Phase transitions of interacting particle systems

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PHASE TRANSITIONS OF INTERACTING PARTICLE SYSTEMS

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PHASE TRANSITIONS OF INTERACTING PARTICLE SYSTEMS

Norio Konno Muroran

Institute of Technology,

World Scientific

Japan

Singapore••New NewJersey Jersey ••London L Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

PHASE TRANSITIONS OF INTERACTING PARTICLE SYSTEMS Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. ISBN 981-02-2076-6

Printed in Singapore.

To my parents: Kazue and Masako

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PREFACE

Recently interacting particle systems have been studied widely from many aspects, for examples, mathematics, physics, chemistry, and biology. Many researchers get interested in this field. Fortunately some good books on it have been published, for example, Liggett (1985) Interacting Particle Systems, Durrett (1988) Lecture Notes on Particle Systems and Percolation. This book restricts attention to the phase transitions of the interact ing particle systems, in particular, their critical values and order parame ters. As for concrete systems, we study basic contact processes, ^-contact processes, diffusive contact processes, long range contact processes, and uniform nearest-particle systems. Unfortunately, with the exception of the critical value of the uniform nearest-particle system with infinite range, the rigorous critical values and order parameters are not known. So we turn our attention to the bounds on the critical values and the order parameters. T h e n our basic question in this book is as follows :

How can we get good bounds on the critical values and the order parameters ?

T h e above question is very basic and many researchers have been trying to get better bounds rigorously. So we give not only our bounds but also other's ones in this book. In the introductory Chapter 1 we consider briefly Markov chains and give some examples. vii

Phase

Vlll

TYansitions

of

IPSs

The basic contact process is a continuous-time Markov process on state space {0, l}2*, where Zd is the d-dimensional integer lattice. This process is interpreted as a simple model of a disease with the infection rate A. T h e value 0 is regarded as a healthy individual while the value 1 is regarded as an infected one. T h e dynamics are specified by the following transition rates : for r) G { 0 , 1 } Z , at site i £ Z , 1 —► 0

at rate

1,

0 -► 1

at rate

Ax

^

r](y),

y:\y-x\ = l

where \x\ = \xi\-\ (- \z#|. Define p\ as the density of infected individual at a site with respect to a stationary measure starting from r\ = 1. We take p\ as an order parameter of the basic contact process. The basic contact process is attractive, so p\ is a nondecreasing function of A. Then the critical value Ac is characterized by p\ in the following way : Ac = sup{A > 0 : pA = 0} = inf{A>0:/9 A > 0}. As for the processes treated in the rest of this book, the critical values and the order parameters can be defined in the similar way, since these processes are also attractive. Then Ac and p\ are not known rigorously. Chapter 2 contains the one-dimensional basic contact process on finite sets. Chapter 3 is devoted to lower bounds on the critical value and upper bounds on the order parameter of basic contact process. In Section 3.3 using the correla tion identities and the Harris-FKG inequality we give the lower bounds on the critical value and upper bounds on the order parameter. Better bounds are obtained by the Harris lemma in Section 3.4. Section 3.5 contains the best lower bound on the critical value of the one-dimensional basic contact process given by Liggett and Ziezold-Grillenberger methods. In particular, we clarify the relation between both methods. On the other hand Chapter 4 treats the upper bounds on the critical value and lower bounds on the order parameter of the one-dimensional basic contact process by various methods. The 0-contact VTOC^$yftffifgff!fy?$$f}%i°f

the one-dimensional basic

Preface

IX

contact process. The evolution mechanism is as follows : 1 —+ 0

at rate

1,

001 -* Oil

at rate

A,

100 -> 110

at rate

A,

101 -»• 111

at rate

0A,

where 9 > 0. Remark that the process for 6 = 2 is nothing but the basic contact process in one dimension. The first three sections in Chapter 5 we restrict attention to a family of the 0-contact processes for 1 < 0 < 2. This restriction comes from the existence of a coalescing duality for this process. After that we will discuss general case, 0 > 1, by using the Gray duality. In Section 5.2 we give lower bounds on the critical value Ac(0) for 1 < 0 < 2 by using the Griffeath method and the Liggett and Ziezold-Grillenberger methods. Section 5.3 is devoted to upper bounds derived from the HolleyLiggett method for 1 < 0 < 2. In Section 5.4 we introduce the coalesing duality and discuss about some examples. The Gray duality is presented in Section 5.5. So using it, we will give lower bounds on Ac(0) for 0 > 1. The diffusive contact process is the basic contact process with stirring mechanism which is given as follows : for each x, y 6 Z d with | x — y |= 1, we exchange the values at x and y at rate D. Then as in the case of the basic contact process the order parameter p\(D) and critical value \C(D) can be defined. In this setting it is known that p\{D) —* {2d\ — l)/2dX and Ac(£>) —► l/2d as D —> oo. In Chapters 6 and 7 we mainly prove that there are positive constants C\,Ci so that Ci 3). Chapter 6 is devoted to proofs of the upper bounds on AC(D) — l/2d. Moreover we give another proof of p\{D) —> (2dA - l)/2dA and show crossover phenomena on p\(D). On the other hand Chapter 7 treats the proofs of the lower bounds on AC(.D) — l/2d. As in the diffusive contact process, we study the long range contact process in Chapter 8. The uniform nearest-particle system with the infection rate A and the distance of range M is the one-dimensional interacting particle system. In particular, if M = 1, then this system is equivalent to the basic contact process in one dimension. In Chapter 9 we consider uniform nearset-particle systems with finite and infinite ranges respectively. Section 9.2 contains the lower bounds on critical vgja^agd^gflj^b^gjds on order parameters when

X

Phase Transitions

of IPSs

M is finite. In the case of infinite range, the rigorous critical value is known. So we give a good upper bound on order parameter near critical value in Section 9.3. Finally Chapter 10 gives a survey of results on diffusive contact pro cesses and diffusive branching processes on the homogeneous tree. I express special thanks to Tokuzo Shiga for introducing me to this fascinating field and constantly encouraging me throughout the whole work. I am deeply indebted to Makoto Katori, for teaching me much about sta tistical physics while collaborating on various projects. Moreover Makoto Katori read various parts of the manuscript and made numerous sugges tions for improvements. I extend our thanks to all these, and to others : Rick Durrett, Tom Liggett, and Roberto Schonmann, who have offered advice in correspondence or conversation. But I am, of course, responsible for all errors that remain. Most of this book was written while I was visiting the Mathematical Sciences Institute at Cornell University during May 1993 - April 1994, a visit assisted by a grant from the Ministry of Education of Japan. I am grateful for not only this grant but also the financial support of the Army Research Office (through the Mathematical Sciences Institute at Cornell University). Moreover it is a pleasure to acknowledge the passive and active assistance of Rick Durrett during this period. Finally I would like to thank my wife, Naomi, and my two daughters, Yuki and Aki, for their patience and understanding.

N.K. Muroran August 1994

CONTENTS

Preface

vii

C H A P T E R 1. I N T R O D U C T I O N - M A R K O V C H A I N S 1.1. 1.2. 1.3.

Introduction Discrete-Time Markov Chains Continuous-Time Markov Chains

1 1 5

C H A P