaker form of the lemma is much stronger than the fact that there is no percolation at the critical point:, which implies only that the pr oba.hility above tends to zero as I 4./5 oc with r_/8 fixed.
7.2.2 Discrete domains The heart t of Sr MAT'S proof is a lemma stating that certain probabilities associated to per colation on ST within a domain D are exactly equal (Of course, this is just a statement concerning a subgraph of the triangular lattice T ) In presenting and proving this statement, we shall often consider the (rescaled) triangular lattice ST together with its dual, the hexagonal lattice 511 obtained by associating to each site v C ST a regular hexagon PL. in tire natural way, to obtain a tiling of the plane By a discrete domain arab mesh S we mean a finite induced subgraph Cs of ST such that the union of the (closed) hexagons 11„, v E is simply connected When considering C,1 as a graph, the mesh 8 is irrelevant, so we may take 1 and view C5 as a subgraph C of T
192
Conlin nul l thew lance  Swirrow s Theorem
In terms of the lattice T On OTT the condition that C Gs) be simpl y connected is equivalent to requiting that both G and its outer bountho y T, 0 ÷ (C), ale connected subgraphs of T. w lane 0(G) consists of the set of sites of T\C adjacent to some site in G In fact, we shall impose the Following additional restriction on out discrete domains G: we shall assume that neither C not U+ (C) has a catvertex. This restriction is it televant to the mechanics of the arguments that follow but simplifies the presentation slightly The inner boundary 1)(0) of a discrete domain C C T is just the set of sites of Cf that ate adjacent to some site of 7' \ C Om additional assumptions ensure that both 0 (C) and 0 4(C) are the vertex sets of simple cycles in T: indeed. viewing G as a union of hexagons, its topological boundar y OG is a simple cycle in the hexagonal lattice If Following this cycle in an anticlockwise ditection, sm. the hexagons 0 e , sites of T) seen on the left hum 0 (C)„ and those on the tight form The condition that neithet G not if + (CI) has a O c (G); see Figure
Figure 7 A discrete domain G (filled uncles and the lines joining thew) drawn with the corresponding hexagons shaded The two thick lines are t he (des in T corresponding to t he Juliet and inner boundaries 0 + (G) and 3 (C) 01.61 The topological boundaly OC of (the set of hexagons corresponding to) G is I he cycle in /I separating shaded and unshaded hexagons
cutye t tex enstues that we do not visit the same ver tex of 0(0) or or eTf. (G) more than once 110111 now on we shall view 0(G) and O H (0) as cycles in the graph 1
7
Billie1101,'S Theorem
193
In preptuation lot S i nn nov l s key lemma, we need a hu Him definition A kranked discrete domain is a dismete domain G (ca Gs) together with k distinct sites ,m; of its boundaly 0  (G), appealing in this ()Mel as 0(G) is crave/sec! anticlockwise. \Ve shall abuse notation by writing simply G lot a hrmarked discrete domain (G;v 1 „ vb.) Palely for convenience, we inmose the additional condition that each marked site v i is adjacent to at least two sites of T \G (This condition is very mild: any site n of 0(G) that does not have two neighbours in T \G is adjacent to one that does.) = sT(G) to be Given a kmarked discrete domai t we define the the set of sites of it (G) appearing between n i and in + , (with o k+ , = as 0(G) is traversed anticlockwise We include both in and in.s. I into In this discrete context, an open missing of C from A; to Ai is simply an open path in G star ting at a site of A; and ending at a site of Aj Given a kunat ked disci ete domain G = (G: (n; )). as ever y tutu ked site i n has two neighbours outside G, we can partition the outer boundary i)± (G) of G into vertexdisjoint paths Ajij = Aj/j(0), 1 < i < k. so that each AY starts at a site adjacent to vi and ends at a site adjacent to cis., Indeed. traversing id + (G ) anticlockwise we may take Ajijj to run horn the second neighbour of c i to the first neighbour of v i+j ; see Figure 8 Note that a site r E C is adjacent to some site of Air if and only if v E Recall horn Lemma 7 of Chapter 5 that a rhombus in the hit/rigida!' lattice alwa ys contains either a horizontal open crossing, or a 501tical closed crossing. but not both this statement,. and its moo!, extends immediately to lmarked discrete domains Although the proof lo t the general case contains nothing new. we give it in lull, since we shall soon use similar ideas in a //101e complicated way Lemma 5. Lel C be a lumoked discrete domain, and let = .11(G) 1 i 1 11'haterer the stoles (t/ the sites in G. this graph contains either an open crossing from A I to A 3 . Of u (lased ree`Otithe from to As. but not Goth Ill pal (imam. the probability that C has on open crossing Pow A, to .1 and the probability that C has tell open crosYng from stt to As sane tO Proof As above. let CY (CO denote the outer boundary of C. which
is partitioned into arcs Ajt Consider the partial tiling of the plane by hexagons, consisting of one hexagon for each site of C U (G). Colour the hexagon ./1,. cot responding to c E C black if n is open, and white
Confunnel invariance chnirnoo's Theorem
191
•
•
ALVA A A A A A V ♦ V VA VA A A V •
•
•
•
•
•
•
•
Figure 8. A 1marked discrete domain (68. e 2 . ea! et) (filled circles and lines joining thew) The outer boundary eP(G), a cycle in T, consists of the hollow circles and the lines joining them. The thick lines show the inner boundary shining endpoints and the 0(G) of C. a writer of arcs A. I < i sjj. corresponding disjoint arcs C a c (CI)
if c is closed Colour the hexagons H,. corresponding to u E frt U At white, as in Figure 9 black, and those corresponding to E AT U Let I be the interface graph for nwd by those edges of the hexagonal lattice separating black and white hexagons, together with their endpoints as the vertices Then every vertex of I has degree two except for loin vertices Th , 1 < i shown in Figure 9, which have degree 1 Orienting each edge of 1 so that the hexagon on its right is black, the component of I starting at y i is thus a path P ending either at go or at Suppose that .P ends at !Li , as tit Figure 9. Then the black hexagons on the right of P form a connected subgraph of G U U At joining :At to Al . Any such subgraph contains a path within CI joining a site adjacent to At to a site tv adjacent to But then e E A, and ar E A:3, so G has an open crossing horn A i to A:3. Similarly, if .P ends at rp, then the white hexagons on the left of P give a closed (*.Tossing of GI from 4, to A i Crossings of both kinds cannot exist simultaneously, as otherwise K5 could be drawn in the plane The second statement follows immediately: as each site is open independently with probability 1/2, the probability of an open crossing
'7 2 Binh nav's Theorem
195
Figure 9 The hexagons Ii corresponding to G U &' (C). with those corresponding to the marked vertices / 4 , vo, ra, N labelled. A hexagon EL., v E is coloured black for i = 3 and white for i 2,1 A hexagon H i ., v E C. is black if t, is open and white if t ■ is closed There is an open path in GI nom A i to A 3 if and only if there is a black path in the figure horn A lt to .11 and hence if and only if the interface between black and white hexagons joins y, to g.1
from A 2 to A., is I he same as the probability of a closed crossing from Au to A., This lemma completes mu brief review of the basic proper ties of critical site percolation on the triangular lattice T In the next subsection we present the first step in Sninnov's proof of conformal invariance
7.2.3 Colour switching We are now ready to present, the key lemma in Smiruov's proof: this 'colour switching' lemma states that the probabilities of certain events involving paths in a discrete domain are exactly equal. We shall shall often identify a site of T or 8T with the corresponding hexagon In particular, in the figures that follow, rather than drawing site percolation on the triangular lattice, we draw 'face percolation' on the hexagonal lattice, since it is easier to shade hexagons than points Let C be a 3mar ked discrete domain, and let; x i , :to. .e, be three sites in (3 forming the vertices of a triangle in C, labelled in anticlockwise
196
Confonnal im yal lance Stnirnov s Theorem
order around this ttiangle Thinking of open sites as black and closed sites as white. we write 131 for the event that there is an open path .joining x i to Ac = ATP). i e., an open path from x i to a site in .4i, and 11'1 for the event that there is a closed path horn x i to Let 8 1 13,113 denote the event that there ate vertex disjoint paths Pi from x i to with P i mid 172 open, and P3 closed; see Figure 10. Note that
Figure 10 1 Ire hexagons {do responding to a :Smarked discrete domain II the Nei tices correc3). this time ‘‘ilhout its outer boundary (G: sponding to I he heavily shaded hexagons al e open and those corresponding to the unshaded hexagons are closed, t hen /3, /3.11 .3 holds
8 1 13Th;, = 13, ❑ 13,} J IL, is just the box product of the events 13 1 . 13, and II 3 . as defined in Chaplet 2 Define 8 1 11 .43 3 = L3 1 ❑ II Q C /33 sinalai b. and so MI [he probability (HMI ibutimi associated to critical percolation on t he iangultu lattice is of commit 'nese' \ Cd if we change the state of every open site to closed. and vice versa Huts Ft/3 1 11 2 11 3 )=P(11,8 2 Ba),
(5)
and so on. so there ate font potentiall y distinct probabilities associated E {13.111 SlIkh nov's Colon y to events of the lot In X i 3",2 3 . X. ). Switching Lemma states tlmt three of these are equal Lemma 6. Lela G be o 3m a t/eed discrete domain and let c 1 . :c xa be sites of G Jointing a triangle in G. labelled in anticlockwise rode) around
7:'
Stith 1101' s
Thew ein
197
this triangle. Then P(/3 1 /32 11 .3 )
1.11(D I
2 83 ) = POL L /32 B3 )
(6)
h t contrast to (5), there is no symmetry of the metal( setup that implies (6). Lemma 6 does not say that when looking for disjoint paths, the colours me it relevant as it makes no statement about IP(B, Bo B3) To prov e Lemma 6. we shall show that P(B, Ilji/33 ) = P(8 1 Ijillja)
(7)
Applying (5). or the relation P(B i 1 ,11'3 ) = 1111 (8 1 which is essent ially equivalent to (7), one of the equalities in (6) Follows immediately The older inequality follows similarly, or by relabelling In tutu. (7) is equivalent to IP(/3 1 11',D3 =
(8)
The idea of the proof is as follows.. Whenevet B 1 FI holds, one can find l i n t el/mist' open and closed paths Q i and Q. , witnessing Bi condition not only on B I ji. but also on the pietise values of the paths cart find these paths without examining the states of Q i and Qo sites outside them Next we shall show that if Fi l l ri B3 holds, then thole is nu open path p, front 3i3 to .4 3 outside the inunermost paths Q j and Qo Ilms, the conditional probabilih that 8 1 11:)/3: i holds is just the probability that the outside' contains an open path horn ,r 3 to A 3 . As we have not yet examined the states of alp sites 'outside' Q i and Q(i. this is the same as the probabilit y that the domain 'outside' (2 1 and Q contains a closed path from to A: 1 , which is the conditional probability that B111'41:3 holds At the risk of scorning too pedantic, we shall present a detailed pool of Lemma 6 using the ideas above Note that a little caution is needed: on the level of the vague outline just given, it might seem that the same argument shows that 1P(B 1 BoB3 B I M) = P(13 1 13,11 .1 / .8 1 13,0 In fact, P(13, Vii ) and P(E3 i BoB3 ) ate not in genetal equal To find the intim most paths witnessing B 1 111i, we shall fellow a return/ interlace between hexagons. Consider the /initial tiling of the plane by hexagons. with one hexagon ha each site of U (G) As holjae, we colont a hexagon H, cm responding to a site v of CI black if e is open, and white if v is closed This time. we colon y the hexagons corresponding to Ali black, those corresponding to At white. and those to At grey As before, let I be the subg l aph of the hexagonal lattice consisting of
198
Coriformal
Smernov's Theorem
all edges between black and white hexagons, with then endpoints as the vertices This time, every vertex of I has degree 2 except for a vertex y where .41' meets At, and one or more vertices y i incident with grey hexagons; see Figure 11. Let P he the component of I containing y. Then P is a path starting at y and ending at a vertex :p i where hexagons of all three colours meet.
Figure A 3marked discrete domain with its outer boundary Hexagons are black, those corresponding to At white, and those corresponding to to A. grey Internal hexagons are black if the cot tesponding site is open and white if it is dosed The interface path P starting at y ends at a grey hexagon
4
Let iv he the centioid of n,r0;r 3 , so a' is a vertex of the hexagonal lattice Let Tr =emu; be the edge of the hexagonal lattice separating xr from tn, oriented towards tv; see Figure 12 We shall prove Lemma 6 via a sequence of three simple claims. In the statements of these, the assumptions of Lemma 6 are to be understood. Claim 7. If Br W., holds, then the Oiler face path P sla t Hag at p travetses the edge e in the positive direction P1'00.1 Suppose that the event B I IV, holds, and let P1 be any open path horn 3$ 1 to A l , and P, any closed path from wo to ;19 Note that PI and P, ate necessmils disjoint. As any site in
is adjacent to a
7.2 Swinton . Theorem
199
Figure 12 The centroid w or a triangle ruts:es in 7', and the oriented edge 7 =71 of the hexagonal lattice H that separates x i from r
site in AI, there is a cycle C in the triangular lattice form ed as follows: follow PI from x i to A i Then follow (part of) At to its end Then follow par t of At, and, finally, follow P, from A., to :1,,} The cycle C' is shown by dotted lines in Figure 11 We may view C.', which is a cycle in the triangular lattice, as a simple closed curve in the plane in the natural way As we go around the cycle C' starting at: 1, 1 , we first visit black hexagons, and then white hexagons Thus, exactly two edges of the interface graph I cross C', the edge 7 shown in Figure 12, and an edge yy'. These me the two edges of H shown with arrows in Figure 11. The path P starts with the edge gy', which takes it inside C' As all grey hexagons are outside C, the path P must leave C at some point, which it can only do along the edge 7, proving the claim, ❑ Let P' be the path in the inter face graph I defined as follows: starting at y, continue along edges of I until either we traverse 7 in the positive direction, or we reach a grey hexagon If P does traverse the edge 7 in the positive direction, then P' is an initial segment of P Otherwise, P' is all of P. Let N(P'), the neighbourhood of P', be the set of sites of G corresponding to hexagons one or more of whose edges appears in P' Claim 8. If P' ends unlit the edge then the set N(P') C G contains (necessarily disjoint) paths Q i from. :f t to A i and (29 from, r, to Ait, with Q i open and Qs closed. Proof. The argument is as in the proof of Lemma 5: if P' ends with the edge 7, then the sites corresponding to the black hexagons on the left of
C'onforrnal Mem inner Stn0 non s Theorem
2(10
meeting A and containing P' 161111 a e()Itilecticd subglaph S of At Any such subglaph .5' includes a suhgtaph 5'' of G containing both and a site of C adjacent to At Every site u E .S'l is open: the corresponding hexagon is black (as 11 E 5), and o E C. so u is open. Finally. any a, E G adjacent to At is in A l , so there is an open path C S C N(P) how c i to A, Similarly thew is a closed path (2, C to A, ❑ Qo C .1V(P) Flom Together, the claims above show that BULL holds if and only if P' ends with the edge "(7 The proofs of Claims 7 and S also given little mote Claim 0. The event 33 I II I,23 holds if and only if 13' ands with and then, is on open pith P3 C C from 3 .3 to A$ using no site of the /31 I I)IY3 holds if and only if P' twig hbou t hood N(P') of P' and them k a closed path P3 C 0 from .r3 10 .4 using no conk with Sill!
N(P')
It static:es to move the first statement Suppose that B 1 11 433 Proof holds, so then: are disjoint: paths P nom to A, with PI open Po closed. atal fi , open Then the pool of Claim 7 shows that. tw ili t horn its initial and final edges. P' lies entirel y within a cycle C' funned by PI, Pi and parts of AI and (i e the dotted c y cle in Figure 11) Thus, even site of AT/3”) is on of inside C' But 133 cannot moss C'. so P3 lies entirely outside L. awl is disjoint from N(P') The I everse implication is immediate hum Claim 8. It is now pas) to deduce Lemma (i Proof of Lemma 0 As noted eat Het . it suffices to show that (7) holds, i e that P(BI II ,Bt) =P(Bi Let 9. he the state space consisting of all 2 1(I1 possible assignments of states (open (a closed) to the sites of C3, and note that the probability measure: P induces the nonnalized counting measure on U For w E R. let P i (w) be the path I. ' defined as above. with lespeet, to the confignt ation w Let 2 he obtained nom w by flipping (changing Croat open to closed of vice versa) the states of all sites in CI\ AI ( 331(w)). The path latly be found stepdwstep. at each step examining the colour of a hexagon adjacent to the cut tent path Hence. the event that
7.2 Stith not, s Theorem
201
p' takes a particular value is independent 0 1 the states of the sites of P'(w) so w" = w for any w E Q In patricidal P' (I) w' is a Injection Thus the map
G \
Suppose that w E B 1 11 0.133 Then, by Claim 9, the configuration Le contains au open path in G \ N(P(w)) from ,r 3 to A3 Hence ‘21 contains a closed path G \ N (1.3' (w)) = G \ tV(TIr2)) hot u :c33 a to A Thus, by Claim 9, co' E B i lV)11 3 Similarly, if w' E .T3 1 11011 ,3 , then ./3,11 1 ,B 3 . As w w' is a measurepreserving injection, ❑ p(B 1 11 .08 3 ) = P(B, ll irlr3 ), completing the proof Let us remark that, while we could have used the Mtn/lace path P' to refine 'howl most' open and closed paths Q 1 limn x i to A i and Q., front .1 2 to A ‘ r, there was no need: it was simpler to work directly frith properties Of the interface itself
7.2,4
Separating probabilities
The next step is to give a kninal definition of the 'separating piobabilities: the limits of these probabilities will be harmonic functions on D From this point On we shall view the discrete domains Go that we shall consider as subgraphs of ST rather than T: this makes no difference to the properties of Gs as a gtaph, but will be convenient for taking Bunts later Let Cs = (Gs: r:i) C ST he a 3mat lord discrete domain .r‘ith mesh 8, and let c E C be the centre of a triangle in ST As heline, we may think of z as a vertex of the hexagonal lattice SH dual to (11 A 'fey idea in Sndinov's proof of Theorem 2 is to consider the probability of the event E?:(c) = { Cs contains an open A r A, path separating c from At }, and the events E (z) and g(z) defined similarly, where, as before. Ar i(C; s) is the path in the inner boundary of Cs starting at ri and ending at or +1 , and At = Aj (Cs) C 0 + (C; (5 ) is the corresponding path in the outer boundary of Cs Usually. we shall take z to be the centre of a triangle in Gs, although the definition makes sense for points z nearer to (or even outside) the boundary of Cs The subscript fi in out notation is shorthand to indicate the dependence both On S and on the discrete domain Gs. Needless to say, by an open Ar Ao path Pin C7,5 sve mean a path P in the graph Gs C ST all of whose sites are open starting with a site
202
Conformal
11)0(11
Smini.ov'■; Theorem
in A i and ending with a site in An Such a path P separates z hom At if, when we complete P to a cycle C in (Yr using the arcs All and At, the point z lies in the interior of the cycle C when C is viewed as a piecewise linear closed curve; see Figure 13 Equivalently, the path P
Figure 13 A discrete domain G's (circles and the lines joinil g thew) %vith the associated coloured hexagons: open sites ate shown by Ii led circles and shaded hexagons, closed sites by empty circles and unshaded hexagons. 'The outer hexagons (those not containing circles) correspond to the u t ter boundary 0 11 (0s) of C,5 , divided into truce ales .4.1 1 , At and A4 at the thick lines Art open A 1 .19 path P and the associated cycle C separating z, the centre of a triangle in C, from At are shoran a thick lines
At if any path in 511 consisting of edges dual to bonds of starting at z arid ending on (a dual site adjacent to a site on) At, crosses a bond in P. nor to revisit a vertex. Tints, Note that P is requited to be a path, the event .Er (z) does not in the case shown on the tight in Figure hold As before, let w E C be the centre of a triangle xr:ror t a in C. with the vertices labelled in anticlockwise order, and let z be the site of 511 adjacent to a that is farthest from x 3 ; see Figure 12 Suppose that the separates z front
72
8711:177101t ' s
'Theorem
203
Figure 11 The figure on the left shows a schematic drawing cif a 3marked discrete domain a5 together with an A i Aii pat h separating z hour A . If the sites on this path are open, then Ei,t(z) holds. The figure on the right shows a connected set S meeting and Ao and separating z from A4 that does not contain an ArAo path separating z fron t At. If only the sites of S are open. theu Es(z) does not hold
event EPz)\E,(a,) holds, and let; P be any open path in C6 separating horn At Completing P to a cycle C as above. the cycle C winds ar ound z but not around w. Therefore, C' must use the edge t i rk2 . If we trace C by following P from Ag to A i , returning anticlockwise along At and At outside Co, then C winds i1/01/11(1 c ill tire positive sense, so we trace the edge m i x, from fo to ,r 1 . Hence, P is the disjoint 1111i0/1 of open paths Pr from m to A i and A hem r2 to A,. As we shall now see, there is also a closed path P3 horn ,r3 to .43 Claim 10. Suppose that 3,,,,r.),:ra,lir and z ate as above (sec FiffittCS 12 and 15), and that the event g(z)\Ej(w) holds. Then B I M Va holds. e.., C, contains disjoint paths P, Mining ;c 1 to Ar = A(ac), with P, and A open and Pa dosed Proof As above. let P he an open A t –A, path in Cis separating z from . We have already shown that P may be split into open paths Pi horn 3,1 to A i and A from to Ag ft remains W find a. closed path P3 joining ,r 3 to any such path is necessarily disjoint horn the open paths PI and Pg Note that x:r itself is certainly closed: otherwise, the Open path PI :r 3 A separates 11., from Once again, we follow an interface As usual, we consider the partial tiling of the plane by hexagons H,, corresponding to vertices v of Go U
Conformal 'now; iance Surirvoil5 Thcairtn
204
U a + (0 A l. where 0 + /Go) = U is the onto bounda t v of 05 We C ohan a hexagon H,. c E 61 ,r, black if l' is open and white if V closed We colour H, black if e E At U and white if u E At; sc.; Figute 15.. Let. I be the oriented interface graph whose edges a t e the
Figure IS Ilse discrete domain (circles and the lines j0 n og them), with t he associated coloured hexagcms 'The inter lace I between black (dark and lightly shaded) and tc kite hexagons has exactly two vertices of degree I, namely yr and y2 An Open À  path P and the associated cycle C' separating z from At are shown by thick lines If :rr i is not connected to :L i by a closed pat I I , then t he while component containing ,r 2 is surrounded by a connected set S of black hexagons: those not on 1" or At" U AT are shown lightly shaded The union of and P contains an A l A 2 path sepal ating from Alt consisting of the clashed lines together with part of P The °limited edge of the hexagonal lattice crossing Tow:, is indicated b y an arrow
7
edges of the hexagonal lattice wit h a black hexagon on the t ight and a white one on the left This graph has exactly two vet tices of degree the vett:ices .111 and rki shown in Figute 15 As x/i is open and
is closed, the oriented int et lace I tont tuns the
7 2 Solo [mei,: Theorem
205
edge J o ftin hexagonal lattice (tossing .10.1 3 with 1 on the left: this edge is indicated with an arrow in Figure IS Suppose first that the component of I containing f is a path. Then it ends at a vertex of I with degree 1. which must be it, But then the chine hexagons on the HI of this path Mira a connected set containing 3). and meeting At As before„ it follows that there is a dosed path ram ,e 3 to .21 3 , as required. Suppose next that the component of I containing f is a cycle, as in igme 15. Our aim is to deduce a contradiction. Let hr. 1m... h, be the sequence of hexagons seen on the right of this ci cle as we trace it once star ting hom f Then each // i is black. h i corresponds to 3.' 2 , h, /responds to x i . and h i is adjacent to h i* , lot each i Ide n tifying a hexagon with the corresponding vertex of the lattice, we do not have Mr an y i: otherwise. the interface cycle would visit w = twice Recall that we lime open paths P I and P. , joining to A t and respectively. and that these paths can be closed to a rude C sunounding c but not w by adding parts of At and A:. No edge of the interface I can cross C. so ever v center h i lies on or outside C Let he maximal subject to h, E P.) u At, and let 5 he minimal subject to s > t and 11,, E Pr U At Note that 1 < r < s < as h i = :ro E Po and to
EPn Let S = .11,„.I} Then S is a connected set of vertices of the graph C ri — for rued from Cs by deleting the edge ..rTro: the hexagons corresponding to S ate lightl y shaded in Figtue 15 As S contains a neighbour of Pn U At and a neighborn of „Pi U At, it follows that .PI U S contains a path F' in (1,5 — r i :r9 joining A i to A, (The part of tins path off P 1 UP) is shown by dashed hues in Figthe 15 ) The corresponding hexagons are black (drawn with dark or light shading in Figure 15). so the path I" is open Fu Iv lies on or outside C. so it separates z from At As P' does not use the edge x i it follows that 13' separates w from At This contradicts mu assumption that g(w) does not hold ❑ h
The proof above is longer than one might: like, and can probable be expressed more simply Note, howeve L that one cannot give a 'purely topological' proof: in order to show that E, (m) holds, it is not enough to find a connected black set meeting A i and A, and separating to front At Indeed, one might expect that the centre c of a triangle in ST is separated fron t At by an open Am An path if and onl y if z lies 'above'
206
Conformal 11111(1rion.ye  Smirnov's
ThCOlern
the unique path component of the black/whiteinterface in Figure 15, However, this is not the case, due to the possibility of the configuration on the right in Figure 11 It is easy to see that the converse of Claim 10 also holds Claim 11. Let w E OFI be the centre of a triangle x i x 2 x3 in Gs, labelled in anticlockwise order If z E bf1 is the neighbour of w opposite x3 , then E,(2)\.E,?(w) holds if and only if B 1 Hd173 holds Proof The forward implication is exactly Claim 10. For the converse, suppose that /3 1 B911 13 holds, as in Figure 16 Then the disjoint open
Figure 16 A schematic picture of the event B 1 B211:3 horn 370 to An together give an open paths Pi from and to path P flow A 1 to As P uses the edge x i :1H the path P separates exactly one of z and w from At As there is a closed path from :r 3 to .43 , the point tv cannot be separated horn At by an open path Thus ❑ ES(_) holds and Er, (v) does not. As before, given a 3walked discrete domain (3,5 and a point E 611, let Es(s) be the event that G5 contains an open A 1+1 A i + 2 path separating z how At, where the subscripts are taken modulo 3 In the light of Claim 11, Lemma 6 has the following immediate consequence
7.2 Swinton's Theorem
207
Lemma 12, Let ii:roxa be a triangle in a 3malked discrete domain
Go, with its vertices labelled M anticlockwise outer. Let w
E 811 be the centre of the triangle, and let. zr, C 2, Z3 E 611 be the neighbours of ay in 673. with z i and x i opposite for each i. Then
P(Ecilt (z i ) \ E (Iv)) = P(ERz 2 ) \ ERiv)) = P(E,Nz 3 ) \ E:J(w)). z 3 . Claim 11 states that the events g(c 3 ) \ Es(w) Proof. Setting z and 13 5 B2 W3 coincide Permuting all subscr ipts cyclically, we have E!(cr) \ = B 2 B 3 111 [ and El(z2 )\ El(w) = 133 8 1 1 , 1 7o The conclusion thus follows horn Lemma 6 ❑ For the centre z E alI of a triangle in C,s, set; 1'n(z) = P(E,i5(z))
(9)
and, if to and , me the centres of adjacent triangles in C6, set:
P(E:5 (z) \ .E7dtv))
h si (w, c) Note that, trivially,
n(z)n(rv)=iov,z)
(10)
As noted by Stith nov, there is a surmising amount of cancellation in (10) It turns out that for z and Iv far from the boundary of L.), the quantity ki5 (w,z) is of order 5 2/3 as 6 —4 0; the exponent 2/3 is the '3arm exponent' appear ing in relation (51) in the next section. In contrast. fA(z) — is presumably of order 6. We shall not be concerned with proving these statements, as they me not needed For the proof of Srniinov's Theorem. Roughly speaking, Lemma 12 implies that the discrete derivatives of the a ate related to each other by rotation For a precise statement, it is easier to work with integrals around contours. We consider only discrete trianguleu contour s in G1,5, i.e , equilateral triangular contours C whose corners are sites of ST and whose sides are parallel to edges of ST, such that all sites on C me vertices of C. as in Figure 17. We orient C anticlockwise Givers such a contour, for 1 < < 3 we define the discrete coolant integral •D
I
) M C ) Clz
as the usual contour integral of the piecewise constant function f(z) whose value at a point z of an edge 51, of ST on C is the value of
208
Conformal Moat lance  Stith not, s Theorem
A discrete triangular contour C The d iscrete contour integral of Figure fi r wound C' is defined using the values et .111 at points of the hexagonal lattice Along :rig. fen example. we integrate j',(ai)
IA(w) on the nearest vertex w of OH inside C, immediately to the left of
aril;
i e „ on tlw vertex of (if
see Figme 17 For example, if C is a
triangle with side length 6, mid a ' is the centte of the triangle C", then
f (z)
. 0u) identically on C. so 1■(13 Inz)dz = 0
of unity Let w = (—I + \/3)/2 denote one of the cube toots
Lemma 13. Let CI; be a discrete 3hooked domain such that no point of C is within disloace a of all three arcs of OGo. where a > 20006 If C' is a discrete triangulat canton; in. Cs of length L. then (1)
(:)(1: —
f,f(z) (lc < AL(61a)"
( 1)
1.2.3 where Mc sumo set ipl is taken modulo 3, o is the constant in Lemma 4. and A is an absolute constant
fm i
Proof Ri x shall prove (11) lo t i = 1; (lir corresponding equations for
i = 2,3 follow by relabelling the domain Let a' E OH lie the centre of a triangle in Cs, and let z be a neighbour of iv in OH From Claim 10 above. if E,C(::)\E's(w) holds for some then them are monochromatic (entirely open (a entirely closed) paths from the three sites of
Or adjacent to iv to the duce boundary arcs of
Cs The point w is at distance at least a (rout one boundary ale. .1i, say, so a monochromatic path horn a point adjacent to i v to As gives an open or closed crossing of the annulus with centre ut and how l and outer radii 6 and a. 13v Lemma
the probability that such a crossing
7 2 Swirnov s Thcomm
exists is at most 2(10006/a)° =
0(010")
209
Hence.
/a)").
hs i (w..z) =
(12)
uniforml y over C. w with a' the celiac of a triangle iu Cs and z st, denotes adjacency in the graph here and below, Let
w:
tv)11,15(
wct c. systs where C' is the set of yet tices of OH
interim of C. and z nuts
over the duce neighbours of w in OH To evaluate — w we view w and
as complex munbets Fix w E C' , and let z i Ca be the neighbours of a ' in anticloclavise cadet, so — w =
w) =
j
(13!
From Lemma 12 we have 16(w, )= M(w..m) = Ow. za)
(,A)
Also, applying the same lemma with the toles of :I , z Ca) =
c . )) =
pet nutted,
Z)
and h(15(a)7 c 3) =
Setting z 4
(
CI , we thus have (zi
tv)q(ut, zj)
— tv)1;;(v
1±i)
j=1
w(zi w)h,v(m.=i i i) .i=1 3
(v)16(w I=i
where the first equality is simpI N a relabelling Of the same sum. the second is Eton, (13), and the third (Li) and the following similar equalities Stumning over w E C"', it follows that S2 =
(13)
210
Conformal invariance  Snail noM ii Theorem
call write
— u01011, z) as 5'; +.57. where
=
5,, =
(z tv)11 61 (vi, z) +
— z)h's(z, u,)
it,,z€C"
and (z — w)11:5 (ev, z).
The sum S7 has CALM) terms each bounded by Ssup„,,, , h[s(u, Using (12), it follows that 87
0(L(S I o)°)
(16)
for each For
(
z E C`) with iv ti z from (10) we have
z — a)his(lut
(a'
Z
)1/1,1(Z, w)
=
=
h ■S
WX/116(111,
(z — w)( —
(It ))
f4(w))
To obtain the must stun this last term over all unordered pairs {m, z} with m + z and E C' Collecting all terms of the Ram :17101"), E C. we thus have :17
=
E E  Jig ?, ) E ('rvz)f,R4v) E E (a' — z)1,(10). wE C" z—w, ze4c,
In the first sum in the final line, the coefficient of each 1, ((tv) is exactly 0 There is one term in the final sum for every edge viz of SH crossing C from the inside to the outside. The edge ruz of 511 may be obtained from the dual edge 1,"// E C of ST by a clockwise rotation through w/2 followed by scaling by a factor 1/ \/[3 (For x, y and w, see Figure 17 ) Thus, z — w = (  14/fr3)(y — :r) Hence, h our OM definition of the discrete contour integral, •
=
(z — . 0.1A(10 = —
.11(Z) (1Z;
(17)
VF3 c
As Si =
+
, the result follows from (15), (16) and (17)
Later, we shall show that if we have a sequence of 3marked discrete domains Co with mesh S 0 that give finer and finer approximations
7 2 5'71thtwo's Theorem
211
to a Jordan domain D3, then the functions Mz) converge to continuous functions f' on D Lemma 13 will imply that the (usual) contour integrals of these functions around a given contour are related by nultiplication by co.
7,2,5 Approximating a continuous domain In this subsection we show that one can approximate a Jordan domain D by suitable discrete domains without changing the crossing probability significantly, proving the technical Lenuna 14 below This result is immediate for sufficiently 'nice' domains, such as rectangles. The reader Amested only in such domains may wish to skip the proof. Let us recall some of the definitions involved in Theorem 2. By a classing of a 4marked Jordan domain D4 in the lattice ST, we mean a jath in OT whose first and last edges cross the arcs A i (/).1 ) and Aa (M), vith all vertices except the first and last inside D. If the sites of a crossing are open, then it is an open Glossing of D.1 . As before, we write Po(D.,) = (D4 T) for the probability that D4 I as an open crossing. The corresponding definitions for discrete domains are simpler: a crossing of a 1marked discrete domain Gs is simply a path in the graph C:s joining A l (C;(5 ) to A3 (Cs) We write P5 (G. 6) for the probability that a discrete domain C,, C ST has an open crossing Let us write dist(x, y) =  :y1 for the Euclidean distance between two points :ET, y E R2 C. and (list (x, A) and dist (A, B) for the distance between a point and a compact set A, or between two compact sets A and B We avoid the more usual notation d(t, y) due to potential confusion with graph distance For two compact sets A, B C C, their Hausdoijd distance is du (A, B)
sup Udist(a , B) : 0 E A} u {dist(b, A) : b e = int : A C
B C A(=)},
where A V) denotes the (closed) eneighbour hood of A. If = (P1)) is a 4marked „Jordan domain and (75 C 6T is a 4marked discrete domain, then Cs is Eclose to Dt if the corresponding boundm y arcs of D.1 and of G5 are within Hausdorti distance e, i.e if dri Ai(G,5)) <5 fm < i < 4 To understand the definition above, recall that .4 1 (.0.1 ) is an open
9 1 9
Conlin mat invariance  5'ntit owl's Theorem
,Icadan cur ve in C In contrast, ttl i (a) ) is lo t wally a set of vertices of G . i a set of points in (51' C C: condition (18) cm ' he ha y ' mete( with this definition. Often, however, we shall view Co as a union o ,) is a piecewise linear curve in C. closed hexagons, so its boundary will IICVO hrthis case, A i (es) is naturally defined to be pat t of OCA) make a difference which definition we use: the two interpretations of :1;(96 ) give sets at Hausdorff distance (5/0 from each other, and the arguments that follow will not be sensitive to such small changes. Thus,: we shall feel fl ee to switch between these viewpoints without notice be a 1marled Jordan domain. Then. Lemma 14. Let some do = 60 (D ) ) > 0, there are families {G' s ,0 < cl cz a 0 } and 1. 0, T. 0 < C f517, with the following propertie s. As < 6.0 of discrete domains cS 0, the domains Gs art o(1)close to D.1. rS
Ds(
— o(1)
P,s( D .1 ) < 1?) (C ) + o( I),
(19)
and, given y> 0, there arc (5 1 , 11> 0 such that fur all r5 < S i , any two points n: and of Cif with dist (w, z) < q may be joined by a path in GI lying in the boll 13.,(w)
Iloughl ■ speaking, L.enuna 11 shows that. to in (WC ThCOlent 2, it suffices to Ivor k with discrete domains Mote specifically, using the fact that (Itis close to .D. 1 , we shall show that Pi (Cn) — FID4), w hich.
together with (19), implies 'Theorem 2 The last condition in Lemma If is a technicalit y that we shall need to ensure that certain functions go we shall define ate tutifin tid y equicontinuous as (5 varies The rest of this subsection is devoted to the proof of Lemma The construction of the domains (.7;t will be broken down into a se t ies of steps, and the proof that they have the required properties into a series of claims Before getting started, we present two simple facts about lot dart curves that we shall use Lemma 15. Let D be a fixed Ionian domain with boundar y such that if w, y E f and dist (.r. y) s> them is an. = ) > and q divide F lies entirely then one of the two arcs into which within the ball B (s') Proof
This is standard.
immediate Iron the fact that 1 = (IT),
72 &nil noMs Theorem
213
v iarre i is a 1to1 continuous Mal) horn the circle T into C Such a nap 7 is a homeomorphisur, so both 7 and its inverse ate continuous 'unctions on a compact, set, and hence uniformly continuous Alter natively, the fact that 7 is uniformly continuous implies that, given s > 0, there exist to = 0 < tr C • • • < < = 1 such that len , 7([T, t. i .r. 1 1) lies in some ball of radius s./4 Any twosetsI F with ] — i 0. ±1 modulo n are disjoint, and so separated by a positive y of F distance Hence, there is an r > 0 such that any two points or an adjacent arcs F 1 . F ) . In either within distance t lie on the same rise, them is an rue of F joining a' to y and lying within 8c(r). ❑ emma 16. Lc/ r 6c a Ionian chive boandinga domain D. and let. E D be freed Given 5 > 0, there is an = g(F.C. ^ ) > o with the following p i ppin ty far every point .r of 11, there is an .r' E D with dist (r. .11 ) < c that may be joined to C by a piecewise intent path P with dist(P, F) > Proof Let as p}(.r i ) be a (minimal) finite set of balls covering r, and Pick one point c t E 13,p)(x t ) fl D for each I, so every E F is within distance 5 of some .t t As D is a connected open set, each z i ma y be connected to C by a piecewise linear path P1 in 1) The minimal distance between the disjoint compact. sets P Ui and C is strictly positive, so there is an q > 0 such that the 2qneighbour hood of P is disjoint
❑
horn F
One can show, that. given a rinun lied domain D., and an > 0, lot sufficiently small S there are 1/ / ta/ ked discrete domains G:5 C ST that am sclose to D.:, such that any crossing of Gs in ST contains a crossing of D4, and any crossing of f).1 in ST contains a crossing of G,"riThis implies that P6(0,7 ) P6(D4 < Ps(Cr{ )
(20)
In fact, it will be cleaner to move a somewhat weaker statement, involving only crossings that do not pass to close to the /milked points Pi. This weaker statement does not imply (20) Howevel, we shall show. using LC1/11 / 1a 4, that it does imply Lemma 11, which is strong enough fon the proof of Theorem 2 The basic idea of the construction is as follows. If is a rectangle with the cm ners uni tized, then we nay take (CT to be a slightl y !tinge' and thinner 'rectangle' in the lattice, and Crq to be a slighth shorter and
214
Conformal Invariance 
no v's Theorem
fatter rectangle The case where D is a polygon is similarh easy The general case is not mate so easy: when we 'narrow' D in one direction we may end up with a disconnected graph, for example. Also, when we extend it in the opposite direction, we may bump into ourselves, and end up with a domain that is not simply connected. These are not 'real' problems, but, nevertheless, it takes a fair amount of work to overcome these difficulties. For the rest of the section, let D. 1 = (Pi)) be a fixed 4marked Jordan domain, with boundary F. Given El > 0, in the constructio that follows we shall choose other small quantities >
>
> 5.i > Er, =
where fi is a. function of so that 6; 4, 1 is much smaller than fot each i. In particular, we shall assume that 10005 i + 1 < say Let s t > 0 be given Our first step is to simplify the crave F = OD mac. the nmrked points Pi As before, A i = A i (D) is the arc of the Jordan: curve F running from P. 1 to Pi± i . The arcs A t and A 3 are disjoint, and so separated by a positive distance The same applies to A 2 and Ai,: Let Zo e D be fixed throughout, and note that dist(zo, A i ) > 0 for each Reducing if necessary, we may assume that. dist(zo, > 106 t , and that, ri.
E1,
dist(A .4 3 ), dist(A 2 , A. 1 ) > 106 1 .
(21)
In particular. dist(Pi, Pi ) > 10s, for 1 < i < ) < 4 Choose 5;9 > 0 with E2
5 7(r,
Ei
where T(', s) is the function' in Lemma 15 Thus, if to and z are two points of F with dist (iv, c) < then they are joined by an arc of F that remains inside M, (w) Let Bi be the open ball of radius am around Pi: and let C1 be its boundary Taking the indices modulo 1, the al c A i of F joins Bi to Bi .m, and so contains an arc F i disjoint from B i U Bi+ joining Ci to Ci + 1 ; see Figure 18 Note that for j = i + 2, i +3, we have clist(F i , pi ) > dist(A i ,A 1 + 2 ) > 105 1 . Hence, each arc F.1 is disjoint from every B1 . For each the set .4 1 \ F i consists of two arcs of F each of which joins a point P1 to a point at distance 69 from Pj ; this arc is thus contained in B_, (P 1 ) Thus, (F i , A i ) < Let be the 'simplified' curve obtained by joining P i to Pi arid P1+1 by straight line segments of length 0, as in Figure 19, and let D' be the interior of I" Let he the arc of F' starting at Pi and circling at PiFr, 69,
E,
7.2 Smilwov's Theorem
Figure 18 Th e
joining
215
to
Figure 19 'The 'simplified' curve r (thick lines), and its interim D'.. The points P1 arc the cent res of the circles C; with radii am (drawn tvith solid lines) The dashed lines ate the arcs of F \ F', each of which remains distance of some Pi , as indicated by the clotted circles..
and note that didri , Ai) <
(22)
for each i We shall modify r in a way that is analogous to replacing a rectangle by a slightly longer and thinner one The disjoint closed sets 1 < i < 4, ate separ ated by positive distances, so there is an Ea > 0 such that dist(r i ,F;) > 10E3 (23) for i
210
Cotd611nal en001 inure Smitnoe •; Theorem
Let s;3/3)/2.
( I)
Where. as before, t is the function in Lemma 15 Let N 1 = ft' ) he the closed s A meighboruhood of P i , and let acc (N1 ) denote its external boomlarw, i.e tire boundary of the infinite component, of C \ N 1 The Jordan curve er inds around F 1 . Flom (23), it does not meet: any other I .) , so it can cross only hr the line segments inside Bi and joining In tact, it can cross only the line segments L i and to RI and Pi+t respectively Furthermore, as Fe meets Ci and (7.';+1 at one point each. and lies outside Bi the curve (N1) meets each of L i and exactly once, at the points Q; E and (2 1; E see Figure 20 Hence, With dist((2e,R) = dist(C4,P j+i ) = E g D'(Nd consists of two arcs joining (2, and Cj i . one inside r and the other outside
r
r
Pignut 20 The curve 1 11 , (thick line) together 1Aith its £ 4 neighbourhood N1 (shaded) 'The extol nal boundary ir(Nd of N., crosses r at two points. the points Q, CI, on the line segments L, and 14 The ca l ve 51 is one of the (N,) joining Q, and Qf we take S, outside r for i 1,3. and two arcs of inside for i= 2 4
For i = 1.3. let .55 he the arc of D=c (AC) outside F', and, a little dissonantly. for 2,4. let Si be the arc of D'(Ari ) inside r (The external boundary of IV; = ft' ) is defined without reference to the rest of F'. so Si is part of the external boundary of N 1 even when Si is inside F'.) These choices for the arcs Si correspond to the operation of replacing a rectangle by a longer and thinner one, by moving the first and third sides of the rectangle outwards a little. and the second and fourth sides inwards a little We shall write R i for the region bounded Iry the closed (nuke formed
7 2 SinirtunCs Theorem
by Si . . and the two line segments of length of these curves
S.1
217 joining
the endpoints
Claim 17. 117e can paramelf Ltd: the open Jordan carves Sm and continuous injections sm. gm : (0.1] sm(1) C with !Mt)) alt t < fin. E Ca + , and dist( sm i (0) E C h
I;
Proof. Let tim(a) and i i (a) be palm/utilizations of the .lindan turves Sm I] and I'm traced in the appropriate directions. Define a map a : let Y„ be a point of I'm at distance [0,1] as follows: fin each a E TOO E Such a point exists as Si is writ of exactl y srr., hom X„ Note that the boundamy of Ni = j" tDefine o(u) be !Ijm(o(u)), to each 1. as there is a unique closest point of i (0) = 0 and o(1) of Q i and (X The straight line segment L„ = X„Y„ meets U only at its endpoints Also. the interior of L„ lies in A'm and hence in R i (As we Ca avetse S. on one side we have the unbounded component of C \ ) Thus, L„ separates Pi On the tithe/ side we have Ni and hence into two pieces The boundaty of one of these contains all points of Si appearing X„. and all points of L i appearing berme Y„: the boundary of the other all point's appearing after these points. As the line segments L. L„, cannot moss, it follmts that o(ad) > o(u) lot > a. So hut, we have found a win to hare Sm continuousl y so that a nearby point traces 1m monotonically. but not necessarily continuously We can trace both curves continuously by 'waiting' on Si whenever the nearby < point on I i jumps Mate fin i nally, wilting 0_(.) = sup{o(44): and 0 + 40 = inf {o(P) : > .r}. as o : [0,1] [0,1] is increasing. we can find conti n uous [Unctions a,(0 and a,(1.) horn [0.1] to [0.1] such that each a; is weakly increasing. amid
0.(11/(0) ^ //2(0 5_ (4+(11(1)) lot even a example, one can take u t (1) = sup{ : a + (A(d) < and a 2 (1) = ni(t). Let = ;i(a2(1)) and g i(I) ai(w2(1)) At points whew cr is continuous. we have la,(1)= 0(1 [ (I)), so, by definition of o, the points ,s;(1) and ni (t) ate at distance exactly EA As both T.Cim and in ate continuous. and each discontinuit y of a may be approached Flom both sides by points at which o is contitotous, it fellows that the points ij i (o4(u l (t.))) are also at distance exactly 5.1 from si(t). these points are within distance 2E,, of each Millet B y our choice (24) of :7: 4 , it follows that the ate of 111 joining these two points lies within
Conlomat Momlance Surrnoo',4 Theorem
9 18
a ball of radius E 3 /3 centred at either point:, and hence within a ball of, radius g 3 /3 centred at s i (t). As in (t) is a point of this at we have (list (il i (t), s,(f 177.1 s3/2
So fa t , the functions v .) are only weakly increasing, so gi and are not injections We can modify the tt ) slightly so that they are shied., increasing, tot example by adding a small multiple of un to tt, and vice versa Using uniform continuity, this does not shift the points 1/1(1) s i (t) significantly, and the result; follows. 0 Let; P be the closed curve formed by Sg, 8: 3 and S. together witl the line segments joining the endpoints of these curves to the points p and let D" be t h e interior of 1; see Figure 21
Figure 21 The curve F  (solid lines) and its interim IF The dashed li ale the curves Each Si is palt of the external boundary of N i tr. It ll : ha i = 1,3. S i is outside i 81 is inside r
Let IT denote the arc of P starting at P1 and ending at P1+1, so FT consists of Si together with two straight hue segments From Claim 17, can find corresponding parametrizaticms 1'; and FT that remain within distance particular, we
S3 fit
for 1 < i
4. Also, piecing together these pain/netrizationsrye obtain
219
7 9 Son; non's Theorem
an t ett izations of the Jordan cm yes 17 and such that c otresponding points ate within distance 5.3 Roughly speaking, we shall take CC to consist of all sites v of ST vhose corresponding hexagons H„ a t e contained UnfoitunatelY, this set; need not be connected, so we shall have to be mot e careful Suppose that 0 < 35 <
(26)
Et/10)
wLcre q is the (Unction appearing in Lemma 16, and let D (7 be the set of sites v E ST such that H„ C D  Reducing 5, if necessary, we may some lu that < 05 4 /100,
(27)
vhele 0 is the smallest of the angles at the cot nets Pi of 171 Recall that. ca is a (fixed) point of D. with dist(co,r) > 10n, i.e with (z 0 ) C D Thom the construction of 11 ', we thus have Biie ,(zo) C winds around zo, it follows that V also winds mound :9 , so D' As zit) E in fact. B 8 _, (cry) C D Hence, the hexagon containing cu is contained in DBegat cling DS as an induced submaph of ST let Cloi be the component. of D:1 containing (the site corresponding to the hexagon containing) cu; see Figtne 22 We shall take 67 as one of our discrete apptoxitnations
r
Figure 22 A pint of C;;;" (viewed as a union of hexagons). together \in.( ' auof hexagons contained in D  0:5 is other (small) component of the set the component of containing the point; zu
to I/ To make G s into a 1marked discrete domain (Gis : we take vi to be the VCI ex of 0,1 closest to Pi
1.
220
Confotwal
)11141 I )(Wet
Sitthlt017 ti The orem
Claim 18. Ire have d ji (OCC„F`) < si/10. Proqf Suppose first that :r E 00 ,7 Then ,r is incident with hexagons H e EC6: and H„, The hexagons and H„, ate adjacent:, and, H, E DI. so by definition of 0 611 , we must have H„. 0 D:.;" Thus, 11„ meets and dist (it, < 25 < £4/10 Suppose next that :r E By (26) and Lemma 16, these is a point'. x' E D with dist(e,,r) < £ 4 /10 and a path P joining to co with dist(P,1 1 ) > 361 But then P Os) C D contains a path of hexagons joining zy to 3.!. so x' E G7 As :c 0 D D 07. the line segment ;ri/' ❑ meets 067, so dist.Cr, 0067 < .s.. / /10 Recall that ri. 1 < r < ate the ales into which the points P1 divide so [7 stints and ends with shot t line segments slatting and ending at P1 and Pi _e j : the test of 17 is the onve Si As any point of Si is w ithin / distance/ E. 1 of U. it follows how (23) that if .r E mid y E 17 with i ) ate at distance dist(r.y) < 106, then, swapping .r and rj if necessa t y, we have j = the point :r is on the line segment of 17 ending at Pi.+1 , and y is on the line segment of 17+ scatting at P1,1 As these line segments meet at au angle of at least 0. how (27) both e, and y ale within distance 105/0 < se/10 of Pi . i e well within the ball 131
As is a component of the union of the set of hexagons contained iu a siumb connected domain. it is simply connected Let us Imo/ the bounda t v of 0 7: anticlockwise Each boundary vet tex v is within distance 2S of a point of F. which must lie on some If o' is the boundary vet tex alto c, then a' is within distance 25 of a point of some IT from the remade above, if i j. then = {/.k± I}. and .1. y E B.,;,(124.), say hi ti the/ wonls. the closest boundary me r7 can only switch when we ate very close to some Pt . But in this legion. we know exactly what r looks like: it meets 13,„,„(Pk ) in two line segments From Claim 18, the boundaty of 0:11 comes within distance £ 4 /10 < £3 of a Hence 00W meets B,„(Pk ) in a single ant, and, as we trace the bomicht l y of OCC, the closest curve r7 switches horn to r I when we trace this ate Recall that c i is the closest vertex of 0 s11 to the point P1 0 CIA: Thus. v, is a boundar y vertex of 07 lying i t / /3, 00 (Pi ) Let .17 = 1 ,2 , v3 , ) be the bouudanv tuts into which the c 1 divide the boundan of QC.: Clain ' 18 and the comments Ame imply that d it (Ar. [7) <.z. i fth
(28)
'7 2 Stun nov s Them
CM
Using (25). it fellows that d li (AT F) < 2E 3 Hence. for 1 < i < appealing to (22). .11) < 2E 1 (29) abet words. C.;;; and D, 1 are 261close Claim 19. Let P he a crossing of 07. a path in 0:5" from Ai to AT If P does not pass within distance si of any Pi , then P contains a crossing of D Rom A t to A3 , and P inte i sects any crossing of D from • to A 4 that does not pass within distance ,;t of any Pi Proof. Let ,r E AT and y E AT be the endveilRees of P As :r E 0:7 the point r! lies in D, the intetiot of From (28), disti(x,r7) < e4/10 As :c is far horn Pi and Pi. it is close to a point of S i , and outside D Recall that R i is the domain in the complex plane bounded by and ri We have ,e E P t arid y E 13 3 . so P joins P i to R3 Let P' he a minimal subpath of P joining f i t to P3 Mien the first edge of P' leaves R I , which it must do across ri and hence across F 1 C F. entering D Similatly, the last edge of P' flosses 17 3 P can only cross F' on the boundary of R i , f?3 , so the rest of P' lies inside D'. Far limn the comers Pi , a point is inside D. i e, inside if and only if it is inside 17 Thus, all other edges of P are inside D. and P' is a crossing of D from A I to A3 mw mussing of D horn A, to .1 4 retraining fat from all Pi includes a crossing of 0,7 hour AT to AT which. by Lemma 5, must meet P ❑ Let 0;7 be defined as 0T. but with all subscripts cveled, so ice take and 84 outside F'. and 8 1 and 84 inside r
89
Claim 20. Let Di and E > 0 tie given IfE i mid CIW 07 Ulc constructed as above. then
>0
is chosen small enough.
Pitt()  E < (DI < &tact)+ E ate separated by some positive distance Opposite arcs of Proof c > 0 horn (29), it follows that if Er is small enough, then any pair of opposite arcs of D4 , 0T or 0;1 are at distance at least c/2 From Lemma A, it follows that if El is small enough, then the probability that am of these domains has an open crossing joining opposite arcs and passing within distance E l of some Pi is at most s Assuming no such
999
Conformal invariance Smic no t t's Theorem
crossing exists. then by Claim 19 any crossing of G7 front Ai to A. (losses D, 1 front A i to Aa, so R5 (D. 1 )
Ps(C; ) —
The second statement follows sit/Wally.
0
The final property that we require of out domains Gt is that nea t fps. points may be joined by a shot t path in the domain We prove this for Gs; the conesponding statement for G ict then follows by per tnutit i g the marked points R. Claim 21. Let y > 0 be given Their is an q = q(D4 , 1 ) > 0 with th following property. If E l is chosen small enough, then any two points w and z of C 5 with dist B., (w).
Z) < n OIC
:joined by a path. iu G7 lying hr
Proof We shall assume that ti < 7/2
We may assume without loss of generality that m and z lie on the boundary of Gs (viewed as a union of closed hexagons) To see this, consider the line segment wz If this lies in 67, we me done Othet wise, let x and g be the fist and last points of this segment on the boundary of so 'on'. gz C G q Then disiff, y) < y < 772, so it suffices to joins and y E OCC by a path in 67 lying within 13_ /2 (y). saw Thus, Claim 21 fat tv, z E (16 follows front the same claim fit iv', E with 7 replaced by I/2 We may also assume that the line segment tvz does not meet OGri again: other wise.. listing the points c1 in which this segment meets 00,7 in order, it suffices to connect each to the next by a path in Thi n(ri ) C B, (w) Then the union of these paths contains a path in B,(w) joining ay to We shall choose 11(.0.1 ,7) .50 that g < 7/100 and 5r/ < r (F. 7/3),
(30)
where r(r ,E) is the function appeming iu Lemma 15 We shall choose ai < If the line segment iv: lies inside Gs, then we ate done, so we may assume that it lies outside As tvz meets the simple closed curve 8(77 only at its endpoints, it divides \ GIs into two components, one of which is bounded: see Figtue 23 Let R denote the bounded component. Let B be the at c. of 06T which. together with iv:, bounds R If B C L32(w),
7 2 Sunni
Theorem
99:3
Figure 23 Pmt of CC (hexagons). including two points at and z on its boundary with dist(w, z) < 'Hie shaded legion R. is bounded by B C OCC, a piecewise linen/ curve joining w to z„ and the line segment The curved line is part of F  ; the point :r' E is within distance 25 of the point x E DC", and B C I? is the arc of from le ' to z'
then we ate clone, since B C joins ar to c Thus we may assume that d ime is a point ,r E B wit h dist(t, > 7. One of the hexagons meeting at say H1 , lies in U. while anothet, flo, does not, and so lies in 17; the hexagon tr■ is drawn with dashed hues in Figure 23 As Hi and Ho are adjacent, it follows from the definition of G ,7 that 17 . meets Ho. at a point x', say Let w' and z' be the points at which we leave 1? when we trace the cm ye F front a' in the two possible directions, and let B C 1? be the ate of E horn Iv' to z' containing x' As r cannot cross 80:5, the points t ri and z' lie or/ the line segment to:: Iu particulat , dist(uV , c') < As noted above, it follows front Claim 17 that we may parametrize F and so that co t responding points ale within distance 5. . 3 < E t Rom the construction of f" (see Figure 19), it follows that we may parametrize F and r so that cot responding points ale within distance 27 1 < 2q Let w" and z" be the points of F cottesponding to the points ed and c' of r Then dist(w". z") < 5q, so. IA' (30) and Lemma 15, one of the two arcs of E joining w" and z" lies in 13,)/3 (17" ) The points of I"
Confot mal in vat iamt Suth6ov s 'Thcorem conesponding te,/ this ate Mini an am B' joining er/' and lonmining within 1.3_,,,3±,:,(6 ") C d o(w) This ate cannot be 13  , which emit /tins the point: 2 r 13. 1 ,(r) Thus. B  U = [The calve B  U /IC:/ / is contained in B. and so does not wind mound curve 13' U tv'z' is contained in B,(w). and so does not wind atoned r Hence. I3 U B' = P c does not wind mound x, cunt /Mid ing E
Cri T
CD
❑
The pool of Lemma 1/1 is now easily completed It remains to cuisine that CI ,7! and (.7i" satisfy all the conditions in the definition of a discrete domain The conditions that the domains and theit outer Imundmies have no cut v e hex 11110 he et/Fenced by fit st teplacing tS bti et/10, and then I mincing each hexagon by a 'hexagon of hexagons" As noted eat Mu the condition that each marked vertex has at least two neighbours outside the domain can be ensmed by moving each walked vertex at most one step nronnd t he boundal v Proof of Lent Illa .14 \Ve have shown that given a small enough Ei > 0, = E5(EI). the construction tot described abm, e is valid lot all it < Considering a sequence of Values of m O. we may thus pick domains Cit lo t all rt smaller than some no in such a way that Cs is defined using h m(rr) — 0 as el — 0. a value s 1 (6) The tcyuir etn ent that CST and /7 1 ale o(1}close as — 0 Inflows nom (29) Condition (19) is given In' Chihli 20, and the final condition of Lemma 14 is gum/it/O'er! by Claim 21 ❑ Let Os I emai 1r that the proof of Lenuna 11 shows that when defining Pi ( /7/, TL it does not inattel exactly how we neat the houndmv proof of SmirnoWs Them cm will be valid tot any definition of a c t ossing of D., lin \vhich a flossing of the 'longer. thinner' domain C:;" horn 01 1 (LC ) to ../1/ ! (CC) t hat stays tar ham t he cot nets guru antees a crossing of D., how .ffl to 21 3 . and prevents a mussing of DA 110111 1 2 to .4 I . FOr example, we could define a ("tossing of DA 111 ST from A i to .4 ) to be open path in D Ma/ling and ending at points within distance 1051 of Ar and .!1!
7.2,6 Completing the Proof of Smirnov's Theorem Let D4 = (D: P). P./ . P i ) he a 1ntai keel hot dan domain. rind as berme let 7(D 1 ) denote the limiting ctossing ptolabilit S lot D I predicted by Ca t ilv l s fonnula Thus 7(/). 1 ) is a cottfbnual MN/a/iron of D I which.
7.2 Smirnov's Theorem
225
following; Ca t lescm„ may he defined as follows: let be the unique conto the equilateral triangle that maps k/rut/II map from , P2 and 8 to the vertices (1.0). (1/2. fil/2) and (0.0), respectively, so /naps p, into a pout (r, 0). 0 < ar < 1 Then rc(Ds) = Let C ST be the discrete domains approximating Ds whose existence is guaranteed by Lemma 14 Note that there is a function E = 0 as r5 r 0, such that with s
0:sl. is aiclose to Ds
(31)
Of coutse, the same holds for Cl,t, but, as we shall see.. it suffices to consider G's. To prove Ibeotent we shall show that
Ps(G)
rr(D.1 )
(32)
as /5 0. where Ps(CCir ) is the pH/It/Wilily that the 4marked discrete domain contains an open crossing joining its first and Hind boundary at es. Indeed, (33) P,s(Gt ) r(D1) can be lamed in the same way as (32) and then P i (Dr) 7i(D4) follows front (19). In fact. WI iting = P2 . P3, P I , PI ) lot the dual' marked domain. the consti net ions of the apptoxiination CI" to D4 and the apt/it/xi/nation 61,1 to DI given in the previous section ate identical Hence. (33) follows hum (32). the /elation m(D. 1 )+ 7,(D1) = 1 (see (4)).. and Lemma 5 ha this leason we consider only C;" Tot the moment, we shall regard the domains D., = (.D: Pt , P2. PI) and 0,T (67: e t .r u:t vs) as 3mallred, by fmgetting the fourth mailed point.. Pi of v., We write = .4 i (D// ), 1 < i < 3, fig the boundary mcs of the 3tai Iced domain .D 3 = (D; P}, P4 ) \\'e use corresponding notation 16/ the brt/Id/ay arcs of 67 Let f,l(z) he the 'et ossingunder' probabilities defined by (9), for the domain Gs = = , 173 ) Thus, lot z the centre. of a triangle in CI . th e have C5(z) = P(E75(.:)).
where Es(z) is the event that these is an open path in 67 born = A 1+1 (67) to .:4; 4.2 separating z horn Ai (an arc of the outer bor/ditty of 67) Hoe, the subsoipts ale taken modulo 3 Let us extend to a continuous function /115 on the closure D of D as follows Recall that is defined at/ yin ions points of i e..
226
Conformal Maw lance S t mirnov'a Theorem
the centres of various triangles (faces) of the lattice 81/' At the centre E 611 of any face of 45T meeting D. set: g,C(z) equal to the value of JA at the closest point ay where /,( is defined. (H there is mote than one such iv, choose one in an arbitrary way ) Note that, from (31), we have dist (II'
< E +
26.
(3
say \\1e extend hour the centres of the triangles meeting .D to the entire' triangles as follows: first, take the value of fio at the corner :r of such a triangle to be (say) the average of the values at the centres of all triangles incident with a and meeting D. Then divide each face of ST into t lace triangles meeting at the centre, and define chi on each of these triangles by linear interpolation Finally, 'forget' the values of 9 i5 outside D to obtain a function with domain of definition D \Ve shall Use two proper ties of the interpolating functions f./15. each !Ali is a continuous function on D Second, for any and c E D. there are points le t E 811 at which fA is defined. with 115 (w) < gfc(z)< fRui f ). and dist (net„). dist (iv', z) < 2E.. where = 5(5) is as in (31) The first property is inunediate from the definition of q The second follows Prow (34) Claim 22 The f (Indians
MY'
uniformly equiconlinnowi
Proof It suffices to show that t he functions fL ii ate tudIcainly equicout iummns Given ii > U, we roust show t hat t here is an q > 0 such that. for all to EDCC With dist (c. u . ) < q Mid all 8, Ale !Mee (c)— (415 001 <.1 ht doing this, we may choose an) 6 1 (ii) > 0 and restrict our attention to the functions for S < 11 1 (0): whatever the values of (z) tSr (it) the linear inter polation in the definition of id ensures that the > functions gA. e > 0r ( i3). are uuiforruhy ecpticont imams (indeed. indica rub, Lipschitz) As no point of the 3marked J o rdan domain 3 D3 lies on all three arcs = (D3 ). there is a constant e > 0 such that
max dist (II', A i (D3 )) > fi n
(35)
aro. rc E 1) Let us choose 'y > 0 so that < c/ltl and (63/c)'< d/2. where o > is the constant in Lemma I By Lcuuua 11, if we choose Sr small enough. I Iwn t here is an I/ > tl such that any two 'mints g of GT. 8 < S I . at
7 :2 Sminuols Theorem
227
distance at most q are joined by a (geometric) path in G il that stays within tire ball 13s, CO Reducing 6 1 , if necessary. we may assume that for all 6 < 6 1 , where 6(5) is as iu (31) c(6) < Suppose that 6 < 6 1 . and ar z E D with dist(w,z) < ti/f. It suffices to show that g,15 ( z ) < 0, 15 (w) +13 From the second property of the interpolating fitnctions q listed above, there are ///, z' E 67 with ( to') and dist,(tu,a0 < ti/4 and dist(z, z') < 0/4 such that gA(tv) > z' may be < 0(z)). Note that dist(a",zn < q The points 9,i(z) joined by a path in CI; lying within 13,(u') As (715 is 2connected, it follows that there is a path P' E 811 from w' to z' all of whose vertices are the centres of triangles iu CT. with P' C B22(1/P1) Suppose that EA(z') \ EA(iii') holds Then there is an edge :ry of P' such that EA (0\ EA (x) holds. But then, by Claim 10, the three sites of 07 immediately next to :c are joined by monochromatic (i.e all sites open, or all sites closed) pat hs to the three boundary arcs of CT One of these paths must: end at a distance at least c – – > c/2 front ni', say, where e is as in (35) Note that dist(r, w') < We have shown that if EA (.2')\ EA (tv') holds, then there is a monochromatic path crossing the anutilits centred at a / with inner and outer radii 3 7 and c/2 By Lemma 1 and our choice of 2.. this event: has probability at most
2 Hi It follows that
–
' ') < ci Thus. (a
< /Pc ' ) C ( 0'1 ) + <
+
❑
as required
The functions ifs take values in [0,..11 Hence, these Functions are uniformly equicontinuous and uniformly bounded Thus, by the ArzelftAston Theorem (see, for example, Bollobtis [1999, p. 90]), any subsequence of (g, A, q) contains a subsequence that converges uniformly continuous to a limit IP), with each The last two (rather substantial) pieces of the jigsaw puzzle needed to complete the proof of Theorem are collected in the next two claims The first describes some crucial proper ties of the possible limits (g 1 . 0 2 . ,q 3 )
Claim 23.
Suppose, fa, sonic sequence
 0, the G inks (11
228
C'onfot mat mew lance
converge toilfu l oily to (g 1 // l1 ) any contom C in D. we have
nov 5 l'hemmti
avtk each
oE
5C1
Then
or
(36)
A i we ham
11'(o)=
who du 'opal
coalMoon
g(c)do.
g i+I (o) do = w an foi (my point
f
0
and gi+1(z)1
(0)
= 1.
(37),
ipts are taken modals 3.
Proof Thioughout the proof
WC consided onl y values of in the sequence /5 lot d„ to /noid nimbi/is/Atte notation We stmt with (3(t): since the g i ale continuous functions on a compact set. they ate unikandv continuous Thus. it: suffices to consider equilateral triangular contours C with sides parallel to the bonds of T: an arbitrate contour C' in 11 call be approximated by a sum of such
SS„. wilting
CO/MOMS
Let C be an equilateral t tiangulin contour ill D with sides parallel to t he bonds of T Poi each J there is a discl et e triangular contain Cs in ST tc id lin distance ci of C. Using (31). as C is contained ill t he open set, D. lot ri sufficient1Y 5116111 we Iliac C C7 ,c Recall that the discrete contom chi? [ 15 (i:)d:: is defined IA int grating a function uhosc /dues al given IA values of at lattice points wit hin dist mice and, at these lattice points. = rhii Since t he fittict ions ti,C a t e unifin nthequicontinuous. it follo/‘ s hat /)
dz
!Edo)
o(I.)
as S — whete the second it/leg/id is t he usual contom ioteguai Since the ri1c = q converge indlo t l i th to we have ( lc as 61 = d„ — deduce that
r
=
(I) f:/(:)(1
o(1)
Combining the relations /Write with Lemma 13. we
0
g'(1)
d o
0(1)
Since licit het integral depends on S. this proves (3(1) FO 1 (37). it suffices to prove the case i = 3. sad, flu a fixed o
E
7 2 Smil noWs Theorem Since the g of .43,
at e
229
continuous We mar assume that is not an endpoint disc(_. .*1r). (list (c.. A 2 ) > a
(38)
for sotuw (I> 1) 0. the 3marked domain (3iis .sclose to D 3 with E = E(6) — 0 As 6 Shifting GITT with co e D and co — Hence. thew a t e points E 8H is the centre of a ttiangle 1w at most 26. we mac assume that uniformly, coo fro in C1sThus. f,((cs) is defined. and. as !LI — latch)
fhis( c ) = g (c ) + o(1)
(39)
Is — 0 Flom (31) we may choose a porn ft's On the 'mutably of CT,T (viewed and hence as a union of hexagons) so that, its (5 —+ lb we have ws first (ms, c o) 0 Let :I's E C',.,1" lie a site of the inner houndm v of Coll in whose hexagon tux lies, so (list (,ro. ws) < 6 Note that it c = P l . then we may take rs to be e 3 The :Hausdoi II distance between the discrete boundary arc A;(67) and the emit:imams air A i (Da) tends to zero 0 Thus. from (38).. lot 6 sufficiently small. the site lies on the ate A3 of C1,7 (viewed as it 3marked domain) and the point: ws on the cottesporaling ate of the bow/du t y 30W of the union 0,7 of hexagons Suppose that D:51 (.7. 3 ) holds Thc‘n time is an open path limn Al(0,7) limn At (Gill ), and, in pal t Rada', sepal ating to .'10(03l ) sepat ating Such a path must moss the line segment: cows It follows that horn ws some site of within distance (list(:,, ws) + 26 = o(1) of c 3 is joined by au open path to snow site at distance at least a — o(i) hum c3 this event has mobabilitv o(11), so Lemma =,i) = 111'(El i3 (c 3 )) = o(1) Using (39). it follows that cia (:) = Recall that 3! 3 is a lumndat V site of A 3 (0) within distance 6 of ms Let Es be the event that some site in Baistfrc ?co+ t oc(zo) is joined by an open path to A i (CI3 )U Aii,(CC) As above, LP(E3 ) = obi ) Let 0,15 be the 4marked domain obtained front the 3marked domain 03 l by taking to as the 'Muth marled point Note that if c = so ,r 3 = el , then we recover the original Imarked domain If E3 does not hold. then no open path horn A t (gli ) to A;1 (0) can tome within distance 26 of the line segment /tows. Hence E,i(z:s) holds if and only if Cfs has an open (dossing h om A [ (Cs ) = (0;C: ) to .13(6"3): see figure 21
Conformal
230
ar lance  Mirwou's Theorem rs A!,
Figure 24 A 3marked discrete domain 61 1,1 with boundary ales A i , 1 < i < 3, and the corresponding 4marked domain 0,1j with boundary 8104 < i < 4, obtained by marking a point ass on .4 3 . If no open path joins A I U An to the line segment ,rozo, then an open path from .1 ' to A3 separates z,) from :At if and only if it ends at a site of AC,
Hence, P(Gs has an open classing limn A', to A'a ) = d(zs) + o(1) (10) Similarly. if E6 does not hold, then EA (As) holds if and only if C's has an open crossing from ,41, to .4 ,',, so has an open (flossing from A1, to A',1 ) =
(zs) + o(1)
Using Lemma 5. it follows that fd (z6) + ffi (7. 6) = — o(1).
Appealing to (39) again. g 1 (z)+ 92 (z) = 1 follows, completing the proof of the claim ❑ As before,
+ fr3)/2 be a. cube root of unity
Claim 24. Let A he the equilateral triangle with vertices n i = no = w and va = w2 . There is a unique tnide (2 1 ,62 , y 3 ) of continuous functions and satisfying (36) and (37). MU theunore on.D taking values in ( 0, y i (z) = (:z(z)), where h i is the lineal function On A with // 1 (z) equal to twothirds of the distance from z to the ith side of A, and y is the unique conformal map from D to A whose continuous extension to F maps Pi to ye for 1 < i < 3 Proof Let (0', r?, q 3 ) be continuous functions on D satisfying (3(i) and
7 2 Smirh an's Theorem (37), and set = t +0_,92 contour C in D we have g =
9 +
g2 + w 2
w293
g3
231
Then g is continuous on D. For any 91 +w2
gl (a2.1
g'= 0.
ice, by Morera's Theorem (see, e.g Bear don 11979, p 1661), the g is analytic in D. unction f On the b011./IdatY of D, condition (37) ensures that, fors E A i , the value of g(z) is a convex combination of co 1+ r and44) 1+2 , i.e., that g maps into the line segment ig 4. 1 e i+ 0 Since g is continuous, it follows that g maps Pi into Furthermore, as z traces the boundary of D in the anticlockwise direction, g(z) remains within DA and winds exactly mice around this boundary As noted by Bettina (2005], it follows that g is a conformal map from D to A Indeed, applying the argument principle (see, e.g., Beardon (1979, p 1270, the equation g(2) = w has a unique solution for each w E A, and no solution for w outside A Since g maps the craps g and y are identical Pi to Arguing similarly, the function + g2 1:g3 is analytic in D. continuous on the boundary, and takes the constant value 1 on the boundary Thus g2 + ga is identically 1 This means that the realvalued functions g' are determined by g: for example, (201 — g2 ga ) / 3 + (g 1 + 9 2 + ifs)/3 = 211.e(g)/3 + 1/3 As g y, this shows that the triple (g I ,g 2 ,g3 ) is uniquely determined. The triple given by g i (z) = hi(c,....:(z)) satisfies conditions (36) and (37), so the claim follows ❑ We now have all the pieces iu place to prove Sinn noy 's Theorem Proof of Theorem 2 Let P3, P4 ) be a 4mmked Jordan domain Let (.7:5 and G7 be defined as in Lemma 14, for all 0 < S < So, 7r(D4 ), since where So > 0 is constant It suffices to show that Ps(C7) the same argument shows that Ps(G,I) ir(D4 ), and then .P,5(D4,T) g (D.1 ) follows from (19). Let f‘ and g,is be defined as above, using the discrete domains 67 We claim that /4; o y, where h i and y are defined as in (Hahn 24 Suppose, for a contradiction, that this is not the case, i.e that there is there are 1 < e < 3 an E. > 0 and a sequence (50 such that for each and 2 E D with ^
(41)
Co Ilia ! Mal 1111 , 01 )(HUT S1011101 s Theorem
232
B y Claim 22. the functions g are unifounk eqificont Moons on the compam set IT Since they ate also unifinink bounded the sequence ,, , has )1 1111ilin ink COMV/ gent subsequence Li t Claims 23 and 2[1, this subsequence CO/Wetgem to (h i 0 y.11 2 0 ;. // 1 o y) contradicting (A ) The proof of Claim 23 slams that these are points c, [ n with — Pr such that f
,14
)
P,;(67 )
(cs)
o(1) =
o( 1) 1 ) = 11 2 (y:(P.1 ))
o( )
Indeed. the first equation is exact Iv (10) For c oo PI : as noted Mat\ e. the domain Cr[:, appeal ing in (10) is exact IN t he 1m a nked domain 0:[[ in this case As 11' (y(1). 0) = 7(1 ) 1 ) by definition, t he proof is complete ❑ Site poi colatimi on the tiiiiiiLmtm lattice is t he mils standard pc[ i colat ion model For g hick toilful/nal invariance is kiwi\ iv. tittle MC SOUR' 110/1standaid models to \\ inch either Sinn nov's Themem. or its pub! has been adapted Indeed. Canria. Nelk111811 and SidthilVitillti [2002: 2004] haVe ', 1 St ithiitiii0(1 t ith i lismal invariance certain dependent site peu blat ion models nu the tiiaugular Lattice, obtained l, l imning a patIicnlur deterministic cellular automaton horn an initial state given 11\ independmo site percolation The\ establish continual invarance by showing t hat the dependent model is hi a cethriu sense a small per tin bat ion' of t he independent ittoth . 1 and t hen apply ing Themem Chaves Lei 120061 defined a Lather unusual class of percolation mids based on dependent bond percolation on t he I iangulal lattice. mid muted the equiirdent of SmitnoCs theoreur t o t these models by t I anslat ing Stint Smituoc s pool to this context ft remains an blip)! ant challenge to move conEumal invariance for am other standard model. lo t example. lo t site or bond pet colation on the square lattice. or lot Gilbert 's model or random \Mir : Hun percolation in the plane (see Chapter SI lot
7.3 Critical exponents and Sell/ amin—Loewner evolution Thee is a widel y held belief t hat the behaviour of various phenomena'. inducting critical percolation. should be chat arm/ ized by certain `ethical exponents" This opinion originated among theoretical physicists hilt by WA\ mnuy mat hematicians hale been come t red Pot percolation. these et itical exponents should (10 1 )0 1 1(1 on the dimension. but nut on the details 01 the particular nrodel considered. T[his is a eels
7 3 C ritical exponents and Schramnistocuinel evolution
233
substantial topic in its own right; a detailed discussion is beyond the scope of this book Hew. we shall briefly state the main rigorous results for percolation: the existence and values of these critical exponents for site percolation on the triangular lattice follow from Smitnov's Theorem and the work of ',alder. Schramm and Weiner. Our presentation is based on that of Smirno y and Werner [2001]: we refer to the reader to their paper for further details and full references To describe the exponents associated to critical percolation we need a few definitions. \Ve shall consider only site percol a tion on the triangular lattice, although the definitions make sense in a much broader context We write F1, 1, for the probability measure in Whiell each site of the triangular lattice I' is open with probability p, and the states of the sites are independent As before. we write Co for the open erratic; of the origin, i the largest connected subgnaph of T containing (I. all of whose sites are open Fin the critical exponents. we use standard notation: see, e g Reston [1987c] Recall that the percolation probability 0(p) is defined as = fril i ,(C la is infinite)
= 1/2. while 0(p) > 0 if p > 1/2 and that 0(p) = 0 it p < general set ting. the behaviour of 0(p) rrs p It is believed that. ill a that i tends to m hom above follmis a power 0(p) = (p
p()4*()(1)
p
p, limn above.
(42)
where ./.1 > (I is a constant that depends on the dimension but not the details of the petcolat ion model Tinning to the size of the open cluster Co when it is finite. as earlier, we write V (p) = Ep (lCal) for the expected size of (number of sites in) Co As k(p) = x for p > mr . it is rather mote informative to modify the definition slightly.. and consider \ f (P) = E a
so that
\ f (p) = A(p)
( lCallacokic)
tP,,(IC01
if p < pr Again, powerlaw behaviour is expected:
\ I ( p ),
_
as p— pc,
(43)
where is a positive constant. and p p, bran either side. Fru theimme, at the critical probability. the tail of the distribution
231
Conformal hrouriunce Smintov'S Theorem
leol is expected to follow a power law: (it < ro l < oc) = a1/64am as
11
as is the tail of the distribution of theradius 1(C0): (11. S
r(Co) < oc.) = n 1 / 6 , +00) as
a
In the last definition, it makes no difference whether we measure the radius r(Co) of Co in the graphtheoretic or the geometric sense, since these two metrics are equivalent More precisely, we may take r (C0 ) to be the maximum graph distance of a site at E C0 from 0, as before. (a we may take r (Co) to he the maximum Euclidean distance of a site in from the origin For site percolation on the triangular lattice, we have 0(pr ) = 0. so the condition roj C Do can be omitted in (bl) and (15) The ear relation length describes the 'typical radius' of art open cluster: writing for the Euclidean distance between rr E T C C and the origin, lei Co
1/2
5(M ,01)),E7
1111 2 11',({0 —
rol < x})
n
It is conjectured that = IP — Pc1 1 '
for
P
(1G)
The reason for the term 'correlation length' is that ti(p) is expected to be closely related to the probability that two sites at a given distance 1. arc in the same finite open cluster: roughly speaking, this probability should decry exponentially with WE(p) Finally, it is expected that, at the critical probability, we have P P ( 0 
(I) = i41 2d11+"(1) as
(IT)
where (/ is the dimension (so (I. = 2 for percolation on the triangular lattice ) The constants (.3„ 7, 6, (b., v and ri defined above are called critical exponents, provided they exist We have used standard notation for these exponents (though 6, is also written as p); the !bun of (47). example. shows that this notation is not the most natural for percolation. In two dhnensions, it is not hard to deduce horn the Russo SevinottrWelsh Theorem that it one of r l and 6, exists, then so does the other, and d2hq=216,
(48)
235
Y3 Critical exponents and .Schramm.Loeumer evolution
' Aced, let x i , ,1], be two points at sonic (large) distance r II there is an open path joining II and to, then each,r; must be joined by an open path to the boundar v of the ball B, i3 Ur 0, say But with probability bounded awa y from zero there arc open cycles in these two balls surrounding the centres that ate joined to each other; relation (IS) their follows using Ian is's Lemma (Lemma 3 of Chapter 2). Mester' [19871; 1987c] established highly nontrivial relationships bemen the various exponents for twodimensional percolation. Firstly, building on Iris work on the 'incipient infinite cluster (Kesten (19861), exists), then so does 6', he showed that if ti exists (or, equivalently. it Fur then/lore, Ire showed that if the exponents ri and 11 (5 ± 1) = with S exist, then so do the other exponents, and Ij In pa/
2 1/ 6I 1 '
=
6— I + 1
sind "
±I
for twoditnensim al percolation the 'scaling relatio + 2,3 = 0(5
)
and
= ti(2
1/ ),
ns'
(J19)
hold, as do the '111 T elsealing relations. dO, = +
I
and 2 q
I
— S+
(50)
It. is believed that (19) holds in all dimensions, and that (SO) holds fo r d < 6: see Chimmett [1999] A yet y° large number of papers have been written about the ethical exponents associated to percolation aud the relationships between them: see, for example, Rudd and Frisch [1970], Wu [1978], Kesten [1981], Aizemnan and Newman [1981], Durrett [1985], Chayes and Chayes [1987], lasaki 11984 Mester and Zhang [1987], IKesten [1987a: 1988] and Hammond [2005] Burgs, Ch ryes, Kestim and Spencer [1999] proved the deep result that the hyperscaling relations (50) hold as long as two assumptions are satisfied: 5, exists, and, at p = N. the crossing probabilities for cuboids with fixed aspect ratios are bounded away from I. (Their results are stated for bowl percolation in Za , but proved in a more general setting.) The latter assumption is expected to hold for cl 6 Bettuning to two dimensions, Selo anun [2000] studied a certain scaling limit of 'looperased random walks' in the plane, which we shall riot define. He defined a family of random curves iu a domain in the plane, whose distribution depends on a real parameter which he called the stocli astic Locivrod evolution with parameter h, and denoted SLE„.. The
230
Confirm& i mam iance 5'mirnotc.s Theorem
random cunt S L E„ is often known b y the name Schwalm Locione i con: lotion. SatWI W I showed that if. as conjectured. the scaling limit of the looperased random walk is confounall y invariant. then it roust be SLE), Fun thermore. he showed that SLE6 is the only possible cordon wally in_ variant 'scaling limit: of el itical percolation on a lattice in a sense that we shall not untke precise Stith nov [2001a] proved that critical site percolation on T does indeed have a scaling limit. and that this limit is conformalle invarinrut and thus equal to „SLEn Considering the face percolation on the hexagonal lattice corresponding to site percolation on the triangular lattice. it liAlows that. for p 1/2. the longrange behaviour (i.e . limiting helm ion ic as the lattice spacing tends to zero) of interfaces between open and closed (black and white) regions converges to SLE1 In a series of papers Lawler. Schramm r i nd \Ve t net [2001c: 2001d; 20026; 200221] studied the behaviour of SLE: ire particular. the y deter ironed Nal •cl it ical exponents' associated to SLEi ; Combining these results whin those of Slid/ /10V [200!a]. SLIM mixt and \Vet nett [2001] established the existence and values of the various critical exponents fin sit e percolation on T Theorem 25. Poi site parrolation on the biangulat lattice the critical opponents 3. ; ti and q di/toed implicitly by(42). (13). (.16) and (.17) and flu' inners 5
13
=
15;
4
5 and= TT
0
These values for the critical exponents coincide with the predictions of theoretical physicists: see Kesten (1987c]. Sion 110V and \Victim' [20011 and the references therein As noted above. 6, = 1112 = 5/48 follows relatively easily, and d = 91/5 161 IOWS ftun a the results of [(estctu [1984 The proof of Theorem 25 is based on crossings of annuli lb sa y a few words about this proof. denote tu ALt the event that in the site percolation on r restricted to the disc /3 R (0), there are (at least) distinct open clusters each of which connects a site in B, (0) to a site near the boundary of B E (0) For fixed i. the asymptotic behaviour of P IT:, (A; ??) does not depend on as long as n is large enough for tins probability to be positive \\:n it ing A llt for rrü, ??. say, S i nn no\ and \\hiller noted that. b y the results of Kesten 119874 Theorem 2,5 fellows horn the tur, relations P1 / 2 ( A IR) =
(51)
.1
ideal cdpoilelds and Schramm doewnci cuolution
237
wd (52)
P l/ q (1 ;?) = R5/
exists and takes Clearl y . the [list of these total ions states siniph that, the value 5/iIS Lawler, Schlanun and \ Vet net (20021)] showed that relation (51) Eelhaws flow SItlit 410V ' S COltrOl inal 'mini lance tesults and then results on SLE6 Smiinov and Wei net (20011 then punted (52) and thus 'Theoem 25 In fact_ Smir uov and \Venni' moved a none genital statement about ct ossings of the tumulus A); 1?) emitted tit the origin. with inner and ante! tadii / and R respectively As helmet we think of open Paths as black and closed paths as white Given a sequence c = (c;)1=! E ,If of colones. let H,( 1 , I?) he the event that A(t. I?) contains j vertexdisjoint monochr intuit ic paths P i whew P1 has cutout et, each 1); starts at a site of .4(7 1?) adjacent to the inside of the annulus ar id ends tit a site adjacent to the outside. and the initial sites c i al the c((, , v 1 around the cliche of tachus P1 appeal in the cyclic oi del (As the paths do not (loss.. then final vertices apnea/ in the sane cyclic order al ound the outer tit cle ) Let Ge (1. I?) he the emus corr esponding to 11(1 17). hut defined in the hull'annulus =
E C : t
<
< R lite(z) >
It is not haul to sett that. For crit ical site percolation on It the probabilit y of the event C,(1 .1?) does not depend on the terms of the sequence c, only on its length Indeed.. the woof of Lemma 5 allows us to deli ie a I?) From the inner tilde 'lowest' (clockwisemost) open missing of e."(;.1?) to the ante! citele. 'whi t /revel such a crossing exists Having Found such a classing. Pi , we may then look lot a lowest open at closed crossing above PI in the same wa y. The lowest classing P i may be found without examining the states of sites above Pi . so when we look fin the next clossing, the inobabilitv of success does not depend on whetIon it is tin open cat a closed crossing that we seek It follows sin/HaiIv that ^us(C',(; R)) =
R)
lot sonic (ti (t, lit ) that depends on the length of c but not the actual sequence Sinn ion and Wet nei (2(101 showed that. , it j > 1 is fixed and d is huge enough. then (1 ) (t .1?), 1?
(53)
238
Conic)! Mal inner iance
Theolvin
on 's
x■ This exponent JO + 1)/6 is known as the jarnt exponez in the halfplane
as R
Returning to the full annulus, one can show (see Aizenman, Duplantiét (fIc ( t, R)) is independent of the colours and Allat011y 119991) that P i r, in the sequence c, provided C contains at least one B and at least One H' This is related to Lemma 6 above: if the sequence contains two terrns of opposite colours, then it contains two consecutive terrns, and one can start by searching for an Innermost' pair of paths of opposite colours Then, working outwards, each remaining path may be found as the lowest crossing of a certain region, and the probability of finding the next path does not depend on the colour \VI iting bj (t, H.) for P i/2 (He (t,R)), wirer e c is airy sequence of length j containing at least one B and at least one H', Snriu nov and Werner [2001] showed that if ) > 2 is fixed and r is large enough, then Di (r,B)
RH12 0/12+00)
(54)
as R ce This exponent (12 — 1)/12 is tire (inallichromati i) 'harm exponent in the plane The values of these exponents, and of the halfplane exponents, were pr edicted correctly by ph y sicists: see, for example, Salem and Duplantior [1981], Aizennum, Duplantier and Akaror/y[4999] and the references therein. If one is careful with the exact definitions at the boundary (rather than glossing over them as we do here in our brief description of the results), then the events .4 /;:i? and fle(r,R) coincide, where c is the alternating sequence of length 2k: the k black paths correspond to the k open clusters joining the inner and outer circles of A(r,/?), and tire white paths witness the fact that: these clusters are disjoint. Hence, the case 4 of (54) is exactly (52) \\T ifton': going into the details, we shall sty a few words about the proofs of (53) and (54), and hence of Theorem 25 As noted ear lie, t he key elements are the result of SMitlIOV12001a1 that the (suitably defined) scaling limit of an interface in critical site percolation on T exists and is equal to SLEG , and the results of Lawler, Schramm and Werner on the behaviour of SLE6 Putting these ingredients together, one can show that a d ( i . R)
(R 11).1(i+1)/ti+o(I)
(55)
as R, r — x with .1711 fixed. (In fact, to obtain (55), one first needs an 'a priori' bound of the Ram a 3 (1 R) = 0(17 I —I ) as R cc, for constants I
and > 0; see Smirnov and \Verner 120011 and the references therein )
7 3 Critical exponents and SeltrantraLocumei evolution
239
To obtain (53), one also needs 'approximate multiplicativity' that, for < 1 < 13, (1,1(1 1, 1 2) 65(T2, 1 3)
e(a10r,13));
see Kesten [1987c] and Kesten, SidO/aViellIS and Zliang [1998] When Considering paths of the same colour, this relation is fairly easy to derive front the RussoSeyrnomWelsh Theorem Fortunately, for halfannuli, One can assume that all paths have the same colour: The corresponding relation for hi is much harder; see SatittION" and Werner [2001] Finally, let us note that, in the annulus, the restriction that not all paths be the same colour really does seem to matter It is likely that 1Pr/2(lf,(7,R)) a, cc with t and j fixed, where c is a sequence of length j with = .13 for every i However, these 'monochromatic' exponents 7; are very likely different horn the multichromatic exponents (j2  1)/12 above " rm. ticulm, Grassberger [1999] reported numerical evidence that = 0.3568 ± 0.0008. The numerical value, or even the existence, of ;; for j > 2 is not known, although Lawler, Schramm and Werner [20026] showed that 7,, exists and is equal to the maximum eigenvalue of a certain differential operator Critical percolation is not the only discrete object known to have a cargo, mally invariant scaling limit described by SLE: Lawler, SCl/181M/1 and Werner [2004] have shown that one may define certain natural scaling limits of looperased random walks in the plane and of uniformly chosen spanning trees in a planar lattice; these are related to SLE2 and to SLE8 , respectively. The brief remarks hr this section hardly scratch the surface of the the theory that has grown out of conformal invariance and the study of SLE For a selection of related results see, for example, Schramm 120014 Lawler Schramm and Werner [2001a; 2001b; 2002a; 2002c; 2003], Kleban and Zagier [2003], Beflara [2004], Dubedat [2004], Than [2001], Morrow and Zhang [2005], and Rohde and Schramm [2005], Informative surv eys of the field have been written by Schramm ' [2001b], Lawler [2001; 2005], Werner [2004; 2005], Kager and Nienhuis [2004] and Canty [2005].
8 Continliuni percolation
Shortly after Broadbent and Hanttnetsley slatted percolation theory rind Eras and Bányi 11960; 1961a1, together with Gilbert 119591, founded, the theory of random graphs. Gilbert. 119611 started a closely related area that is now known as continuum percolation The basic objects of study are IV/1(101U g cometric graphs, both finite and infinite Such graphs model, for example a network of transceivers scattered at random in the plane or a planar domain, each of which can communicate with those, others within a fixed distance Although this field has attracted considerably less attention than percolation theory, its Minot tante is undeniable: in this single chapter, we cannot. do justice to these topics. Indeed, this area has been treated in hundreds of impels and several monographs, including Hall [1988] on coverage processes Monts 119911 on random Vorolmi tessellations, {tester and ROY [1996] on continuum percolation, and Penrose 120031 on 1. rtmlotn geometric graphs These topics are also touched upon in the books by Mathison [1975]. Santal6 119701, Stoyan, Kendall and Mecke 11987: 19951. And kutzutnian [1990] and Molchano y [2005] In the first section we present the most basic model of continuum percolation, the Gilbert disc model or Boolean model, and give sonic fundamental results on it, including bounds on the critical area In the second section we take a brief look at finite random geometric graphs, with emphasis on their connectedness The most important part of the chapter is the third section, in which we shall sketch a proof of the analogue of the Hatiis Kesten result for continuum percolation: the critical probability for random Volonoi percolation in the plane is 1/2. lyre shall frequently circuit/1ns sequences of events (A„) with E(.4„) as n :xy, using standard shorthand, we say that A„ holds why or with high probability in this case. As usual, an event holds almost snafu, or as . if it has mobabilibt
8
The Gilbert disc model
211
8.1 The Gilbert disc model for r > 0. the yerfes set of the standmd Gilbert diNC model, or the Boolean model. G,, is a set of points distributed 'uniformly' in the plane, with density 1 To obtain G,, join two points by an edge if the distance between them is at most i The trouble with this 'definition' is that it is not clear how we can choose points unifinntly, with a certain density n fact, it is easy to turn this hopelessly loose idea into a definition of a Poisson process in the plane, the 'proper' way of selecting points uniformly Let A be a positive real number.. and let PA C be a random countably infinite set of points in the plane Let us write bi A (U) fin the number of points of PA in a bounded Bore! set U: note that i t A (U) is a random variable \Nje call PA a homogeneous Poisson process of intensity (density) A it the following two conditions hold (i) If U,. ..U„ are pair wise disjoint bounded Borel sets, then the random variables p A (Ujj ), pA(U„) are independent; (ii) For every bounded Borel set U, the random variable / A (U) is a Poisson random variable with mean AlUi, where IUD is the standard (Lebesgne) measure of U It is easily seen that there is at most one random point process (a random countably infinite subset of the plane) satisfying these conditions; in fact, as pointed out In fiónyi in the 1050s, condition (ii) alone defines the Poisson process P A In the other direction it is not hard to show that !hele is a point process satisfying conditions (i) and (ii): for example, one can use the following concrete construction For A > 0. let {Ne i : j) E 2'} be independent Poisson random variables, each with mean A Thus,
for k = (I, )) E
0. 1
Let
P ( X i ,i = h ) = (CA\k /,•! be the unit square with bottom left vertex
Q;; = {(i.!/): I ti .< i
1. j < y < p± 1
every j) E E.2 , select Nu points independently and unifinudy from then the union of all these sets has proper ties (i) and (ii), so we may take this as the definition of PA Although we shall not study it here, let us remar k in passing that a Poisson process • P f of intensit y satisfies (i) and (ii), except that the
212
Continuum prErrolation
is a lionnegative _el: mean of µj(U) is given by 11 , /, where integrable function RenitIllilg to Pa, note that if Z is a Bore! set of measure 0, then the probability that PA lets any point in Z is 0; hence, in what follows, we shall assume (lint PA fl Z = th lot all measure 0 sets Z that we consider. For example, we shall assume that no point of P A is on a given (fixed) polygon; a little wore generally, given a plane lattice, every point of PA will be assumed to be an interior point of a face For A > (1 and 7' > 0, let a, a be tine random geometric graph whose vertex set is P. with an edge joining two points of Pa if they are at distance at most 1; see Figure I We call C, A the Chlber t model with parameter s r and A, of the Boolean inadel with parameter 5 r and A. With this notation, the standard Gilbert (or Boolean) model is C. = C
Figure I. Pail of Hie graph C a (dots and lines) Two vertices ale joined they are within distance i e , if each lies in the shaded circle centred on the other The model GC , A has numerous variants and extensions First, let us rescale by writing H, ,A for C.), A. This resealing is not entirely pointless, since it allows us to define a random subset of R2 in a natural way, by writing D = .D, ,A (P A ) for the union of the discs of radius 7' about the points of Pa; see Figure 2 Note that the probability that there are two points of out Poisson process at distance exactly r is (l, so it makes no difference whether we take open discs or closed discs There is a. onetoone cot respondence between the compor ter its of C2,. a = (1,,, ',CPA ) and those of D, A = D, a(PA ); in particular, as P A has (a s )
8.1 The Gilbert disc model
243
Figure 2 Pail of the jit atilt G2, A (dots and lines) The shaded region is D,. A. of (.32,. A toe adjacent if and only if the conesponding shaded discs meet Two VW tices
no accumulation points, Go t A has an infinite component: if and only it 1),. A C R2 has an unbounded component Note that although D, A is veto close to H, A. the two models are not isornol pine, since the disc of radius I about a point may be covered by other discs making up D„ A The advantage of consider ing rather than Go, A is that the complement: of .D, A, the 'empty (or vacant) space' 1:3, ,A = 1R 2 D, A not coveted by the discs, is just as interesting as the original set D,.A particular, we can study the the component: structure Of E„ A as well: we can look for an unbounded vacant component, i e , an unbounded component of E,,A We may define variants of the graphs (1, ,A and 11, A, and sets D, ,A and E, A, by replacing the circular disc by an arbitrary centrally symmetric subset of R2 . Thus, given au open symmetric set A C R 2 for 11 C p2 we write G A (11 7 ) for the graph with vertex set 11 1 hi which two vertices x and y ate joined if x—y E A. Equivalently, for x E R2 , set Br B+x, where B = and join two points x and y of IV if 13, n B = Again, the union jrciv B„ reflects the structure of G .4 (11') If A is the disc of radius r, then we may write G, for GA (IF) In two of the most natural variants A is taken to be a square and an annulus Of course, the model also generalizes to d dimensions in a natural way. Taking for W the point set of a homogeneous Poisson process PA of intensity A, and letting A be the disc of radius r centred at the origin, we see that G A (PA ) = G, ,y More genet ally, in G A (1r) both A and 11/
211
Continuum percalat iaaa
may be chosen to lm random: one of t he simplest cases is when we assign independent identicall y dish United nonnegative random variables r(x): to the points .c of a homogeneous Poisson process 'P A , and take the unit of the discs 13„,x)(3.), E PA Clearly. all the models above extend trivially to higher dimensions There me many other 'natur al ways to define random geometric graphs, some of which we shall mention' in Section' 2 If we condition on a particular point c: E 112 being in PA , then the degree of c in G, ,A has a Poisson distribution with mean riT 2 A lit rad the sir tutu e of G, ,A depends on the parameters 1 and A only through the expected degree: for Aurg A t 11): , the graphs A„ and C„ A, have the same dish ibution as abstract ,midair' graphs Thus, when studying fin exannile, the various critical phenonwna concerning these giaphs, are free r1) change either / on A, provided we keep a = / 2 A COnSrallr In view of t his, ice shall \\ /ite (7(a) fu, any of the random graphs C, A with:: o = at 2 A, the canonical neptesentative being C, we call a the degree of 0, A The quantih a is also known as the connection arca. OF ShIlply, area: G, is ((ethicl bi joining each point x E P, to all othei points of P, in a disc with area hi an obvious sense. the random graph CI, A models an infinite corn mimic:at:ion ! Rawer k in which two transceivers can cornmunicate if their', distance is at most 1 As in the discrete case (when studying percolation On lattices or hit (icelike infinite graphs), we sa y that (.7, A percolates if it has an infinite component N'Ve unite 0(t A) = 0(a) for the probability that the component of t he or igin is infinite To intake sense of this definition_ we shall condition C, ,\ on the or igin being one of the points; equivalently, we shall assume that the origin is in 'P A This assumption does not change the distribution of the remaining points of C, A The first question concur Mug I his ; continuum percolation' is when the 'percolation mobabiliti s 0(t, A) = 0(a) is strictl y ppositive. It is trivial. that 0(a) = 0 if a is sufficientl y small and 0(a) 1 as a Since 0(a) is monotone ha:teasing, there is a critical degree or ci Meal area a,: if a < 0, then the probability t hat C7(a) maculates is 0, and if a > at„ then this probability is strictl y positive Also, as inn the discrete case. Kohnogolo y 's 01 law (Theorem I of Chapter 2) implies that if a < 0„ then every component of G(a) is finite a s and if a > a, then C.;(0) has an infinite component a s Note that I he critical degree a t cot responds For percolation As we shall see later in to the critical probability Theorem 3, the natural analogue an of :/ci is equal to a,.
S I The Gitlie,t disc model
245
Our main aim in the test of this section is to give hounds on the n itica t degree a c . An easy way of bounding a, is by e0111pat ing C, A to a discrete percolation model with good bounds on its critical probability Perhaps the most natural model in the face immolation On the hexagonal lattice in which each face is open with the Sallie probability indeipen(entry of the states of the other faces, and closed odic' wise; two faces ate iwighbouts if they share au edge (As every vertex has degree 3, someshat misleadingly, this is equivalent to sliming a vertex ) This amdel is t ome usually desetibed as site petcolation on the triangular lattice Theorem 8 of Chapter 5 tells us that if p < 1/2. then a s there is no face petcolation, and if p > 1/2, then a s we have face percolation To obtain a face percolation model F l om PA. let A be a hexagonal lice, with each face a regular hexagon of sidelength s. Define a face percolation model on A b y setting a face to be open if it contains i t least one point of PA. and closed ()diet wise. so each face is closed with limitabilit y c AA and open with probability t  C AA , \viten, A = 3452 /2 is the area of a hexagonal face (see Figure 3) Front Theorem 8
010 Figure 3 Comparison between Gilberts model and face percolation on the hexagonal lattice, i e site pwcolation on the hi:log[11w lattice: a hexagon is shaded if it contains one or more points of the Poisson process The co p esponding site in the triangular lattice is then taken to be open If the hexagons have sidelength s. then AB = 2 ‘,/s. so AC = of Chaplet 5, if 1  c A1 < 1/2 then we do tot have face pet col ion, and if 1  c AA > 1/2 then we do All that remains is to thaw the appropriate conclusions alma the disc model G, A Note that any two points in neighbouring faces ate at distance at most ((2A 2 1 2 ) i/2 s s ■/171, and any two paints iu nonneighbouting faces are at distance at least H Consequently, if
246
Continuum percolation
1 — e A•1 < 1/2 and r < s then we do not have percolation in C,,A, and if 1 — c AA > 1/2 and r > sVT .3 then we do, Taking A = 1 and substituting A = (3/:73/2) 5 2 , we find that if (3 ■41/2)s 2 > log 2 then a„ > 7r5 2 , and if (342)52 < log' 2 then a c < 137,s2 , i c ,
log 2 Mil1o°'c, 9< a, < 34 34
We have given these inequalities to show what one can read out of the approximation of a Poisson process by face percolation on the hexagonal lattice, although the lower bound above is worse than the trivial boon 1 implied by the simplest branching process argument. On the other hand. the t i pper bound, given by Gilbert [1961] under the assumption (unproved at the time, but proved now) that the critical probability for site per colation on the triangular lattice was 1/2, has not been improved much in over four decades The slightly better upper bound in the theorem below is due to Hall [1985b], while the lower bound is the bound from the seminal paper of Gilbert [1961]. (In fact, the numerical value for the formula given there was 1 75. !) Theorem 1. Let a„ be the critical degree (area) Pi the (tither (Boolean ,lice model G,. Then 671
271
612
< (lc < 10 588
Proof Let us start with the lower bound. Let Co he the vet tex set of the component of the origin in G, = G, , r, briefly, the component of the:: or igin We shall use a very simple process to find the points in Co one by one, but in order to achieve a concise formulation of this pr ocess, we describe it in a rasher formal way. We shall constr uct, a sequence of pairs of disjoint (finite) sets of points of the plane, (Do, Lo),(D 1 , say The points in D, are the points of the component Co that ale dead at time t: they belong to the component Co, and so do all their neighbours; the points in L 1 are hue at time t: they belong to Co, but we have made no attempt to find their neighbours. To star t t he sequence, we set Do = 0 and Lo = {X 0 }, where X 0 = 0 is the origin Next, let No be the set of neighbours of Ko, and set Dr = {X 0 } and L i = No Having found (D,, L i ), if L, = 0 then we terminate the sequence; otherwise, we pick a point K, horn L I , and define Dt+. = Dt U{Xt } = {X 0 ., X, } and L 1+1 = u L,\ {NJ, where Nt is tire set of neighbours of Kt that are not neighbours of any of
8 1 The Gi(bell (list model
247
is locally finite (no disc of radius l' contains the points in D, Since infinitel y many points of P i ), the sets D, and L, al e disjoint finite sets By construction. fun then mole, if =0 then Ca = U C Since \
I.V
0 1
At
CU
Ar
i=0
laVe
ID / I =
t—1<
(1)
Let 1 7, be the disc of /minis I with centre X t , and set U, = (1,0 Conditioning on the points Xo. .X1 . we find that IN / l is a Poisson, andom vat iable with wean II ./ \U,_ 1 1 Since the centre K t of I", is in, a disc V,. s < t — I. Figure I tells us that
Figure 4 Two (lists V, and I; of radius r with effi g ies A. and Xf, such that. E The (nett of Pr \ V1 (shaded) is maximized if dist (X„. X t ) ttct I, as shown In this ease the area is ft = (if + 41+t 2 Now. to bound let Zo. Zi. be independent Poisson random variables. with E(2 0 ) = a7 2 , the men of the disc Dr,, and E(Zi ) = b fur i> 1 Then inequality (1) implies that POCol > <
P
i k\i=0
II 6 <
it follows easif\ that 1
il l ^
IC
as A
i .
)
918
Cortiirtir II III Ile IrOla 1011
Since r = 71  <
we have b <
+
= 2if + 3 \ /73
the ni tnat area is indeed at least as huge as claimed
Pot time upper bound on a u , Flail (19851,1 tweaked Ciilbe i Ls argument r km2 that gave die trivial hound 't' 3,111 — 19 89 by replacing the cells of a hexagonal tessellation be 'rounded hexagons' To give the details of this argument, consider the lattice of hexagons, each with sidelength I For each hexagon H, let H' be the rounded hexagon in 11, the int ersection of the six discs with centres at t lie midpoints of the sides and touching the opposite sides. 05 iu Figure 5 By coast/ uct ion, each of these discs
Pigmy 5. The shaded part of each hexagon is the legion within distance the midpoints of all ti sides The labels correspond to those in Figure
I of
has marlins 15. so am two points of two neighbouring rounded hexagons ale at distance at most 2 N/if Hence. if the probability that a 'minded hexagon contains at least one point of the Poisson process PA is greater than 1j2 then. appealing to Themeni 8 of Chapter 5 as berme, we find that ary/ii.A Petcolates with strictly positive plot/ability therefore, if the area of a rounded hexagon is a then C .—A" < 1/2 implies that a, < 1.27A Hence. 97 log 2 <
Finall y . the men a 01 a rounded hexagon H' can be mead out of Figure 0 First the area of the sector ACD is 3c/2 To calculate lip . note that sinyi, = r. }ir; sin(57/(i) = so cos; = ■/15/4 Also.
S / The Glebe') disc inodd
249
Figure The points A and C' are midpoints of two opposite sides of a hexagon II; 0 is the centre of H and 131' is half of a side. Thus, A0 = 01' = V3/2. 013 = BC = 1/2 and LCOB = 7/6 The shaded region is part of a circle wit It centre A, so AD = We write y for LIDO and r5 for so that ,42, + = 7/6. The shaded domain DOC' is one twelfth or the area of a rounded hexagon II'
sin 12
4'5T \ir")  I). while t1'= sin — ) = )f)Fin t lumina e. sin c
OD =
sin(57/6)
II
esio( ) = 0 2709
AD — (17)
and so the area of the triangle .40D is (Bc, The mea a of
OD) = 31.'3(15). —1) OD /I
' is twelve times the area of DOC'. so 13\41.(Ji :32
so that u = 2 167
— 3r1 tncsin(
and a r. < ID 588
)
91:3( \/7) —1 8
. as claimed.
The simplest form of the branching process a/glutton' above implies that if .1 is any bounded open svnunettic set in 1P1( 1 . then the critical degree fin C T (PA ) is at least i IGdI [1985b] improved s lower bound as well: this inunoventent involves a more substantial modification of Cilbez argument than the tweaking of the upper bound p l esented above. Indeed to obtain his improvement.. Flail compares disc percolation ' to a multi t y pe 'munching ocess, with the 't y pe' of a child defined as its distance hour the father. Them em 2. The radical degree for the Gilbeil (Boolean) iliac model is ()realer than 2.181.
259
ermithumn percolation
There has been much numerical work on the critical degree For the Boolean model, and on its square yin iant Simulation methods have been used For the disc by Roberts [1964 Don't) [1972], Pike and. Seager [1971], Seager and Pike [1.971], Flenrlin [1976], Haan and Zwanzig [1977], GawL: inski and Stanley [1981], Rosso [1989], Lorenz, Otgzali and Heuer [1993], Quinlan:ilia. and Torquato [19991, and Quintanilla, Torquato and Ziff [2000], among others, and fort the square by Dubson and Garland [1985]; Aron, Drmy and Balbog [1990], Garboczi, Thorpe, DeVries and Dm., [1994 and Baker, Paul, Sreenivasan and Stanley [2002] (Not suupr is night., several of the bounds obtained happen to contradict each other ) For the critical degree of disc percolation, Quintanilla, Torquato and Ziff [20001 gave lower and upper bounds of 1 51218 and 4.51228; for square percolation, Baker, Paul, Sreenivasan and Stanley [2002] suggested 1 388 and 4..396 In studying the critical degrees in these two models, Baliste" , Bollobris and Walters [ 2 005] gave rigorous reductions of the problems to complicated mune] ice! integrals, 1\ hich they calculated by Monte Cat lo methods Indeed, their method was the basis of the discussion of rigorous 99% confidence intervals For site and bowl percolation critical probabilities in Chapter 6 The basic idea is to use results about kindependent per colaHon to prove that a certain bound On holds, as long as a certain event E defined in ter ins of the restriction of 0, to a finite l egion has at least some probability Thi (For example, they consider the event E that the largest components of the subgraphs of (7, induced by the squares [0, ((2 and [1 . 2/1 x [0. L] are part of a single component: in the subgraph induced by 10,2(1x (0, t] Considering events isomorphic to E, there is a natural way to define a Lindependent bond percolation measure on Z 2 such that ever V bond is open with probability P(E). Another possible choice ha E is described in Chapter 6 ) Unica tunately, one cannot evaluate P(E) exactly: however, there are Monte Car lo methods for evaluating P(E) that conic with rigorous bounds on the probability of errors of certain magnitudes; see Chapter 6 for more details of the basic method Using this technique. Balister, Ballobris and Walters [2005] proved that. with confidence 99 99%, the cr itical degree for the disc percolation is between 4.508 and 4 515 and that fin the square percolation' is between 4.392 and 4 398 The results and methods we discussed in ear lier chapters for discrete percolation models are easil y applied to the Gilbert Boolean disc model to 'rime the Ind( 11/01 letiti of the infinite open cluster above the critical
9_5 251
8 I The Gilbw t disc Model
degree a c. and exponential decay below a, We start with a a result of Roy [1990] giving exponential decay below a„. which implies the analogue of pH pr for Gilbert's model This result holds in any dimension; lot notational simplicity we state and prove it only for dimension two Theorem 3. Let 7c. f 2 A = a < a c . where a„ is the critical degree for the Gilhevl model, and let ro (ar A H denote the Immix, of points in the component of the origin in G,
A
Then
P0C0 (G, A )] >
<
where c„ > 0 does not depend on n In pal ticular a
ur= thr
{71/ 2 A : IrtaCo(G,
L. where
A lp = Do}
Proof 'The result is more or less immediate from klenshikov's Theotent
(see Theorems 7 and 9 of Chapter 4), using the natural approximation of continuum percolation by discrete percolation Roughly speaking, we shift the points of the Poisson process PA slightly. by ;rounding their coordinates to multiples of a small constant S If the shifted points are at distance at most r  then the original points must have been connected in C, A On the other hand, if the shifted points are at distance more than r + 20.6. then the original points cannot have been connected in C, A Let us fill in the details of the argument. Pick A i > A and rr > so that a < < o r , and 6 > 0 such that r + 2015 < r I Let 6722 = (62.7) 2 lie the square lattice with laces Fit = {(,c, y): IS < < (i
< y <(J+ 1)6}.
Let A(6,./ j ) be the graph whose veil are the faces Flci , in winch two faces are joined if the maximum distance between two points of their union is at most r i We couple G,, A, with independent site percolation on A(S. L I ) in the obvious way: declare a site F15:, of AO, to be open if and only if it contains at least one point of 'P A , Note that the states of the sites (i e laces) are independent, and that each is open with probability pr = 1  cAla Suppose that there is an infinite open cluster in A(6,1 j ). If :r and y ate adjacent open sites of AR I I ). then there is at least one point of PA, in the faces of ,5& corresponding to A and g. and any two such points are joined in CI, , ,A , . Hence. G,, A, has an unbounded open cluster But or < a„. so this event has probability zeta Thus, p i < pli(A.(6.11))..
9'0
Continuum pettolalion
The gul l) ) ) A O I I I satisfies the conditions of Menshikov's Theorem, Theorem i of Chapter (As usual we tepid A(6.1 1 ) as an oriented graph In replacing each edge by two oppositel y oriented edges ) Set p = 1  e A
❑
Unlike Theorems 3. this does not follow easilr limn the co/les/Raiding discrete results: the Brent that them is at most one unbounded (occupied there is no 01 vacant) cluster is 'within increasing not decreasing . stotight fo t w i nd wa\ to bound its probabilit y b \ that ( )I an event iu a disown. , app t oximation to ()illicit 's model However. the Button Keane proof of the Aizenman ICesten Newman uniqueness result lot lattices (Themen t 1 of Chilling 5) can easilr be adapted to Gilbett's model (hice we know d i nt (with probability I) G, A has a unique infinite component. it is reasonable to expect that inside a big box' them is onl y one big component. Pen t ose and Pisztoi a [1990] showed drat the heuristics. if moped r stated, ate indeed tine. but they hare to work rather Laud to more 1110111 Ot Ile/ results tot percolation on lattices have analogues lot cuminmn pe t ( M i tt km Fro example, Alexander 0996] proved an rutalogne (111? Russo SCVIIRMI SVelsh liniment for occupied eh/stets in G, A.
8I 7Ite Gillett disc model 253 e
101
13, A
N.
C(
tesponding result lot vacant clusters was plover( by
Bo y [1990] Even in the plane. t he 011)00 model has 1111111(1 otts tur d extelisions Em example. instead if a dim... We way use a convex domain IC C R2 to define a random set ti
D a . = U(
IC )
i=
whew PA } is a Poisson process of intensity A in the plane This process percolates if D k. A has an unbounded component Note that K is not assumed to be s we genet alize the I/11(1 etat ion of the Gilbet t model shown in Figure 2 The co t tesponding giaph is exact Iv G A (P A ) as defined eat lief wit h A =I( —k In pm ticulal. fhe expected degree of a vet tex is A times the m e t' of IC — IC \ (IC rot the critical intensity of this percolation process. lonasson [2001] proved the beautiful result that, among convex (10111/1illti IC of area A, (K ) is minimal fin a triangle Ro y and Tairenuna [20021 showed that an analogous result holds in higher dimensions, with a simplex replacing the triangle To conclude this section Nye note that thew turf natural models with critical degrees close to the minimum First, as shown by Pent ose the finical degree for long purge pe t colat kW " ill Z2 (ot J') tends to 1 as the 'range' tends to infinity (See I3ollobas and Kohayakawa [1995] for a combinatorial pool.) 'Molesurprisingly. let A : the annulus with twin 1 and I + e (anti so area rr(2s. E 2 )); then the critical degree for (Py ) tends to 1 as E 0. This was proved. independently by Flanceschetti, Booth. Cook. Aleeste/ and Duck [2005]. and I3alistet. Bollobtis and NValters (2001) In the lattei. Impel. it was shown that the of tesponding asset t ion for 'squaw annuli' is false: let S, be t he squa t e annulus with the buret square having sidelength 1, and t he outer 1 0 < E < 1. say 'Then the critical degree for Cs_ ('PA ) is at least c > 1. with c independent of E Tinning to hall pc/potation in higher dimensions (corresponding the Boolean disc model in dimension two), let (14(!`i) be the critical degree (volume) in E a , so that = (4.2) Peru ose 1[996] showed that HY) d (pc., Batiste/.. Bollobris and N Yakut s (20011 proved a general result about models with ethical degrees tending to I t he minimum possible value: t his implies tiiyiallv the results fin annuli and highdimensional balls
254
Continuum percolation
Ilathet than asking fin an unbounded component, one may ask Mr all el almost all of out space to be coveted by a collection of random sets Usually, the random sets a t e not translates of the same set, but are chosen with a certain probabilit y distribution For example, we may take independent identically distributed compact sets K I ,16, in Rd and take the random set E U(,r1 NJ, in I
when! {.1 _ is a random sequence of points in 1P d Conditions impl y ing that E is the entity space R I were given by Stovall. Kendall and Mucky [1987], Hall [1984 Meustet and Ro y [1994 and Mo[clumpy and Scher balm\ [2003]: At linn a, Roy and Sat hat [2001] gave conditions for the sets to covet all but a bounded part of
8.2 Finite random geometric graphs As random geometric graphs fiequently model 'reallife' networks, e g a network of transceivers dist/ Hutted in a bounded domain, it is mum al to stud y finite random geometric graphs.. with the quintessential example being the restriction of the Boolean model to a finite set To define this, be the restriction of a Poisson process P„ of intensity n to the let squaw [0, Fur r > O. let G, (1;,) be the random geometric graph in hich two points of V„ ate joinecl if their distance is at most I It is easy to check that C, (1 7,, ) is close to the model C.', (U„).. in which U„ consists of a points chosen uniformly noun the unit square, and two points are joined as before Clearly, scaling makes no dillerence to the model: multiplying I by I and taking the testi idiot/ of the Poisson process, P„ 712 to ID, ([ 2 we get a model isomorphic to GC (1/1„) Fat example, it is natural to replace the unit striate by [0, jir] 2 and then take the tesniction of the Poisson process of intensit y I to this squaw AV hat does matter is the relationship between the expected degree and the expected number of points In fact, to obtain a mathematically more elegant model, we shall choose our points front the torus .11 obtained horn [0,11 2 by identifying 0 with C. To he precise, let G, A) he the random graph whose vertex set is a Poisson process PA on ff7 with intensity A, in which two points of r The expected number of yer'PA are joined if their distance is at and fort tC/2 . that a E PA, Rees of this random graph is n = t he expectation of the degtee of a in CI, (Iti, A) is d = n 2 A Once again,
8 2. Finite noulom geometric graphs
255
the scaling is irrelevant: a and d determine (3, (r71, A) so, with a slight abuse of notation. we may write (17„ d for this model. The main advantage of using the tor us, and so C,, e h is the homogeneity of the model: however, to all intents and purposes, for r1 = il(n) o(n) the models Cri „ (tin) and (3, (U„) are interchangeable provided d =7112n. The first question we should like to answer about the finite graphs C„ , a is the following For what values of the parameters n and d is the random geometric graph C„ a likel y to have isolated vertices? This was answered by Steele and Tierney [1989] Later, Penrose [1997; 19991 extended this result by giving detailed information about the distribution of the minimal degree ti(G„ a) of C„ ( 1 It is fascinating that this result is the exact analogue of the classical result of Lidos and R 6/ IVi [1961Id on random graphs (see also Bollohas [2001, Theorem 3 5 .1) Here we present only a weaken for w of Penrose's thecae'''. Theorem 5. Let rl = d(n) = log p ± A; log log n fixed nonnegative integer. If o(n) then Nri(G„ and tf 0( )
< —
o(n) where Ar is a
I
then (8(G „ > lr ± —
Pi vot :10 simplik the calculations. we sketch a proof h=0 Thus. we set e = log p + o(n) : and let A and t satish = A1.1= and d = 7112A, so that C„ 6', (14, A) We write X for the number of isolated vertices in C u m, A) The probability that a fixed disc of radius 1 in ti
contains no points of our Poisson process Pt is c'nA It follmvs horn basic properties of Poisson processes that (2) ilit t A u) = ne' 1A pe ti ccr Indeed, dividing T? into (C/r) 2 small squares Si of side ar < 1/2, by linearity of expectation, 1E(X 0 ) is (I/s) 2 times the probability that Si contains an isolated vertex of CI, OTT A) (No Si can contain two such vertices ) As E 0. this probability is asymptotically
so
li ltt/s) 2 AE 2 C AS2 —
r". (1
Coolinuum pc, colotion
256 imph ing (2) II (1 —
then (2) implies t hat P(S(G„
=
> 1) <
u( 1)
Now, suppose that (1(11) To prove that in this case (7„ (7, (T./. A) is ye t v liken to have isolated vet tices. we need mutt het Oat her trivial) step IVe knou that
ni
= Iti(X0 ) — Ix: It is easil y checked
that t he second moment Itt(„V,.;) of Xo is oh, = (1 + o( 1 ))/// i ; hence. by bi ltebvelly% l ts inequality —
INO(G„ a) > = Pau =
u( 1
The genet al case can he p i (‘ ed along the same lines. \\ it 11 a little mote calculation. Im fact. it is easy to pttne that Fin d = ((Oil in the pp/ Op into lunge, thenuttbet _Ar. of Net t ices of degtee A. has as y mptot ically Poisson distribution: t his implies Yen good bounds on t he mobability Il il ei(G„ a) > Although this is not inintediatek obvious, it is not hind to show that hemetn 5 does nuked cal y ore/ to t l i e model (3, defined on the souffle l athe/ than the lotus, i e that the dumndan effects do not matte] ou t nex t aim is to state a considembh weight jet lesnit of Pentose establishing a close connection between t he mope ' ties Of •4runitecteduess and of l i ming minimal degree at least s e star t br defining t he hit ling
radius lot at at hitt tu
■ ptopett y and an al bin set Chen a point set
P and a monotone inneasing propel Iv Q of graphs lie a I n Opelty Q such that if (7 has Q and C' is obtained In adding an edge to G then C' also has (9) the !Oiling !who .: pQ (P) of (9 on Pis defined as pQ (P) = whiff : C,(P) E Q} Note t hat if Q C
(hell P(2(
> 1 ) (2'(P)
for
PVC?
ty set. P In pm t ic
ulat . the hitting radius of sconnectedness is at least as huge my t hat of
htrc ing minimal degree at least s: ) lot
f PC I
o
P
A basic Jesuit it t lie them \ of randrun glitplis is the result of 13ollobtis and 7itonmson fltS51 that lot almost even I //mIum graph process t he hit ting time of sconnectedness equals the hitting title of having minimal
8 :2 Finite random a t:met t le 'myths
257
degree at least s (see Bollobas [2001]. Theorem 7 1) Penrose [1999] (see also Penrose (2003. pp 302 3051) showed that this result car ies over to the Boolean model on the tor us Theorem 6. Let PA be a Poisson process on the ton If with intensity A Then
(PA) = ps>,(17 A )) = 1
lint
❑
Thus, for A huge enough, if we strut with a set PA of isolated points, and add edges one by one, always choosing the shortest possible edge to add. thew with high probability. the very moment this graph has minimal degree 5, it is also sconnected Combining this result with Theorem 5 we can identif y the critical degree for sconnectedness Theorem 7. Let s he a freed nonnegative i ntent ' , awl let d = (1(0= then 0.(n)— log + (s — ) log log f t ± of )
IP'(G„ :I is sconneeled) —4 and if (1(0 —
then d is ssivnneetcd) —
As shown b y Pennose [1999]. all these results can's over to random graphs on the snuffle and. nuttati dis nuttotis to the / i fdimensional cube 10, lf"' and torus The analogous problems For the simpler graph delined using the If.„distance, rather than t he Euclidean distance were studied by Appel and Huss() [1997; 20021 The random graph (7„ d above is connected if and only it . the minimal weight spanning nee in the complete graph on the same vertex set. wit h each edge weighted by the distance between its endvertices. has no edge longer than d. Such spanning trees, generated by a (possibly nonuniform) Poisson process in the mdimensional cube [0,1]"'.. have been studied by a number of people, including Henze [1983], Steele and Slump [1987]. and Kesten and Lee [1996] Our next aim is to studs the connectedness of one of the many models of random geometric graphs related to t he Boolean model. Let be the restriction ()I a Poisson process of intensity l to a square S„ of urea to the A. points nearest to it. Let 11„ t be n and join each point r E
258
Continuum percolation
the random geometric graph obtained in this way Note that o is the, e,rpected number of vertices of H„ , e; also, if H„ . p has or > 1 vet (ices then it has at least km/2 and at most km edges The random geometric graph H„ k is again a model of an ad hoc:: network of transceivers and, as such, it has been studied by many people,: including Kleimock and Silvestet [1978], Silvester [1980], Flajek [1983] Takagi and kleimock [1984], Hon and Li [1986], Ni and Chandler [1994] GonzalesBanjos and Qniroz [2003], and Xue and Kumar [20041. We should like to know for which Functions k = k(n) the graph . More precisely, we should like is likely to be connected as n find a function ko(n) such that, if F. > 0 is constant, then lim P( ,A
ro
if k 5 (1 — 04.0(4
is connecte ) =
if k
(1 + E)ko(o)
Such a function ko(n) is considerably harder to deter mine than the ctiti: cal deg/ ee d for connectedness in the Boolean model given by Theorem 7; since the trivial obstruction to connectedness, the existence of an isolated vet tex, is ruled out by the definition of H„ As we shall now See, simple backofanenvelope calculations give us the order of /,.[) (o) Nevertheless, we ate very far from determining the asymptotic value of 1;0 (0 In the arguments that fellov,, the inequalities are claimed to hold only if o is sufficiently huge. Let us see first that if r > e then k = k(n) = [clog is an upper bound fix ko(n). If every disc of area a = 7r1 2 centred at a point a: E V„ contains at most k other points of S. then ti„ e contains „ as a subgraph, whew „ is the graph on V,, obtained by joining two points if they are within distance r The variant of Theorem 7 for the square (rather than the torus) tells us that the graph C„„ is connected whp (with high probability, i e., with probability tending to 1 as n x) if a = log n + log log o, say. (In the application of the theorem, s = 1, and ev(n), log logn ) Fm this a, the probability that a fixed disc of area a. contains at least A' + I points is at most
ah±1 (k + 1)!
<
1
([/(!)/ <
where c < < c and z > 0 nom basic proper ties of Poisson processes, it Follows that the probability that the disc of area a about seine point of 1,7„ contains more than A. other points of V„ is at most 1E' Consequently. H„ , p is connected hp, as claimed
8 2 Finite modont geometric graphs
259
1(1 — s)logn/81 — 1 is a lower Next, we show that if z. > 0, then k bound for To this end, define > 0 hi = + 1, and consider a family 'D of three concentric discs D I , D3 and D5 contained in S„, where D; has centre x and radius it: see Figure 7 We say that the family D is
WO
Figure '1 A family D = {D i , D. Dr,} of three concentric discs, and two possible discs D, touching D I For k 2. the fandiv D is had., so contains no edges from the vertices in D I to the test of the graph bad for the graph
hold: 1, (or set V„) if the following three conditions
(i) D I contains at least k + 1 points of 11„, (ii) D3 \ D I contains no point of V„, and \ at , the region D, fl (D 5 \ D3 ) contains at least (iii) for every c E + I points of 1 1„. where D, is the disc with centre c and radius dist(x..i) — 1 k is disconClearly, if some family D is bad lot H„ t , then the graph nected. A family is good if it is not bad Let us show that the probability that a fixed famil y D is bad fin H„ k is not too small First, the probability that D i contains at least k +
Continuum percolation
260
points is approximatel y 1/2; an\ at least 1/3 Next soin thin (ii) holds with ptobalrilit
Finally. condition (iii)) holds if it holds tin the points c with dist(i)..r), tar For such a point c, the area of D, n (Dr, \ Da) is en 2 for sow > 2 1111/S. one call cover D 5 \ D3 by a constant number C of regions contaitrs sonic The probability that. R., of area 27/ 2 so that any D, con) a given R, does not contain at least k+1 points oft„ is Y(1) Hence, the probability that (iii) holds is 10(1). and in pm Perrin/ at least 1/2 lot y large Since the events (i), (ii) and (iii) are independent, the probability that a family D is bad tor [I n k is at least frH5/(i The square .5„ contains at least
>) 101(( — x) log n)/(87v) )0 11
// 5 log n
disjoint squirts of sidelength lot so it contains at least this Walk" disjoint discs of radius St Therefore. the probability that even familt D is good For 11„ p is at roost (
+16)
lug!,
ConsequentIt the plObabilli V that 11„ k is connected is at most cii of I) The mg uncut above can he consirkriabli simplified if all we want is that connectedness happens around k = Sflog 0: there is no need to use Penrose s Theorem. Firemen ' 7, which is quite a big gun lot such a S/11)111 sparrow Be that as it mat, the tenni/1(s above tell us that if c 1/S then /1„ r„g „ ) is disconnected whp. and if r then H„ Li l „; ,, i is connected whir :Kite and Kumar (20011 were the first to publish bounds on li tt (n): they Firmed that 0 07. i logy is a lower hound and 5 1771logn is an tippet hound. with the upper hound firllowing from Pemose's Theorem (In fact, the upper bound 3 8597 log y is implicit in CionzrilesBarrios and Quiroz [20031 ) Bollobris. Sailun and \Villiers 120051 considerabl y Unmoved the constants 78 and r in the Pi\ ird bounds above: in pm t they disproved the nal trial coujectuc t hat kay) is asymptotically log y which is the analogue of Theorem 6 fin H„ k Tlieolern 8. If < 3103 Own II„ , 1 „„„i I s ( I l yto li nerlfrl wh i t. I 11( 1 if „ H kfitlfleCh (I why c> 1f log I 0 51:19 fin?, „
S Punic irutdoin geometric gtretlIP
261
rather
involved proof and a nundail of related distills \VC itelet the reader to the ot iginal impel Pot I he
Instead of asking lot the connectedness of certain million' geotnet tic graphs. \ye ma y ask ha the mound domain to he cou r ted by the t andont sets we used to define the graph. These enticing() questions have a long history. and ate prominent in three boolts: Stu\ an. Nendall and Matte [1987]. Hall [1988] and Meestet awl Roy 119961 Here we shall say a few avoids about some of the man y results lust before the dawn of percolation. D ymetzEN 119561 laised a beimHiltl question collect ning cove t ing a citric la /tuition) 11(5 Let 0 < independent IN at iondont <1 Drop airs of lengths 1 1 . lg. (0. onto a circle with perimeter ro t w hat sequences (I ) do mu at cs cowl the entity circle whp? Independent IN of 1)NowtzltN St eutel 119671 wised a sindlat question and proved a 101mula that has trtnmid out to by yet v useful Shepp 119721 gave a delicate at gument to in t hat a necessm \ and sufficient condition is that a
2(
,
Flat to [1973] considered iandoin arcs of the same length (r. 0 < 0 < (Sin pi isingly. this problem had been considered br Stevens 119391swei al years byline Eho t etzlt N posed Ids question. in a 101 1 111a1 on eugenics.) To state Flat to's distill. let m be a fixed nal inal nundaa and (hop the at cs on the (Tide (01 pet inwtei I) al Iiin(loni one la one. stopping ;is soon as CNC( y point of the cl i ck' is co N end at least ni limes \I) /he A„ „, 101 the /Hunk , ' of tics used (Thus. A„ „, is the hit ting time lot baying an mFold covet.) Flat to showed that litre 1111(;V, 06
1 ( log( I /d )
loglog(1/d) th id)) =
(:2"7(
\\ hich is once again itiminiscent of the classical Eldirs Ren y ' result Siegel [1979] plowed results about the distribution of thy length of the !Income ( ' pall of the chyle: late/. Hall [1985a] extended this Jesuit to higher dimensions. The t esult s of Stevens wen' gene/ idized Ir y Siegel and Hoist [1982] The number, total length m i d sizes of the gaps left by the coveti ng arcs were studied b y Hoist and Hasler 119841 and Iluillet [2(103]. among Whets Inparticular,Huillet made use of Stentels identity to Noy(' exact and as\ nip( ot ie results about these quantities A dilated problem_ due to 1 11 ()ns i 119581 conceins choosing disjoint
9 1i 9
C'oatinaryrn pal colal ion
shot t subintervals of an interval at random: this problem and variants of it gave rise to much research (see. e g . Nev (19021, Dvoietzkv and Robbins [196II, Solomon and \keine' . [1986]. mid Coffman, Flatto and. delenkovie. E2000D; these problems are also studied under the name of random sequential adsorption models Turning to random covets in higher dimensions, Machina [1988] studied the threshold lot random caps to covet t he unit sphere 8 2 C le'* Suppose we put N spite/War caps of area rEtp(N) independently and at tantrum on 8 2 , and that
p(N)N
lam log A"
—
Machina proved that if r < 1. then whp the sphere is not coveted completel y, while if c > I then whp it is. Further results of 'Mitcham 11090; 2000 concert the intersection gr aphs of random arcs and random caps: these graphs are the analogues of the gr aphs on the tor us and square we studied ear her in this sect ion In a differ ent vein, Aldous [1989] used Stein's method to prove sharp results about covering a square b y random small squares Gene] alizing several em rim results. Janson ([986] used ingenious and long arguments to prove that. under rather weak conditions and aler iippoquiate inalization. the (random) numbei of random small set s needed to cover a larger set t ends tot he ext Imilevalue dist ill n i t ion exp(c 11 ") as the measure of the small sets tends to 0 TO conclude this section, we shall sa l a few words about the random convex hull problem vet ,mother old problem about random points in a. convex domain Pick a points at 1 ile(10111 horn a convex domain K CIE2. NV hat can one say about the (r andorn) number X„ of sides of the convex hull H„ of these II points? Renvi and Sulanke [1963] proved that: the distribution of X„ depends very strongly on the smoothness of the boundary: if K has a smooth boundary then Etk„) = 0(1 1/3 ), while if K is a convex kgoll titan Er(X„) = (log wrier(' C (1)+ L(C ) is .Eulet's constant and c(K) = o(k) depends on K and is maximal fin regular polygons and their shine equivalents In a followup paper,. Renyi and Sulanke [1901] studied the area and perMiele/ length of the convex hull 11„ These results of Reny' and Sulanke have spurted much research, including papers In Cimeneboom [1988]. Cabo and GI oeneboom 0990, Using [199 ‘0, Hume! [1991: 1909a; 1999b]. Using and Bingham (19981, Brinker and lising 119981. and Finch and [ureter (2004] For
8 Random Vih onot percolation
263
II the normalized random variable (X„ — 27rc i a l/3 )1(n 0' V27o)) tends to a standard normal random wadable, inhere(37/2)"1/3 r(5/3) 0 538 and co is expressed in tams of complicated double integrals Finch and Hnetet gave an explicit expression fin c„
example. Gr Oen/MOM/I pl (Wed that, as
8.3 Random Voronoi percolation In this section we shall considet Net another nuns per colation p i 0Criss associated to a Poisson process in 1Pil l for d > 2 Up to n ow. out Poisson process was used to define a random geometric graph. and the question was whether this graph has nu infinite component or not.. this time we go farther: we consider a graph defined by the Poisson process, and then considet site percolation On this graph To be precise, let 'P lie a Poisson process (of intensity 1. sa y ) in RI E P), the Open Voronoi colt U„ of Fot even: Poisson point: z (i e with tespect to P is the set of all points closet to ,r than to am : other of U. is the closed rolonth cell of Poisson point The closure with respect to P Thus. simply the Voronoi cell of =
(P) =
E
: dist (y. ,r) < clist(y,x 1 ) for: ail .riE
It is easily seen that. lot a Poisson process P. with mobability I each cell lir = ll,.(P) is a ifdimensional convex poh tope with (Mitch main (4 — 1)dimensional laces, and any two Votonoi cells ate either disjoint or meet in a full (4 — 1)dimensional face Also, for any kdimensional face of ljr , them are exactly d + 1 — k Voronoi cells Vit containing it Fot convenience, we shall assume that these conditions always hold We E PI the Voronoi tessellation associated to P. The call V(P) = {Vir. V(P) is 0 lundata Voronoi tessellation of Rd; random tessellation V see Figure 8. The Voronoi tessellation defines a graph Gp on P: two VDT onoi cells are ailjaceni, if they share a (4 — 1)dimensional face, and we join two points X, y E P if their Voronoi cells ate adjacent; see Figure 8. The terminology is in honour of Voronoi [1908] who, at the beginning' of the last centur y , used these cells and tessellations to study quadratic forms In fact, concerting tessellations in two and thee dimensions, Dirichlet [1850] had anticipated \ To:tonal by ovet fifty years, so one may also talk of Dirichlet domains and Ph Oil& tessellations In 1911, these tessellations were again rediscovered (see 'Thiessen and Alter [19111), so in some circles they go under the mane of Thiessen polygonalizations
9 64
Continuum p(vcakifion
l'ignie $ [lie upper figure slums pact of I he Voronoi tessellation I (P) associated to it P0i55011 process P c E* 2 : the points ()I P me also short n If the inert t hen their common edge is pall of Voronoi cells associat ed I he per pendiculat bisector of ry the lower figures slum t he graph Cc dal ed to I . (2): I 0 points ()I P arc joined if the cur responding Voronoi (ells meet On the left each hund of Cr is (Hoc ir rising two st might line segments on the right using one I he first ewbedding (4 Gp is r lent lv plaua i II is east/ to shore 1 hit/ t he second is also
S :1 Random Vowing percolai nn
255
Other selbexplanatm y terms are Voronoi dingrmn and Mt idled diagram for the tessellation. Voronoi polygon and Di; ichlci polygon for a cell, and Poisson Voronoi lesscllation tot our raudour tessellation. In what follows, we shall use the terms Voronoi tessellation and random Voronoi tessellation The stink of Voronoi (Dirichlet) tessellations has a vet v long histolv. especiall y in discrete geometr V ill connection with sphere paekings and other problems: See for example. the books by Fejes lath [1953; 1972]. Rogers (19611 and BOrOczky [2000 Although detruministic problems rue outside t he scope of t his book let us remark that I he sphere packing pr oblem asks for the maximum density of a packing ol congruent spheres in Ed In 1929 Bliclihrldt gave t he impel hound 2(1/2 (d 2) for this density Using all ruguinent suggested by H E Daniels. Ilogets [1958] 2`'f2 d/c as impuued this bound to a certain constant ad. With (Id — x Twent y Yeats later_ Kabatjanskir and LevenStein [1978] gave a bound which is better lot > 13: recently Bezdek [2002] proved a lower bound for the sulfate area of Voronoi cells which enabled hint to improve Rogers's bound for all d > 8 There has been much research on random Vol ° tu g tessellat ions as well. first in civstallogiaph y and then in mathematics. For solids composed of diner ent kinds (A cr vstals. Delesse [I818] estimated t he fraction of the 1, (flume occupied by the it Ir CI Nstal: a cent in N Into. Chaves [1 956] gm() statistics fin these estimates. Johnson and Meld [1939] gme a model for crystal growth: the cells of this model depend not onl N on a Poisson process. but also OH tire 'arrival times' of the nuclei These cells need not even be convex, although the \ ale stardomains lion their !nuclei Meijering [195:3] int t oduced Voronoi tessellations into crystallography, without being aware ol the considel ably earlier papers of Dirichlet and Voronoi Since then, this model (attributed to Meijering. rather than Dit bidet or \known) and the related Johnson Mehl !node] have been much studied Gilbert [1962] ' moved results about the expectations of the surface area the nunther of faces. the total edge length. and other parameters of a cell (poi Itedion) in a landon) \Ammi tessellation In pm titular, he noted in passing that Enlels Formula implies that in the plane the expected numbei ol vertices (r1 a pol ygon is ti Since the 1960s. a considerable both of results has been proved about the 'typical cell of a random Voronoi tessellation: see Moller R9911 and Stu \ an Kendall and Med:v[1995] ha man y results. Here, we shall
266
Crontinuuni percolation
note only some of the MOHe recent results concerning planar tessellations Haven and Chine (2000) gave a formula for the probability that the cell containing the migM is a triangle, and Calka [2003] gave an explicit formula for the distribution of the number of sides Calka and Schreiber [2005) proved results about the asymptotic number of vertices and the area of a cell conditioned to contain a disc of radius r as Continuing the work of Foss and Zuyev [19961, Calka [2002a; 2002b], made use of the result of Stevens [1939) mentioned in the previous section to study the radius of the smallest circle containing the cell of the origin, and tire radius of the largest circle in the cell Somewhat surprisingl y although Gilbert introduced Iris disc percolation 'nuclei, and studied the random Volonui tessellation as a model of Crystal growth, he did not pose the problem of face percolation on a nanchnn Vilronoi tessellation. However, a little later, in one of the early papers devoted to percolation (hear y, Frisch and Hammersle y [1963] called for attempts to pioneer 'branches of mathematics that might he called stochastic geometry or statistical topology'. Eventually, this challenge was taken up Inv physicists; for example, Pike and Seager [1971] and Seeger and Pike [1974] per fornwd compute" analyses of the percolation and the conductance of the Gilbert model and some of its variants Detailed computer ' studies of these models were carried out by many people, including Hall!) and Zwanzig [1977], Vicsek and !Kw tesz [19811, Gawliniski and Stanley [1981], and Gawlinski and Redner [19831 among others Percolation on random Vonmoi tessellations was studied by Hatfield [197$], Winter Feld, Scrken and Davis [1981), .1cmuld, Hatfield, Scr iven and Davis [1.9841, and kraut* Striven and Davis [1984]. In panticuIan, based on compute ' simulations and the cluster moment. method of Dean [19631, %\i interfeld Striven and Davis estimated that 0 500 ±0 010 is the critical probability for random VO/Olkai percolation in the plane (to be defined below) Nevertheless, no proof was offered even for the simple fact that the critical probability is strictly between' 0 and I As we shall see later. this is not entirely trivial, unlike for lattices such as the example S in Chapter I In a series of papers, ValddiAs1 and Wier man [1990; 1992; 1993] studied firstpassage percolation on random Vononoi tessellations More recently. Gurvnel and Gliffitath [1997] proved substantial results about random \km onoi tessellations: in the problem thew consider. the laces are coknued at /widow with I colours, and then the colours change ac
S.3 Random. Voronoi percolation
267
cording to the deterministic, discretetime rules of a cellular automaton The question is what happens in the long run Here, we consider cell or face percolation on the random Voronoi tessellation Vassociated to a Poisson process P on le of intensity 1: each dditnensional cell, or face, V. x e P. of V is open with probability p, independently of the other faces See Figure 9 for part of a random
Figure 9 Random Voronoi percolation in the plane at p = 1/2 Voronoi tessellation of the plane; the 01)(1/ faces are shaded Ecprivalently, we shall study site percolation On the random graph Op, in which two vertices ;r, q E P we adjacent if their Voronoi cells and meet in a (d — 1)dimensional face In this viewpoint, given P, each site of Gp is open with probability p, independently of the other sites
Co PI ipuu pt pet (plat lop
268
Since P has (a s ) uo ticc mu t ilation points, the graph Op is a s locally finite, infinite m u ll connected This set ting is rather different front that of previous chapters: we are now considering percolation in a random NI vi ton went. i e.. stud y ing a random subgraph of a graph that is it sell tandom The fundamental concepts of percolation on fixed paints cat ty me t vet N. natm all to this set ting: we shall go met the basics in detail. Let us define a little mo t e formall y the probability measure we shall
1), Px be a Poisson process on 1P1 with intensit \ \ (Usually,
use Let .
and without loss of genet alitv, we take A = 1 ) We shall assign a state,
open of clo‘ced, to each point of P so that, conditioning on P. the states of the points me independent. and each .1' E P is open with probability p \Viitiog P + and P lot the sets of open and closed points in P, respect kel t% lion a fin nad definition it is simpler to (fist define P .' mid and then set P to be then union To spell this out. let P 1 and P be independent Poisson processes on R d w i t h intensities p\ and (I — respectively Prow the basic properties of Poisson processes, or from the (roust uct ion given eat Her . it, and P + ate disjoint a s tun Cher mot e.. given P U P each point of P is in with probability p. independentl y of
is emu 10 check that
the other points in P
w rite
ho the probabilit y measure
associated to the pith (P + ).
and Pp ha Pr p : to avoid clutter we
suppress the dependence on the dimension d. which will always be cleat hour the context. The ptolmbilik measure is defined on a state space fl consisting of oarfraturdions = (A  I .
). whet e X + and A
are disjoint discrete (i e without accumulation points) subsets of The J7field of measurable events is inherited how the constr net ion of the Poisson process \\Then talking of P and Op. we shall often ignore events (It plobabilih 0: ha example we say that
Op
is infinite rather
I han infinite as. Matt\ of the basic concepts of per colation ttanslate naturalk to the andom Vo l carol context. although in some cases there are two descriptions one gt apht hear etic and one geometric. Let us colour a \aaonoi cell
black il the point c E P is open. i e if E P . and while other
wise \ Ve say that r E
is black if
a'
lies in a black cell. i e . it c E F
tot sonre z E E R d is whitc if it lies in a white cell: see Figure 9 'Chas, a. point .r E Pl a is black (white) if and ooh if a point of
P at minimal distance flows is open (closed) Note that points I hat lie in the laces of the Volonoi cells ma 't be both black and white Ic w hat
8.`f Random inonoi percolation
269
follows tic shall mostl y write IC lin a point of P. awl o for a point of ifi'l As usual an open cherlci is a maximal connected subgraph of the graph Cp all of whose sites are Open Alternativel y a black elastic ! is a maximal connected set of black points in Rd i e . a component of the set .rt E : black} ()In twstin i ptions on P ensure that two Votonoi cells l', arid meet if and only if they share a (d— 1)dimensionallace. Hence. open clusters and black clusters i e if tool onb if :de E E(C7 are in onetoone t espondetwe Let Ei„ he the (tient that Op contains an infinite open duster Note that PALL,: ) is au increasing lunction of p Also. slime P lots ne accumulation points. E r., is exact Ir the dent that t here is an unbounded Itlack dust 01 FO/ ant L. the event IL, is independent of the values of P t"n [—L and P11— L. LP' Hence as P I' is the union of tlw independent Landon) the event E„,_ \ be viewed 0,1 E sets P ± + 1 ). Kohnogwor s as a tail c/,0111 in a product limitabilit y space 04 law (Hutment I of Chapter 2). PI 1,(E.,) is () or I fin an y p Since such Pi,(E.,) is inciensing. it foliar\ s that there is a fin = E that if p < Iilp> pn
(3)
Since Op is locall finite, infinite. and connected as noted in Output] I. there is a critical probabiliti ////(Gp) = pi i (Cp) associated to site percolation on (3 p Note that ',OC R ) is a buolow variable, as it depends on the Poisson process P How (3) if p < mi have we lime (E, Hp) = 0 a s . and. for p > ps i C'p) =1as Pe Thus
Pn(C'p) = Pit 0 s i e the I nidcun satiable pil(Gp) is essentially deterministic We sh a ll call p H the ci !tient probability fro random i'wonoi putrid& ion in noting that p it is also the critical plobabiliti rot site percolation on almost //Vet \ VOInnui tessellation associated to a Poisson process in P1't \\c next tin n to the percolation purbabiliti ll(p) l\ it h probtrhilicr 1. Use or igin lies in a unique Voionoi cell io E P. If co is open, let Cf; he tire open cluster in Op containing the site to Also let Co be the Had, diode/ of the might i e the component of black points in Trid awl Co are taken to containing the or igin II Co is closed then both
270
Continuum pc [rotation
be empt yNote that Co is the union of the cells
I
E eff
Set
0(p) =
p
i
)
IP (ci `
1P1, (C is
bounded)
p < p li (ap) and 0(Gp; > 0 for p > pu(ap), Since 0(G p) = 0 flt on i (T) we have 0(p) = 0 if p < p it and 0(p) > 0 if p > A i , just as for site or bond percolation on a fixed graph Let us note that, unlike pu(ap), lot a fixed p > [ (1) the percolation porbability p) associated to ap is not a constant but a tandom variable Let us consider d = 2 fin simplicity; we shall see late/ that y 11 (2) = 1/2. so. Cot p > 1/2, we have O(n) = E(O(Cdp i p)) > 0 It is easy to see that, fin any 1 I bete is a positive probability that ap contains k disjoint triangles 1), each of which surrounds the origin When this happens. we . Nap: p) < ([ — (1— p) 3 ) 1'. : if all Once sites in any are closed, then there is a closed cycle stalemating the et ight, and Co is bounded It follows that, even fin p close to 1, the /widow vaiiahle 0(ap:p) takes values Arbit/tu ily close to 0 with positive pr obability In pa l ticula b 0(C my; p) 0(p) with positive probability, so 0((dp; I)) is not almost surely constant In the other ditect ion. it may happen that there is one (11101 /110115 \colonel cell centred on the origin. with h un dreds of neighbows mound it: in this (1118e, lot any fixed p > 1/2. the !widow vat intik O(Gp:p) can presumably take values al bit/ to Hy close to 101
11.M C1
As we saw i l l Chaplet I, for percolation on lattices. it is I t kin! that the ethical probabilit y is bounded au a \ bout 0 and I As we teinalIced (nutlet , the cot tesponding tesult fin 'widow Volonoi pet colatien is not emit el% trivial The approach that [list sin lugs to mind is comparison with independent percolat ion 01/ a lattice: rye applied this to Gilhett's model in the previous section Unlit' Innatel y. the tessellation 1  (P) can and will contain at bitIwily large cells Thus. given any two points .r. y E Fr t . there is a positive probabilit y that .r and y lie in the saute Voionoi cell so the events • r is black' and 'y is black are not independent One can get mound this problem by continuing talld0/11 T onoi percolation to a suitably chosen 1independent percolation model, and considering the event that the lest/jet ion of (P iH.P1 to a certain local legion I? knees eel rain points in Eild to be black. whatever P + and look like out side I? \ O/
Lemma 9. ho, mum d > 2,
We
ha vc p ii (d) < I
S 3 Random l'oronoi petrolation
271.
Pion{. \\le shall l escale, considering a Poisson process P = PA oil IRd with density A much huger than 1 For each bond c of the graph E d , let L 1 , be the corr esponding line segment of length 1 in Ra , and write R,,
[4 112)
E
:
ELF. dist (x, q) < 1/2}
lot the (closed) 1/2neighbour hood of L. Fixing the bond e fin the moment., we call covet L. by three closed balls Bcf , i = 1,2,3, of radius 1/1 whose interiors ate disjoint; see Figure 10 Let but radius 1/2
13,, be the corresponding balls with the same centres
i = 1.2 ,3. of ' adios 1/1 covering the Elgin ° ID Three (closed) balls C H i . with unit line segment and the corresponding halls 1/2 (labelled only foi i II each 71, point. Of p and no , contains any point of P  . then evei \ point of IL, lies in a black (open) Voronoi cell
. the event that each B. contains at least one point of and no 8,,, contains a point of P For .r E B, , every point of B, than any point outside 13,. is Hence, whenever L.T,. strictly closet to holds. eve' v Point E C U Be is black. The event ET, depends on the restriction of (P c , P ) to the region R, and R I are disjoint, c and E E(Za ) do not slime a vertex, then Let U. l e
or meet only at boundat v points of both. We may ignore the probability zeta event that the bulimia! V of any Re contains a point of P = Pi UP Thus, Ur and U1 ate independent In fact. it S. T C E(27.,") ate sets of edges at graph distance at least I how each other, then the sets Uics and UrET 17, tneet onl y in t !wit boundaries, so the families {tic c E S}
279
Continuum pmrolation
and LT, C T lie i ndependent Thus. taking a bond o //I Di to be open if mid oni U, holds we have defined a 1independent bond pet colation model on Ed The probability of the event U,. is P A(Lr„) =
— exp(—pAv4) exp((I — p) And),
whew nit is the volume of a [Ttlimensional ball with radius 1/1 and //,/ < 3 x Tl ed is the wItune of the union of the balls 13,2 Choosing A huge enough that 0 9Ac, / > IQ say.. and then 9 < p < L close enough to I that (I — p)An d < I/10. we have EA p( )
= (1004
—
> 0 8639
'Thus. in the Iindependent bond petcolation model .11 on 271 , each bond is open with plobabilitL at least 0 8639 Hence. hom the result of Ballstet BolloMis and Walters [20(15). Lemma IS of Chapter 3. with positive limb/Minty the origin is in au infinite open path P in III lViten this event happens. tlw get/met/it path 11 E 6 ( C P/ i consists entirely of black points. so the black component Ca of the origin is unbounded Hence lift the chosen valves of p and A. 1l(` have O(p) = showing that p H = m i (d)
it (C0 is unbounded) > (1 p
hi the other ect ion. we shall compare tandom Volonoi maculation with a 2independent site percolation measure on L'': the argument is due to Balistm. Bolkrbris and Quits [2005] Lemma 10, For welt tl > 2 we ham p it (if)> Mont
rid C
0..
II, be the In peicrube
and. lif t t > 0 let 10 denote the /neighbourhood of Let S, he the event that son c ball of (adios 1/6 with trentte 11, contains no point of P = U' PT and let U, he the event that sonar black component. (union of open Votor toi cells) meets hot H,, and a point of the boundat ■ Off/L I "1 of 11[ I ' ' ° Note that if Co is unbounded then U, holds tot every r such that Co meets
i
8 ,f Random Voconoi percolation
273
Clearl y . the event 5, depends oak on the restriction of ('P". P  ) to H;, I/6) Also. U1 . is the event that Hal e is ti black path in 11;, I/6) joining 0(11,(1/1 1 Hence U, depends onl y on the colon's of the points x (of P:) ) with a E Hil l/(1) If S„ does not hold. then the closest point E H;. 1/6) is within distance 1/3. so the colom of .v is of P to any determined by the testi iction of (PUP  ) to /4. 1/2) Thus, U,,\ Sy. U (U,,\ S e ) is determined b y the restriction of and hence U, US,. = (P + . p) to /1,'/2) > 2 lot smile then the interiors If ti E :S U ate such that ler  the sets HI, 1/2) and 11;,.112) are disjoint. Let (7 be the graph with ver tex in which e and a' ate adjacent if nu  icT < I for even i. so set U S, holds. U is a (3` ) f)regular graph faking (' to he open if we obtain a 2independent site percolation model elf on G. As A — and p — 0 with Ap — 0, we have PA 1 ,(U, U S,) — By Lenuna II of Chapter 3. it follows that 'bete ate A > 0 and (I < p < I such that, with probability I. there is no infinite open cluster in Al 13ut then Co is filth (' with probability I t so fiii( d ) >
0
❑
The arguments above ale vet'. crude: we Mute included them only to give an indication of the basic techniques needed to handle the longr ange dependence in random Nroionoi percolation Pot huge fr. home\ el. it turns out that the method of Lemma 1(1 gives a bound that is not that tat from the bulk: the towe l bound in the result of Brdister, 13ollokis and Quas [2005] below was obtained in this way The upper bound is much mote difficult Theorem 11. If 4 et sufficiently huge, then the el Meal prob ab ility pH(d)
fin random Voronoi percolation on R d ,tatetfics 2  (1 (9il log d)" i < p H (d) < ( 1 2  (/ ‘71/ log& mhate C Is an absolute constant.
We now tu t u to the main topic of this section, the critical probability lot miion' Voronoi percolation in the plane If P is a Poisson process in the plane. then with probabilit y I the associated Voronor tessellation has three cells meeting at even' ve t tea: we shall assume this alwa ys holds how now on Thus the graph Up is a triangulation of the plane.. and Eidel s lot implies that the ayetage (tepee of a vet tex of Up is 6 Thus. J on avetage', ap looks like the triangular lattice This suggests that random Voionoi percolation in the plane mat be similar to site
27,1
Continuum percolation
percolation on the triangular lattice Of course, critical probability is not just a function of average degree For example, it is easy to see that fm any (small) e > 0 and any (large) d, h > 3, there is a k connected lattice A with all degrees at least d such that p7;1(A) > 1 — ctThe numerical experiments of Winter Feld, &liven and Davis 119811 mentioned earlier suggested that p H (2) = 1/2 There is also a compelling mathematical reason for believing this: random Voronoi percolation in the plane has a selfduality proper ty that implies analogues of Lemma 1 and Corollary 3 of Chapter 3, the basic starting point for tire proof of the Hart Theorem, and of Kesten's result that p7.1 (T) = pfr (T) Tr, 1/2, where T is the triangular lattice (Theorem 8 of Chapter 5). Let I?= [a, /4 [c, (II be a rectangle in t he plane By a black horizontal crossing of R. ■ve mean a piecewise linear path P C 1? starting on the lefthand side of H. and ending on the the rightLaud side, such that every point :r E Pis black, i e lies in the Voronoi cell of some E P with t: open; see Figure 11 We write H(R) 111,(1?) for the event that
Figure 11 In the figure err the left, 11(R) holds: the thick line is a black horizontal crossing of R. In the figure on the right, /1(R) does not hold
1? has a black hot izontal crossing Equivalently. 11(1?) is the event that the set of black points in I? has a component, meet ing both the left and righthand sides of I? Ignoring probability zero events, the event .11(R) holds if and only if there is an open path P =n rr2 e t in Cp such that meets the lefthand side of R, 1:;., meets the righthand side, and, for 1 < i < t the cells V„, and 1 7,„. , meet inside I? write 11„(1?) for the event that I? has a white horizontal crossing,
8.3 Random Voranoi percolation
275
and Vi,(/?), VAR) for the events that I? has a black or white ve rtical crossing, respectively Lemma 12 *
Let I? be a 'rectangle in R2 events Ilb (R) and V (B) holds
Then precisely one of the
Proof The proof is essentially the saute as that of the corresponding result for bond percolation on Z2 (or site percolation on the triangular lattice) Defining the colours of the points in 1? from the Poisson processes (P + ,P – ), colour the points outside I? as in Figure 12
Figure 12. the shading inside the (square) rectangle R. is flour the Poisson process 111 ,(R) holds if and only if the outer black regions rue joined by a black path, and V. (I) holds if and only if the outer white regions are joined by a white path. Tracing the in/Efface between black and while regions shows that one of these events ❑ 11;st hold Let I be the inter face punk i e the set of all points that are both black and white (lie in the boundaries of both black and white regions) With probability 1„ the set I is a drawing in the plane of a graph in which every vertex has degree exactly 3, apart front Four vertices of degree 1 As in previous proofs of this type, considering a path component of the
276
Continuum pereolai ion
interface shows t hat one Of HI V?) awl (I?) must Ind If boll; hold, A5 can be drawn in the plane Cotollaxy 13, If S 1 4, anti 9quarc /2( 11 ( 3 )) = 1/2
x [g. Ii (
>
O.
Mc
Proof Dv Le1 1 1111a 1 2 . lb/ U < p < 1 we have Pp(llt,(5))+P i,(Vi v(S)) = Flipping the states of all sites now open to closed or vice V(11S11. se that lil y ( Vw(S)) Flt/)(11,(8)) Fioni the sin/ i n/env of the Poisson pro cuss , = Ifi.„),(III,(8)) Thus. P),(//b(8))+IPI_,,(111,(5))=. 1 Taking p = 1/2 the result follows 0 I t ;eedinau 11997j plover' ;01 analogue of Co t 011 8 1 N' 13 rtniceining 'essential loops in 'widow \lo t onoi percolation on the projective plane lie untuoks that p 8 (2) = 1/2 In' notnett v consideurtions floweret, although Co/011m v 13 is t he 'uison wh y. p 8 (2) = 1/2. I his Elkin' ohserration is even further how a woof than in t he case of bond percolation on Z2 Fin the Val ions lattices w hose et it ical ptobabilitv is known exactly, ant of the ninn y plaids of the trainer 1Kesten Theorem can be adapted with out too n i nth work 1 7 o/ /widow Votonoi percolation . the situation is different Some of the tools used to stud y percolation on lattices do cal v met t o t his cont ext , but ot het s do not In pal ticitlat e is no ohs 101111 was to wink .Nlenshikov's I hern CM. and no direct equi snleut of t he Rosso Se% mow AVelsh Thiess is known. although, as we shall see late!. ;t weak loin; of t he hatterresult has been plover! Let us tarn w our main /11111 in t his section, a sketch of the proof t hat /111 (2) = 1/2 We star t w ith Hat rim's Lemma, t he first of the basic / esults for ordin a ry percolation Burt do extend to I he \ Tot onoi setting slate its analogue for Vor onoi per colat ion, we need t o define 'increasing (1 \ cuts' in t his context We call an event, E defined in toms of t he Poisson processes (Pt P) blockinctrusing, or simpl y increacing, if. fin ever } configuration = (Xt. XI') E and ere; configut at ion = (X.1). ) ivith C X1E and Ail AT, we have E In other words. E is preserved lw the addit ion of open points, t he deletion of closed points 1 he definition of an increasing function j (P + P) is analogous: / is hue/easing if adding open points or deleting closed points cannot decrease the value of .1 Since a point of 1R 2 is Hack if t he (a) newest point of p is open adding open points and deleting closed points can change fhe colon/ of a point
S Random Voinnui ' ,ovulation
277
2 how vylire to black. but not vice versa any event defined by the existence of cer brill black sets is incwasing: an example is the MVO event It is easy to see flow Hat is's Lemma that, with respect to P p . incwasing events rue positively correlated, at least if they are sufficiently well behaved In fact. all increasing events are positively correlated To see this, let us first ()onsider the case of a single Poisson process of intensity I on 1?2 Let us wtite (Q 1 iF ) lot the corresponding probability space, so Str is the set of all dist:tete subsets of F 2 . P i is the pr obabilit y Wen:it/In defining the Poisson process We shall write E i for the associated expectation. Let I. 2 he two bounded measurable in I lie sense functions on 0.1 1 , P i 1. and suppose that each is that ir) C u/ implies ,b (w) < (La') Following Roy [1991) let E k he the afield gene/rued hi the rolloir ing iulou ttation: set g = 2 k , divide Hu. a] 2 into do squares of sidelength and decide fru each whether or not it contains points of Y E Thus Er. pm (Rim 's Q., into 2 4 " pmts. with each part consisting of all E 1 that have at least one point in each of cei lain small squaws. and no points in certain other small squates Any two discrete sets lie in differ ent pat Is of E i k lot large enough so t lie sequence Er. is a lint at ion of M t , Pi) As E l (pr 1. ma y be viewed as an increasing flint( i011 On a discrete product space. bout Lemma I of Chapter 2 (a simple coiolltu t of Hauis's Lemma) we have E R
11(R)=
i =
l/n,
C2
I.
EI(EW/i :0EM2 1:0)
(EMI E k)) E 01(1/4 I Ek))(*)) Ei(m)Ei(f/2)
Since i is hounded mid Er. is a lint at ion, lir, the Mat ti Theorem (see, [or example. 'Williams I1991D g
E t ( g i I L k) 22'
le Convergence
C/a
almost er el \ here. Hence la dominated comet gence. the lefthand side of (5) converges to (g i go). proving t hat Et
( g i q2)
POE' 02)
(6)
whenever th and are nwasumble hounded and Met easing The cot tesponding result flu the colour ed process follows immediately \Vt. ) state this 418 a lemma giving onl y the character istic function case  although the proof Fin increasing functions is the smile we shall not gu
Conti truant percola lion need the result. Hem. then is the equivalent of Harris's Lemma for taudom Voronoi percolation in the plane Of course. a corresponding result holds ha random Vol own percolation in Ea
Lemma 14. Let Ei = E i (P + .2  ). = 1.2. be two blackMe events. Then lo t any 0 < p < 1 use hone IPy (E,
n
IP,,(E1)Pp(E2),
where iti the probability measure minor:toted to fandom Volonoi percolation in the plane Pl oy/ We shall w i ne E
i e )Cei at ion with respect. e to EL p Let, J; be the elm/atter istic function /if E h so h is blackincreasing Fixing F lot (guilt h. I (P r .P  ) is increasing in P l.. so IA O. Eli 12 I P
P HE(/ I P4
^
fat every possible 2  Taking the expectation of both sides ((wet Ps), GUif2)
ri( E t (r I P  )E( ig 2))
But as 1' 1 . Ei ale dee/easing in P , so are the functions g i = E(1; I P). Applying (6) to 1 —y 1 sad 1 go. which ate incteasing functions ° LPthat ( thl Y. we see th a t E g ig2) > E(gr )E(q) EP4 Tr P .p a/H.21 P  1) ^ E(E(li P))1E(E(./2 P)) =
11': (
)E( /2)
Combining I he last. two inequalities, we see that Et J i Li) > as el/Mucci
f i iEt (p),
❑
The uniqueness theorem of Aizumnan. Kesler ' and Newman [1087], Theorem 4 of Chaplet 5. also carries over to the Waonor setting (in any dimension). Just as fur the Gilbert model, the proof given by Button and Keane [1989] goes through. although a little cu t e is needed with the details: we shall not spell these out
Theorem 15. Fm any A > and any 0 < p < 1 thene is alumni ninety at most one infinite open cluster in Op
❑
In Chaple t 5. we presented a proof of Hai / is's Theorem due to Zhang This moo( retied on the basic crossing lemma (Lemma 1 Of Chapter 3).
8.d Random Voronoi percolatton
279
Harris's Lemma, and the symmetr y of the square lattice In his rutpublished MSc thesis. Zva y itch [1996] pointed out that Zhang's proof carries over to the random Voronoi setting, giving the following result. Theorem 16. Fm random Voronoi percolation in the plane, 0(1/2) = 0 Hence, pn(2) > 1/2. ❑ Unfortunately, none of the many proofs of Kesten's Theorem seems to adapt to the random Voronoi setting: there seems to be no easy way of proving the analogue of Kesten's Theorem Nevertheless, using an argument which is considerably more involved that any of the proofs of Kesten's Theorem, Bollolgis and Riordan [2006al did prove this analogue
Theorem 17. Fm nindom Voronoi i noculation h i the plane, par = 1/2. As we shall see at the end of this section, the proof of this result also gives an analogue of Kesten's exponential decay result, 'Theorem 1.2 of Chapter 3, and so establishes that the natural analogue of pr is also equal to 1/2 As /Mira/ Iced above, the wool of Theorem 17 is rather involved, so we shall describe only its key ideas: for full details we refer the reader to the or iginal paper Perhaps surprisingly, the lack of independence in the random Voronoi model turns out not to be the main problem (or even one of the main problems) Indeed, it is easy to see that fig rectangles I? and say, separated by a moderate distance, any events E. E' depending only on the colours of the points in B. and B.' respectively are almost independent In particular, there are independent events E. E' that almost coincide with E and E': this property turns out to be just as good as independence We make this idea more precise in the lemma below Hem, we think of s as the scale of the rectangles I? and 1?!: we shall take s cc with all other parameters fixed From now on, we fix the irrelevant scaling of the Poisson process 'P = TA by setting A =1 Given p > 1, s > 1 and a ps by s rectangle B, C R2 with any orientation, we write F(Its ) = F,(R.,;) for the event that every ball E B, (a:), contains at least one point of P = P + U where = 2y/log s (It does not matter whether we consider open or dosed balls; up to a probability zero event, which we shall ignore, this makes no clifference )
Continuant Navarreton
28(1
Lemma 18. Let p> I be mistant let C IR 2 he a / pi ha s rectangle, > 1 god set r 2v/logs Theo (R,) = 1",(R,,) holds with probability rilso if E(R s ) is any event defined 0 0 1 0 in terms — o(1) as s of the colours of the points in R s . then E(R,)11F(R 1/2 ) depends only on the rest/idiot) of IP 1 .P — ) to the rag a hbourhood R, Progl The firs t statement is immediate horn the properties of it Poisson process Indeed. we 1111W covet Rs with 0( s2 / I2 ) = s2 ) disjoin t (half open) scrum ps S, of sidelength r [ ■122 The area of each 9, is /2/2 2 hogs. so then number of points of P in has a Poisson dist) Haitian with mean 21 0g s Hence. (he probability that a particular 8, contains II() points of P is exact Is C 2 Thus. 1eitlr in obabilit \ I ) [ A vi v S. contains a point of P. and FL/11,1 holds The second statement is immediate Itonn the definition of 17(13.4:11nr argument is as in the p i vot of Lemma 10 above Indeed E(R...) chanty depends oils on the restrict ion of (Pt P — ) to the neighbour hood ./ef) of R, [(Ili's) holds then the closest point of P to ^ 111V .r E R, is r point of B, (r), and hence a point of ri in Mc') Thus the colour of ,r is det e i mined by the closest point of the restriction of (P I . P — ) to [((;)
Of course thew is nothing special about rectangles in Lemma 18; choosing t he slim let site of the rectangle am 010 'scale parameter s will be convenient lac er Also Lemma 18 has an analogue in am dimension, concei ving cuboids say mi ning now to the specific stud y of dimension t \so. the lundiunental quantit y that wee Awn!: %vitt; is the probabilik of a black crossing of a rectangle. Let /„(p, s) =
(HO. ps] x [0, si))
he the P/3 piobabilits that a ps bs s Ieelangle has a black Innizontal crossing start with two easy observations about the behaviour of ji ( p s) Fi t st /p(m)s) 5 1,(112. s) whenever p l < pg, as the cor i esponding events ale nested 'The second obsei sat ion is that / /,(01
—
)
/ ),(m s )hAt)2. s )1),(
s )
(7)
foi all m > 1. The argument is exactl y as in Chapter 3: Iet R 1 and R 2 he m s ha s and / srs he s rectangles intersect lug in an s by s squaw
S Bondi n
own porcolotion
98
Figure 13 1 vo rectangles Pr and lip intersecting in a square S. 11 P and P_ are Hack f t .arzontal crossings of fir arc P2 , respectively, an 1 Pr is a black vertical crossing of then P t U U Pr contains a black horizontal crossing of Pt U Po
S. as in Figure 13 The events H(R I ). 11(R ” ) and F(S) ate increasing Thus. la Ha t is s Lemma. in the Form of Lemma H.
PgH(R
niR re 2 )nits1)
> Pp(H(RWPI(H(R2))11'1,(1.(3))
If 11(R I )na(R 2 )n r (5) holds, then so does II (R U As re,(1"(8))= P(H(8)) =
(see 1:',.igitte
1
s). I he telatimi (7) follows
The itatin al analogue of the Russo Seymour Welsh (RSW) 'Thewern would state that if fp (1, s) > s > a then p (p. s) > ird > for some function q(9, E) not depending on s (so. in particulin fr it .(2. ^ ) is bounded awa y flout zero) Such a statement: has not been proved lot ardour Voionoi nett:oration ft might seem that the proof of the RS \Vtype thethent te e presented in Chapter 3 would early me t to the random Voionoi setting. The strut is indeed promising: thew is no problem defining the leftmost black vertical crossing' Lt (P) of R whenever 1 1 (R) holds: we simply follow a blackwhite interface as befote. The problem is the next step: the event LT(R) = P is independent of the stales of the points of
P to the light of P. but not of II/eh positions: we can find LP(R) without looking at the colours of the cells to the light of 1,1"(R), hut knowing LI ; (1?) tells tts whew tlw centres of these cells are, and that thew ate no points of P in certain discs, puts of which are to the light of .LII (R) Thus, there is no simple way of showing that the limitability that LI: (R) is joined by a black path to the tight side of R is at. least P(11(R)) Of course, a general problem wit h iandoin Voronoi percolation is that no two regions ate independent, although wellseparated legions ate asymptotically independent The first step towards the proof of Theotent 17 is the following result:. although this is considerably weaker than a direct analogue of the RSW Theorem. it tutus out to be sufficient lb/ the determination of pa (2)
Continirilin percolation
282
Theorem 19. Let 0 < p < 1 and p > 1 be fixed If lint inf, 0, then lint sup,_, f,,(p, ^ )> 0
(
s)
Lemma 12 states that the hypothesis of this themem is satisfied for p= 1/2: thus Theorem 19 has the following consequence
Corollary 20. Let p > 1 be fixed. There is a constant co = to(P) 0 such that . for every so them is an s > sa with f il s(p, ^ )> Co. The proof of Theorem 19 is rather lengthy, so we shall give only sketch Proof We proceed in two stages, throughout assuming for a contradie2 tion that the result does not hold Thus, there is a constant cm > 0 such that 40,
(8)
for all large enough s, but, for some fixed p > 1, Ip(IL, s)
(9)
0
Here, and throughout, the limit is taken as s pc with all other parameters, for example p. fixed. In this context, an event holds with high probabititw or why), if it holds with probability 1 — o(1) as s cc. 'The assumptions above imply that. Mr any fixed E > 0, fp (l +5,5) as s
0
(10)
xi Indeed. taking > (p —1)/5 and using (7) k times,
fp (p, ^ )
.n,(1
/,76, s)
fff ,(1
s)fp(1, (b,(1 +
fp (1
(k — 1) 5 , s) >
s) fp (1, s)) k f„(1, s) 71i. 0,
contradicting (9) Condition (10) imposes very severe restrictions on the possible black paths crossing a square, fOr example: roughly speaking, no segment of such a path can cross horizontally a rectangle that is even slightl y longer than high Thus, fot any E > 0, wisp all black horizontal crossings of a given s by 5 squat e 5' pass within distance E s of the top and bottom of 5': otherwise, with positive probability we could find a black horizontal crossing of on e of two 3 by (1 — E)s rectangles, contradicting (10) The next observation is that wh i p any black horizontal crossing of air s by s square 5' starts ar i d ends near the midpoints of the vertical
S 3 Random I lmonoi percolation
283
sides. Indeed, if there is a positive pr obability that some black crossing P starts at least es/2 above the midpoint of the lefthand side, then, reflecting S in a horizontal line and shifting it vertically by es to obtain a square there is also a positive probability that some black horizontal crossing P' of 5' star ts at least 65 12 below the middle height of 8', and hence below P. By Harris's Lemma, crossings P and P' with the stated proper ties then exist simultaneously with positive probability But, whip, P must pass within distance es/3, say, of the bottom of 8, and hence, below S' This forces P and P' to meet; see Figure 14 As P' passes
8'
Figure 14 Two squares S and S'; their horizontal axes of symmetry are shown bye clotted lines Each of the horizontal crossings P. P' of S and 5' passes near the top and bottom of the square it crosses. If P starts above P' then t he paths cross: P ' must leave the region bouncied by partt of the bounds ' v of S' and the initial segment of P up to the point it
within distance Es/3 of the top of we find a black vertical crossing of a rectangle taller than it is wide Using rotational symmetry of the model., this contradicts (10) In a moment:, we shall state precisely a consequence of (10).. For now, we continue with our sketch of the argument Suppose that P is a black horizontal crossing of the square S = [O, x [0, s] Let Pr be the initial segment of P, starting on the line x = 0 and stopping the first time we reach s = 0.99s Similarly, let I2 be the final segment of P, obtained by tracing P backwards horn the = s until the first time we reach x = 0 01.s; see Figure 15. Arguments similar to those above show that each of Pr and P, star is and ends yen \' close to the line = s/2. Fur thermore, one can show that, whip, since each Pi crosses a
281
Continuum petrol& an
Figure 15 A squa t t S (outer lines) and a black path P crossing it (solid and dot ted curves) 'The path P i is the initial segment of P s:opr ing at Hie inner vertical line The he al segment P. of P is defined similarh Ii ;uglily speaking, each Pi crosses a square smaller than 5', and so cannot a vroach too close to the top and hot ton of .9 Thus ;24. P: her wise a slightly flattened rectangle in S would have a black horizontal crossing T he dotted p i nt at in fact passes very close to the top mid botto m of S
rectangle of width II 99s, it remains within height ±(0 195 4 5)s of its stinting points. It follows Hat Pr and A are wh p disjoint: otherwise, P = Pr U is a Hack crossing of 5' that does not conic within height 0 004s, say, of the top and hottom of S: the probability that such a crossing exists is o(1) by (10) But then each of Pr and P, similarly contains two disjoint crossings of rectangles of width 0 98, and so on
i
ft is not hard to make the vague rugument just outlined pwcise, although it, turns out to be bet ter to work in a r (retail*, [0. x [—Cs. Ci], say, of bounded aspect ratio 2C > I. In Bollobris and Riordan [2006a1, Hie following consequence of (10) is proved Claim 21. Let C > 0 he ficed and lel 1? =
be the y by 2Cs rectangle [0.s] x [—Cs.Csj Pot < j < 4. set [js1100.(1 90) ,4/1001 x [—Cs.Cs] Assam:My Mat (10) holds why eve, y black path. P (71),;sing i < aqierc I? ha/C.:on/oily contains 16 disjoint black paths P. 1 each Pi crosses ,/onte 1?) Ito/ icantally. In other \t'olds, the path P contains 10 disjoint paths 1 i winding
S3 Random 1 /01'0110i percolation
285
backwards and to wards in a manner similar to the paths in Figure 15 Of course, in proving Claim 21, one must: be a lit tle mo t e careful titan in the vague outlirie above Never tireless, the proof is fair ly straightfor ward: it uses only Harris's Lemma, certain symmetr ies of the nuclei, and (H)) The last condition is applied to show that \\lip none of a fixed number N = N(C.e) of rectangles with aspect ratio 1 + s is crossed the long vu}' In a black path The conclusion of Claim 21 is clearly absurd: 1? has a black horizontal crossing with probability at least f,(1, s), which is hounded away from zero Taking the shortest black horizontal crossing of R. this somehow winds backwards and forwards almost all the way across R. containing segments that star t and end in almost t he save place lint somehow never meet each other Also, the shot test crossing of R is almost, certainly at least 16 times longer than the shortest crossing of a slightl y narrower rectangle As the length of a crossing of R, cannot scale as more than 52 , this is impossible All this sounds convincing: nevertheless. it is not so easy to deduce a contradiction The problem is t hat, while the constant 16 in Claim 21 can he replaced by any l a rger constant it cannot be replaced by a function of s tending to infinity: we can only appl y (10) directl y ton bounded number of rectangles. Otherwise.. t he o(I) error probabilities il(t1/11111late En get around this, we use alutostindependence of disjoint regions to .seurre the error probability' Tire idea is to consider the length (as a piecewise linear path) 1,(IR) of the shortest black horizontal crossing P of certain .s by 2.9 rectangles R. when such a P exists We take L(1?,) = x it I?, has no black horizontal crossing. Even fen widely separated rectangles. L(R s ) and L(131,) are not independent, so we modif y the definition slightly to achieve independence As in Lemma 18. let P(R s ) = (R s ) be the event that e ye/ y ball of radius r = 2Vlog s centred at a point in R., contains at least one point. of P, and set 7(1?,) =
L(R)F(Rs) holds 0 of her wise
13v Lemma 18. 1,(1?,) = 1,(R.) whp Furthermore. 1.(R,) depends only on the restriction of (P + ,P) to the t meighbom hood of 17, Hence. if R s and 1r, me separated by a distance of at least 2r o(a). tire random variables 1,(17,) and Z (TC) are independent Roughl y speaking, we wish to relate t he distribution of i,(R J. which
286
Continuum pc/colahop
depends only an to the distribution of L(I? s /2 ), using Claim 21[ fact to leave a lit t le elbow room, we relate L(I?„) to L( Bo ). say For 71> 0 a (very small) constant, define g(s) by g(s) = sup{ :
CL (R s ) <
<
where I?, i s any s by 2s rectangle Recall that that L(1?,) < at if and only if I?, has some black horizontal crossing, an event of probability f,,(1/2, 2s) > > cy > 0, horn (8) As L(R.,,) L,(1?,; ) Min, it follows that 0 g(s) < cc if q is chosen small enough. and then s large enough, which we shall assume from now on Also, the sumer/m i n above is attained. so Pii(L(Rs) < 1/( s ))
( 11)
Claim 21 tells us that, whp, ever y black horizontal crossing of R.,[[ including the shortest. contains 16 disjoint crossings P, of slightly rower rectangles. Using the fact that whp every crossing of a 0 96 9 by 2.5 rectangle is actually almost, a crossing of a square, we can place a bounded number of 0 47s by 0 94s rectangles 1 < i < N, such that; w hp, each Pi includes crossings of some pair (R, R,;) of these rectangles, with and Bk, separated by a distance of at least 0 01.s Here N is an absolute constant.. For each such pair (16,Ra), the variables L(B' ) and L(R k ) are independent, so (1.1 (11 ) )
7(&) < g(0 17s)) 5 Pi, (11(Rj ),11(Rk ) < q(0.47s))
= F n (2:(R') < g ( 0117s )) illp ( Z ( R k) < g(0 47s)) 5
112
from (11) Hence, the probability that i(IL) ) + 2,;(1?k ) < g(0 .17s) holds for saute pair (Hi , Bk ) separated by distance 0 Ols is at roost N9 and hence at roost 11/2 if we choose ri small enough The proof of !lemur 19 is essentially complete: horn Me remarks above, whp every black horizontal ossing of B. has length at least 16 times the minimum of L(R 1 ) + .L(Rk ) over separated pails (B1. Rk). Thus, L(R,) < 16g(0.47s))
g/2 o(1) < q,
so g(s) > 160 47s), and g(s) grows faster than s [1 , say As g(so) > 0 for some so, it follows drat there are arbitrarily huge s with (33 < g(s) < cc. But then. with probability bounded away horn 1, the shortest black hor izontal crossing of IL, = [0, ,s1 x [0,2s1 has length at least [[; 3 'This is
$ 3 Random Voronoi percolation
287
impossible, since, will). R. meets only 0(s2 ) Vorouoi cells, and each has diameter at most 0(log s) ❑ As noted in the original paper, the proof of Theorem 19 outlined above uses rather few properties of random Voronoi percolation: certain symmetries, the fact that horizontal and vertical crossings of a rectangle must meet, and an asymptotic independence property. For this reason, the proof carries over to many other contexts; see, for example, Bollobris and Riordan [2006b] \Tan den Berg, Brouwer and VrigvOlgyi [2006] proved a variant of this result for 'selfdestructive percolation', a model of forest growth taking into account forest fires, which they used to prove results about the continuity of the percolation probability in this model Using Corollary 20 and Harris's Lemma, Theorem 19 easily implies that 0(1/2) = 0, giving an alternative proof of Theorem 16 Of course, this result can be proved more cleanly by adapting Zhang's proof of Kestert's Theorem; see Zvavitch [1996] However, Corollary 20 is a good star ting point for the proof of the analogue of Kesten's Theorem, namely Theorem 17 The main idea of the proof is to use some kind of sharpthreshold result to deduce that, for any p > 1/2 and any so, there is an > so with (12)
f„(3,5) > 0.99,
say Before sketching the (rather lengthy) proof of (12), let us see how Theorem 17 follows The argument:, based on 1independent bond percolation on 2Li 2 . is very close to an al gument given in Chapter 3 Given a 3s by s rectangle I?, with the long axis horizontal, let G(R.„,) = II(R s ) n V(5 1 ) n V(S2), where S I and 5, are the
two
s by send squares' of R see Figure 16
g le Figure 16 A 3s by s rectan
such that
G(R s ) holds
B y translational and rotational symmetr y. Pp ( I ( .52)) =
(Sr =
p( ER S1
)) = (
ji,(3, ^ ) > 0 99.
288
Continuum purolation
so P 1,(0(.11„)) > 0,97 Let
G(Rh), G(R.,) n F(1?„), and define elkfil,C) similarly for a 33 by 5 rectangle RI, with the long side vertical From Lemma IS, Pp (F(Rs )) 1 as s x Thus, if s is huge enough, P, (G(R, )) > 09 (1;3) for every 3s by s rectangle Rs . By Lemma 18, the event 61 (R s ) depends only on the restriction of the Poissiu processes (P lP ,P) to the Tmeighbewa hood of B. Choosing s larg , enough. we may assume that = 2Vlog < s/2 For each horizontal bond 0 = ((a, b). (a + 1, IR) of V let = [g as, 2as + 3s] x
2b9+ s)
be a 3 3 by s rectangl e in 1E 2 Similxtly for e = ((a. b). (a,b 1)). set ,R, = 12as, 2as + x [21m. 2bs + 33] Let us define a bond percolation model AI on 1 2 by taking a bond c to be open if and only if G(R„) holds. If e and I' ale vertexdisjoint bonds of 1 2 , then the rectangles and RI are separated by a distance of at least s. Hence. if S and T me sets of bonds of V at graph distance at least 1, then the families {C(/?) : 0 E and {6(R„ ) : e E T} depend on the restrictions of the Poisson processes (P4),P) to the disjoint regions MI) and U rE7 RI B t , and are thus independent Hence, the probability mensme associated to AI is 1independent From (13), each bond is open in 21 with probability at least 0 9 so, from Lemma 15 of Chapter 3. with positive probability the origin is in an infinite open path in A7 Such a path guarantees an infinite black path meeting [0. SP (see Figure 11 of Chapter 3), so 1F),([0, s] = meets an unbounded black component) > 0. The increasing event that even point in [0. SP is black has positive probability, If this event holds and (0, meets an unbounded black component, then Co is unbounded From Harris's Lemma, this happens with positive probability so 0( /)) > 0 As p > 1/2 was arbitrary, and 0(1/2) = Theorem 17 Follow s Our aim for the rest of the section is to sketch the proof of (12). that lot any p > 1/2 and an) so thew is an s > 3 0 with f),(3.. 3 ) > 0.99: as we
Random l'oronot percolation
289
have seen, this implies 'Theorem IT As a starting point. Corollary 20 is strong enough: one o l d\ needs / 1/ 0(3, s) > err for ma l e sufficiently huge s In a discrete setting, we could easily use the Ftiedgutkalai sharpthreshold result (Theorem 13 of Chapter . 2) to deduce that, for this s, we have fp (3, s) > 0.99 It is not unreasonable to expect it to be a simple matter to adapt this argument to random Voronoi percolation, by choosing a suitable discrete approximation, as in the proof of Theorem 3. say However, as we shall see, there is a problem In Chapter if, we presented various methods of deducing the statement, fin bond percolation on V that corresponds to l i,(3, ^ ) 1 for p > 1/2 fixed One of these. the method based on the study of symmetric events.. originated in the context of random Voronoi percolation the others do not seem to adapt to this setting As in Chapter :3, to define symmetric events we shall work in the torus, i.e.. the quotient of Pf 2 by a lattice Mote precisely. we shall work irr the torus 72 = L o = R2/(10sZ)2 i.e the sox face obtained from the square (0. l(Is) x (0,10,s1 Ire identifying opposite sides Here s will be out scale parameter; we shall consider rectangles R in 1'2 with sidelengths Afs„ where 1 < k < 9 The notion of a random Volonoi tessellation makes perfect sense on the torus: we star t with Poisson pr ocesses P + and P  of intensity p and I p on I'm which we may take to he the restriction of our processes on Pi2 to (0. 10sI2 • The definitions of the \Unman cells and of the graph Op. UP ate as berme. We shall always consider s huge It is easy P = y disc in of radius \/log S. sa y, contains a point to check that whp ever of P, so the diameters of all Vinonoi cells are at most 2 Ylog s = o(s) In particular., no cell f ivr aps mound' the torus The rectangles R we consider also do not come close to wrapping around the tot us Thus, events such as .11 (1?) have almost the same probability whether we regard I? as a rectangle in R2 ca in T2: 7:2
11= , (H(R)) =
(II(R))+, o(I)
where Pi, and Pr;' = F,, ar e the probability measures associas s — ated to Poisson processes (P lil .P) on '12 = Trotand on R 2 respectively There is ji natural notion of a's y mmetric' event with respect to the measure P2„ namely, an event that is preserved by translations of the torus The is one of t he two ingredients required for the application of the Ft iedgutKalai shat pt lueshold result: the other is a discrete pr oduct space
290
Continuum percolation
Let 6 = 6(s) > 0 be such that 109/6 is an integer, and divide T 2 = Tios up into (10s/6) 2 squares 51 of sidelength S in the natural way We shall approximate (Pt P) (defined on the torus) with a finite product measure as follows: a square S i is bad if [Si n > 0, neutral if i n 0 and n P 4 > 0; see Pfl P + = 0, and good if 18 1 n Figure 17 Thus, open points (points of 7) ± ) ate good, and closed points 1S
1
= 1Si
1
1
a
Figure 17 A small part of the decomposition of the torus into squares Sr. Points of P" are shown by dots those of P. by crosses A square is good (heavily shaded) if it meets P 1 but not r , neutral (lightly shaded) if it meets neither P 1 nor P , and bad (unshaded) if it meets PM The 'crude state1. of the torus is given by the shading of the squares Usually, almost all squares are neutral
are bad, and the presence of a bad point outweighs that of a good one; we shall see the reason for this choice later For each 4, the probabilities that 51 is bad, neutral and good ate it had Pnemrrat Prood
 exp (62 (1  p))
82 ( 1 — p),
exp(82) = 1  0(62 ), and eXP (
( 11)
 62 (i  p)) (1  exp(52p))
respectively, and these events are independent for different; squares By the trade stale of Si we mean whether Si is had, neutral, or good; the erode slate CS = C5,5 of the torus is given by the crude states of all (10s/6) 2 squares Si It is cleat that if S is sufficiently small as a function of s, then CS essentially determines (I) + , P), and thus Op (defined on the torus).
8.3 Random 1/monai percolation.
291
Indeed, for s fixed, the afields generated by CS =CSo are a. filtration 72 of that associated to PI, In fact, fen the discrete approximation to (essentially) encode the entire graph Gp, it is enough to take 6 = a(811). Indeed, it is easy to check that if S is this small, then two things happen wisp First, no Si contains more than one point of P, so CS essentially determines (p' 1 ,P – ), up to shifting each point by a distance Second, there is no point of T2 such that the distances to the four closest points of P lie in an inter val 2 0.,(5] Thus, shifting points of P by at most \FM does not change which Voronoi cells meet, and so does riot affect the graph structure of Op As we shall see. we shall not be able to take S this small, so the graph structure of Op will be affected by the discrete approximation, although only 'slightly' The possible crude states CS of the torus correspond to the elements of Q" = (1,0,11", where a = (10s/S) 2 is the number of squares Si, and for w (w 1 ) E Q" we take (+4 to be –1, 0 or 1 if Si is bad, neutral or good, respectively. Let us write Pp_ r,. for the product probability measure on Q" in which each w i is –1, 0 or 1 with respective probabilities 1,–, 1 – p_ –p+ , anti pi. Then the discrete approximation to P. given by C'S corresponds to the measure on Q" More precisely, if E is any event that depends only on the crude state of the torus. then P„ (E)=
(F),
(15)
where F C Q." is the corresponding event. and pg„„d and p are related by (11) It will clearly be convenient to use a form of the Fr iedgut–Kalai sharpthreshold result for powers of a 3element probability space (1,0,11 In this context, an event E C .C2" = (1, 0, 1}" is increasing whenever w for every then w' = (c.,1) E E An event (w i ) E .E and Ji > E C Q" is symmetric if it is invariant under the action on 4" of some group acting transitively on [rd, {1,2, . . From (11), both p kid and pg,„Ki will be very small, so the sharpthreshold result we shall need is an equivalent: of Theorem LI of Chapter 2 As noted by Bollobiis and Riordan [2006a], the proof given by iedgut and Kalai [1996] extends immediately to give the following result: Theorem 22– There is fai absolute constant Ca such that, '11' < q_ < p_ < 1/c, 0 < p+ < q+ < 1/e, E C 0, 11" is symmetric and
Contiaaoat ',cicalaI ion
292
increasing and Ft : win { ri+ — where p„„,
(E) >
p_ — q  }
then
(E) > 1 —e whenever
tid log( 1/E)p„,„, log( 1
)/ log tt.
( to)
nrx{g+.p_}.
When we come to apply this result, the hunt of (16) will /natter; this is in sharp contrast to the situation lot bond percolation on 2: 2 , Indeed, when we used Thement 13 of Chapter 2 to (move Kestells 'Theorem in Chaplet 3, the hun t of the era responding bound was not it/tiro/taut All that, mattered was that, with s: fixed and II tending to infinity, the inctease in p required to taise P,,(E) front E to 1 — E tends to zero Mete even though we are using a strange/ tesult, (with the extra factor p,„„, log(.I/p,„;,„) working in out la yout). flame is a limit as to how' small we can take (5 while still getting a useful result front Theorem 22 Vie shall eta/pant (E) with Pi t,(E) fot a suitable event E. when/ p > 1/2 is fixed. Row Corollary 20„. the hate/ probability will be at least some small constant E. and to prove (12) it will suffice to show that (E) 1 — E. In doing this, we shall make a discrete approximation as above. Et ow (1+1). when we apply Theorem 22, we shall have —
p_ — a_ ,= itS 2 — 6 2 /2 = e(a)
Also, a will he constant. p„„, = 8(e5 2 ), and n = (10s/(5) 2 = e(r2/8.2) Substihdiug these quantities into (16)„ we see t hat this condition w ill be satisfied if and out if a > lot some constant > 0 depending on p As p ant/ °aches 1/2 front above. the constant ^1 tends to zero Thus, we cannot aff o rd to take a lei ibly fine discrete approximation; recall that = 0(S  ) vonld be needed for the alma oxiination (access not to affect, Op at all With for  a 5 0 1 8 11 positive constant, out discretization pottiest; will inhoduce ! defects' Mime, given the 'rounded' positions of t he points of P. we do not know which Voionoi cells actually meet The density of these defects will be small (a negative power of s)„ so one might expect them not to matte!. 'r Ids tutus out to he the case, but n eeds a lot of wo rk lo prove Benjamin' and Seta anon 11998] env/untie/ ed the same p i oblent in ram ing a certain 'confoi mal oval fiance' pi upc/ tv of random Vet/m ini per colation in two and since dimensions This is not conformal inwat lance in the sense of the conjecture of skizentrunt and Langlands Pouliot and SaintAubin [199f] discussed in (Maple' 7 Instead, in two dimensions., what Idenjamini and Scht alma proved is essentially the following Let
8 d Random Voronoi percolation
293
C IP1.2 he a (nice) domain. and let S i and 5"g be t g o segments of its boundm y Consider t he Vol onoi percolation associated to a Poisson po t cess O f intensity A on 1?. using a cot tarn metric ds to form the Voronoi cells, rather than tine usual Euclidean metric. Then. as A — a fixed confor 'nal (locally anglepreset ving) change in the met:tic ds does not change the probability that there is a black path horn S i to So by tame than o(1) Let us note that this is also a statement about detects: each \Towing cell is very small. and, locally, the change in the metric is multiplication he a constant factor, so there \\ill be very few 'defects' whore different. Vor (anti cells meet for the two metrics: in two dimensions the expected in three dimensions there number of defects is bounded as A — are more but the density is still vei l: low the result of Benjamini turd Schr arum is that these defects do not affect the crossing probability significantly: even though there ate very few defects the proof is far nom simple In proving Theorem 17 we have a large] densit y of defects: on the of het hand. we have the freedom to vary the probability p, replacing an err bitril/ V p> 1/2 by (p+ 1/2)/2, say. Roughl ■ speaking. we shall show that the effect of the defects is smaller than the effect of decreasing p in ti l ls way Let us nog make some of the above ideas precise Let 1? he a rectangle , in 2 NVe wish to define an event E depending on the Poisson processes (P c . P) on 72 :Mtn that, whenever E holds. then 11(1?) holds even if we move the points of P a little. Given rt > 0. rte say that a point A t E I" E P + with dist (.r. z') > dist(1 . , z) is 1i robustly blink if them is for all o f E P; in other words, the nearest open point of P is at least a distance closer than Ow neatest closed point If rte move all points of Pat most a distance if/2. then any .c E 1 2 that was ifrobustly black (and hence black) will still he black We SUN that a path P C 12 is itrobustly Huck if every point of P is tfa ()busily black. It tin US out that if 0 < pr < po < 1 and > 0, then we can couple the probability measures Ft.), and so that, whp, for event' path PI that is black with respect to the first measure, there is a 'nearby' path Pr that is if robustly black with respect to the second measure, whew = < Theorem 23, Let 0 < u(s) be any function with q(s) <
and > 0 be given. Let ill = 111c way cough nil in the same
294
Con tinituill percolation
probability space Poisson processes P r* PT and o re/ = nos and 1 – po respectively, such that P;.4" with intensities p i! 1 – and pi are independent foi each and Mt; following global event Eg x fo i every piecewisefinew path P 1 that is black with holds whp as s respect to (Pt , p i ) there is a pieceiviselineal path P that is 4hrobustly black with respect to (PT ), with (In (Pr, Po) < (logs )2
The proof of Theorem 23 is perhaps the hardest: part of the proof of Theorem 17, and we shall say very little about it Essentially, we roust deal with certain 'potential defects', where the four closest points of Pr = U' I– to some point E 11 .2 are at almost the same distance (the distances differ In at roost s –e ( I )) We call such points of Pr bad, Roughh speaking, a bad cluster is a component in the graph on the had points in which two bad points are adjacent if they are close enough to interact. The heart of the proof of Theorem 23 is a proof that, tinder the conditions of the theorem, whp the largest bad cluster has size ° (log s). As the probability that a given point is bad is 5 –e(0 , one might expect the largest bad cluster to have size 0(1); it turns out. however, that there will be bad clitsters of size e(log s/ log log .^ ) Hence, the o(logs) bound needed for the proof of Theorem 23 is not far from best possible. Using Theorems 22 and 23. it is not too hard to deduce (12) from Col olltuv 20. and so prove 'Theorem 17.. Proof of Theo:T.7n 17 Let p > 1/2 be fixed As shown above, to prove that 0(p) > 0, it suffices to show that, given so, there is an s > so for
which (12) holds Let 7 be a positive constant to be chosen later In fact, we shall set = (p – 1/2)/C, where C is absolute constant Let er > so be a large
constant. such that all statements ' if .9 is large enough' in the rest of the proof hold kw all s > sr 13), Corollary 20, there is an s > such that / 1/ .,(9, > co, where co > 0 is an absolute constant; we fix such an s throughout the proof. Let Rr be a fixed Os by 9 rectangle in P., = , so P i p(11(R 1 )) > co. Regarding R i as a rectangle in the torus II" = let E 1 be the event that R i has a black horizontal crossing in the random Vcnonoi tessellation on the torus As noted earlier, since I?, does not come close to (within distance ()(log s) of, say) wrapping mound T 2 , we have a( I% i) = Pi';2(H(Ri))
o(1)
,
8 ,9 Random Volonoi percolation
as s
295
oc. . In particular, if s is sufficiently large, we have
r;9(Er) .^ co/2,
(17)
say.
Let 6 be chosen so that 10.9/6 is an integer, and S 21 < 6 < s14; is possible if s is huge enough, which we /nay enforce by our choice of sr. We shall first apply Theorem 23, and then 'cliscretize' by dividing the torus ¶ 2 into (108/5) 2 small squares S i of sidelength 6, as above Set p' = (p + 1/2)/2, so 1/2 < p < p, and consider the coupled Poisson processes (Pif , P and (Pt PT ) whose existence is guaranteed by Theorem 23. applied with m = 1/2, po = p', and q =16 < c1 Let R. , be an Ss by 2s rectangle obtained by moving the vertical sides of outwards by a distance 8/2, and the horizontal sides inwards by the same distance; see Figure 18 Let E2 be the event that there is a horizontal crossing of Ro that is ISrobustly black. this
i)
E
s,
Figure 18 The 9s by s and Ss by 2s rectangles A and (riot to scale), together with a black path PI (solid curve) crossing R I horizontally and a nearby robustly black path P2 (dashed curve), part of which crosses ki horizontally
Suppose that Er holds with respect to (Pi", Pr); flour (17), this event has probability at least co /2 Let Pr C R 1 C T2 be a black path that crosses R I horizontally If the global event Eg described in Theorem 23 holds, as it does whim then there is a path A with (In (Pr P2 ) < (logs)' such that P, is IS  robustly black With respect to (Pt. If s is sufficiently large that (log 8) 2 < s/2, then any such path 12, contains a subpath A crossing horizontally, so E, holds with respect to (P:t,P.r) If s is large enough, then Eg holds with probability at least 1 — e0 /4, so (E2 ) >
(E, ) — P(Eg ") >
co/2 — co/4 >
co/I
be the event that. some Ss by 2s rectangle it/ ¶2 has a 46robustly
296
Continuum imirolati OP
black horizontal crossing Then (E.)>
P, (Eo) > col
Let us divide T2 into n = (10s/6) squares Si as before, and define the crude state of each Si and the etude state CS of the whole torus as above. Let :p; be the event that the crude state CS of the torus is compatible with E3. Since P(E 3 ) = E(P(E3 CS)) < IF(ED, we lune P;,, (EEO > co/4 Let ',brit' and proud be defined by (14), and let similarly, but with p' in place of p. Thus.
and p'gt.
be defined
= 1  exp (62 (1  p ')) ti S2 (.I  if). and //gond =
exp (82 (1  p'))(1  exp ( 62 1/ ))
(52p
As the event E:63 depends only on CS, it may be viewed as an event E3 in the finite product space Q" = {1, 0,1}" In particular, nom (15) = P7; (E753 ) > co/4.
(19)
The event F3 is symmetric, as E3 and hence Elsi is preserved by a translation of the torus through (18, Pi. ), ) E Z. ft is not hard to check that 1:3 is also increasing From (11) and (18) we have pit „ i
(1p1 )52 , pig ,„„ 1 tpe, /Thad ti (1 p)82 . and it,
p82 (2(1)
shall apply Theorem 22 to 1, 0, 11", n = (10s/8) 2 , with p_ = = p800, 1 , and with e = min{c0 /4, ill In, say Note that z > 0 is an absolute constant Let A be the quantity appearing on the lighthand side of (16) with this choice of parameters From (20), all fern of the quantities p_ and are at most if s is large enough Hence, p„,„, < 62 , and WC
= peso" , a_ = pima and
A = 0(62 log(1/62 )/ loge). where the implicit constant is absolute By om choice of 6, we have 1/6 < s 2f On the other hand, = (10s/8) 2 > s2 Thus, log(1/5 2 )/ log a < 27. Hence. A < Chi ,i 2 for some absolute constant C As C is an absolute constant. we may go back and choose .7 small enough that CT, < (p  p')/2. so A 5_ (P  P/)62/2
8 (( Random Vownoi percolation
997
hot (20). we have q+  v+, P
^" (1) 91)52
Thus, if s is large enough, q+ p+ and p_  q_ ale larger. than A, i.e., the hypotheses of Theorem 22 are satisfied Theorem 22 and (19) then imply I — 5 > 1 — 10 — WO (PII) Retruning to the continuous setting, but considering t h e event Er), which depends only on CS, we have p (E) = p
F ) > 
(")
Suppose that CS is such that LI holds, and let (P P) be aqy configuration of the Poisson process consistent with this particular elude state of the tot us By the definition of E7)') . there is another realization consistent with the same crude state such that E. holds. i e such that there is a 4Srobustly black path P Grossing some Ss be 2s rectangle in 72 Prom the definition of bad, neutral and good squares, it follows that in the realization (Pt P  ). the path P is still black Hence. )\ riling E4 for the event that some Ss by 2s rectangle in T 2 has a black horizontal Cl owing. we have PI, (Er) ^ 1Pi, (Etr ) > 1 
I"
The rest of t he argument is as in Chapter. 3: we can cover 7 2 with (is be Is rectangles /?, , ,R9 5 so that, whenever Er holds, one of the R i has a black Mu izontal crossing Using the 'rahroot trick', it Follows that (H(R ) ))> 1 
Returning to the plane, fir (3/2,45) =
(11(R1))+ o(1)> 0 999
s is large enough Appealing to (7), it is eas y to deduce that fp (3, Is) > 0 99. As noted earlier, 0(p) > (1 then follows easily by considering 1independent percolation Since p > 1/2 was at bitranr, and 0(1/2) = 0 from Theorem 16, the result follows
❑
The argument: above was just a sketch of the proof of Theotem 17 hi particular. we said almost nothing about the moor of Theorem 23 Although purely 'technical', such a result seems rather difficult to plover this is actually a huge part of the work in the or iginal paper
298
Continuum percolation
As shown by Bollobiis and Riordan [2006a], for p < 1/2, the proof of 'Thoth ern 17 also gives exponential decay of the size of the open cluster Co or cot responding black cluster Crg containing the origin; this result follows from (12) by considering a suitable kindependent site percolation model on Z2 Theorem 24. Let leol denote the diameter of Co, the area of Co, or the 711110101 ICT of I/010710i cells in Co Then, for any p < 1/2 > there is a constant c(p) > 0 such that F
Pp (FCol hi every n > 1
n)
exp(—e(p)n) ❑
As in Chapters 3 and 4, this implies that the Hanunersley and Temper ley critical probabilities coincide Theorem 25. Let p i he the Tempe/ley cr itical probability for random Ijoronoi percolation in the plane above which Ej,(Kup diverges, where roi denotes the diatnetei of Co, the area of Co, or the number of Voronai = p i( = 1/2 crffs in Co Then. ❑
We have seen that :random Voronoi percolation can be very difficult to work with. Nevertheless, there is some hope that the conformal invariance conjecture of Aitern/mu and Langlands, Pouliot and SaintAubin [1994] discussed in Chapter 7 may be approachable for this model: the model has much [note builtin syninteti y than site or bond percolation on ain lattice. Also, the result of Ben] amini and Schramm [1998] provides a possible alternative method of attack: roughly speaking, t hey showed that for :random Voionoi percolation, conformal invariance is equivalent to 'density invariance', i e , the statement that, with suitable scaling, the crossing probabilities associated to two different inhornogeneous Poisson processes converge
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Index
2close domains, 211 flregular infinite tree, 6 :Farm exponents, 238 kbranching tree, 6 kindependent percolation, 70, 73, 108, 162, 250, 270, 287 nthroot trick, 41 AizelurianLanglamilsPouliotSaintAtibin conjecture, 178, 181 amenable graph, 121 Archinieclean lattice, 154, 155 aspect ratio, 184 attachment vertices/sites, 10, 157 automorphisminvariant event, 1IS ball (in a graph), 108 Vat) den BergNesten conjecture, 44 Van den BergKesten inequality, 12 in percolation, 78 Bettie lattice, 6 black cluster, 269 black horizontal crossing, 274 blackincreasing event, 276 bond percolation, 1 bond, i e , edge, 1 Boolean niodel, 240, 242 box product, 42 branding process, 8 cactus, 10 Cardy's formula, 184 Carleson's form of Cardy's formula cell percolation, 267 closed bond/site, 1 ColourSwitching Lemma,196 configuration, 2, 268 conformal equivalence, 182
86
invariance, 176 239 map, 178 rectangle, 184 connective constant, 17 continuum percolation, 240 correlation inequalities of Ahlswede and Boykin, 45 of van den Berg and kesten, 43 of Fortnin, liasteleyn and Cinibre, 6 of Harris, 39 of Reimer, 44 correlation length, 234 coverage questions, 261 critical area (Gilbert's model), 244 critical degree (Gilbert's model), 241 critical exponents, 62, 232, 234 for the triangular lattice, 236 critical probability, 5 crossratio, 183 cube, 38 decreasing event, 39 set system, 39 dependent percolation, see kindependent percolation directed graph, 24 Dirichlet diagram/domain/polygon/ tessellation, see Voronoi tessellation discrete contour integral, 207 discrete domain, 191 mesh of, 191 discrete torus, 65 discrete triangular contour, 207 domain in 12, 178 clownset, 39 dual bond, 12 dual graph, 12 dual lattice, 12 dual site, 12
Meted,
320
Nlenshikov, 93 Newman and Schulman 1211 Smirnov, 196 line graph, 3 lognioncit one, 16 logsitpermodular,
edge, sec bond equivalent bonds 138 percolation measures 81 random graphs, 119 sites. 1 85, 138 weighted graphs, 156 exponential deep, 70 . 101 for Gilberts [node/ 251 for Voronoi percolation, 298 extended KagoutO lattice 155, external boundary in IC, 216 in :2 13 fact' percolation, 12, 2 finitetype graph 85 r.K.c; inequality, 61, 16
marked discrete domain,193 /narked domain. /79 Martingale Convergence Theorem matching lattices, 146 mixed percolation, 21, 24 monochromatic path, 191 multigraph, 2
67
Newman Ziff method 175
Gilbert (disc) model, 210, 212 Harris's Lemnm for \Orono! percolation 278 bin usdor tf distance, 211 Hex, game of, 130 hexagonal lattice. 9, 136, 155 bond percolation,1A9 face percolation 131, 215 site percolation, 1(i5 hitting I adius, 256 honeycomb lattice, see hezagonal lattice horizontal dual 51 hype/cube 38 hyperscaling relations 235 increasing event, 39 276 function on a poser. set system, 39 independent percolation measure, 3 influence of a amid in percolation 63 of a variable, /16 inner boundary 192 interface graph, 51 131 IllI, 275 Jordan
(10111:1111,
17!)
Kagons1 lattice, 151, 155 157 Kohnogorcw's 01 law 36 Lemma: Balkier Bollolnis mid Wallet de Binijn anal Ercilis 37 Fekete 36 11in ris 311 A far gulls and Russo .16
1
open bond/site, 1 open cluster, 3, 269 open dual cycle 16 open horizontal glossing 51 open ourcluster . 25, 111 open path, 3 open subgraph, L 80 open vertical crossing, 51 oriented graph.. 21 oriented linegraph 82 oriented path, 25 oriented percoIat ion, 21 , 167 outball. /01 outclass 81 outclass graph, 81 outlilac sites 81 outsphere. Sli outsubgraph 25, 81 outer boundary. 192 partially (mimed set :38 Peierls argument., 16 percolation measure, 3, 80 percolation probability, 5 pivotal bond in percolation, 63 variable 16 planar lattice, 138 Poisson process, 2,11 PoissonVoronoi tessellation me ranc:an Voronoi tessellation poset, 38 power set, 37 radius of a cluster in an oriented g r aph, 86 random convex hull problem. 262 random geometric graphs. 210 25  I random set process_ 176 random Vinonid percolation 263
Inded: pindout Voronoi tessellation, 263 retticamalization argument, 68 itliemann Mapping Theorem, 178 robustl y black path, 293 B;e1W T'lleorem, see RussoSeymourWelsh T'hcoret scale, scale parameter, 279 scaling relations, 235 SchrammLoom/et. evolution (STE), 232, 235, 239 selfavoiding walk, 15 selfduality, 12, 129, 274 sharp threshold, 49 site percolation 1 site. i e vertex I SLE. sae SchrammLocuetner evolution sphere (in a graph), 79 square lattice, 2 bond percolation, 50 site percolation, 133, 162 . 1165 square product, 42 squareroot trick, 41 stardelta transformation, 148. 151 stntriangle transformation, sac stardelta I ransforma I ion stale of a bond/site, 1 stochastic domination, /57 strongly connected graph, 26. 85 subexponential decay, 76 substitution method. 149, 156 symmetric event, 66 plane graph 138 set system, 49 ail event 36 Theorem: Aldswede and Daykin, 45 AiZe11111811 and Barsky, 103 Aizettman, Nester and Newman, 121 Mammon and Newman, 107, 111 Arzelit tend Ascoli, 227 Balister, Bollobeis. Sarkar tend Walters, 260 Batiste!. Bollobas tend Stacey, 171 Batiste') Bollokis and Quas, 273 van den Berg tend Kest en, 43 Bollobas and Riordan, 279, 282, 298 Bourgain, Kahn, Katznelson and 48 Cara/ Intodory, 182 Tot tuin ICasteleyn and °Milne, 46 Friedgut and halal, 48, 49
321
Gilbert 246 Grimmett 152 Grinunett ancl Stace Hall 246, 249 Hammersley, 19, 106 Harris 61, 124 Harris and Kesten, 50, 13, 128 146 Kahn, Nalai and Linial, Kesten, 67, 71, 131, 1.17 Kohnogorov, 36 Liggett., 171, 172 Wester and Roy, 252 lienshiletut, 90. 96, 100 Penrose, 255, 257 Reimer, 44 Roy 251 Russo, 134 Russo and Seymour and Welsh 57 Smirnov, 187, 188 StIlit'llOV and Werner, 236 \ Vierrnan, 149, 169, 162 Zravitch, 279 'Ft/lessen polygonalizat ion sac Voronoi Iessellatiem t ranslat ioninvariant event 120 triangular lattice, 13, 136, 155 bond percolation, 1,19 Grit ical exponents. 236 site percolation, 129. 187 twodimensional lattice, 182 uniqueness of the infinite/ cluster, for (filbert's model, 252 for random Voronoi percolat ion, 278 in amenable graphs, 121 upset 39
t
vertex see site dual, 51 Voronoi cell, 263 diagram, 265 percolation, see random Voronoi percolat ion polygon, 265 / essellat ion, 263 vertical
weighted graph, 138 156 weighted hypercube, 37 weighted platuu hit tic( 138 witness, 42 ill percolation, 78 Zill s foomila 186
List of notation
1,41: cardinality, ALB: symmetric difference, 38 A 0 .41, A 08: square/box product., 12 :1 C B: subset (equality allowed), 38 A P.L : closed 6/leighbourhood, 211 A L : boundary are of a (discrete) domain, 179, 193 At: outer boundary arc of a discrete domain, 193 13i (a binomial distribution, 28 8,4,1:4 Indl in a graph, 108 Br(z): ball in Ia 2 or C, L83 Bh:D: outball in an oriented graph, 104 C: complex numbers, 178 = {/1 E A : — y}: open cluster containing :r, 4 G;f: open outcluster, 25 C': outclass graph, 81 Dr y: occupied set in 0, , A 212 El, etc: expectation associated to Pp elm. 6 B(A): edge set of a graph A, 1 E15.(:): existence of open separating path, 201 E, A: vacant set in GI ,. A, 243 0(o): 0, ,A with ers 2 A = 244 G A (Il!): generalized Gilbert model, 213 Cis: discrete domain, 191 CTT: discrete approxinuttions to a domain, 212 G,„,7 : finite random geometric graph with degree d, 255 G, A: Gilbert model, 212 G,: G et , 212 C r (1,), (1 ' A): finite noulom geometric graphs 254 H: hexagonal lattice, 136
11(11)= 11 1,(R): 17 Ili”; a black
horizontal crossing, 274 lonearest graph, 257 there me k infinite open clusters. 118 P = PA: Poisson process, 211 P A : open points of P, 268 P closed points of P, 268 p : probability ineastit• e associated to bond percolation, 2 P';\ probability measure associated to site percolation, 2 (2, 11, (4: weighted hypercube, 37 (2: mulch: ant of 27 2 168 17,, (:r) = (4 + 1): there is an open path from 9i (:r) to St (:r), 87 S LEN : SchrtuninLoewnet• evolution, 235 5,(4): sphere in a graph, 70 St (r): outsphere in an oriented graph 86 I: triangular lattice, 129 :2 7 2 * torus, 954 :2,, : 2 : discrete torus, 65 1  (A): vertex set of a graph A, 1 VI (11), 1E (R): R. has a black/white vertical crossing, 275 1:r : Voronoi cell of a, 263 natural orientation of 'Si/ 10 H„
(11(!): degree of a vertex/site 24 d(r,y): graph distance, (ii d(::, y) Um an oriented graph: distance from a to Li, 86 dist(x,y): Euclidean distance, 211 flauselorff distance, 211 f/z): probability of EA (:), 207
List
of
f p (p, .9 ): pa by s crossing probability, 280 (4: continuous extension of fi}, 225 It p (rn. n): In by n crossing probability in 52 , 59 1,(m, n): h rr(nr,n), 59 pH: lialffillettiley critical probability, 5 PH (1): critical probability for Voronoi percolation in Rd , 269 pi : lemperley critical probability, 6 pc : p H or p i , 9 r (ex ): radius of open cluster, 61 in an oriented graph, 86 s: scale paranteter, 279 —
or {:r — y}: t here is an open path front x to p, 3 {:r L }: there is a long open path from :r, 79., 8(i :r` — y}: uu open path front St (r) to (a : WI {a rt' IL} = R,,(a): here is MI open path Gen Sj (t) to St (r), 87 A: a (usually infinite) graph, 1 A': dual graph, 12
17 o
I ati o n
323
(A, p): a weighted graph, 138 A il :: open subgraph in bond percolation, A;,: open subgraph in site percolation, 2 TV: oriented graph, 24 out. subgraph, 25 influence, 46 SA: resealed lattice. 179 511: resealed hexagonal lattice, 191 ST: resealed triaiiguha lattice, 187 ll(p) = 0}; (p), etc: percolatirm probability, 4 x(p) = x!,.!(p) etc: expected cluster size, 6 A( 4 ): connective constant, 17 p,,(p): min x pn(a7 , p ), 88 p11(x,p): probability of Rar(x), 87 (CO: inner boundary, 192 (0): outer boundary. 192 it': external boundary. 13 216 D : discrete contour integral. 207
a s: with probability I, 240 whp: wit h probability I — of1i 240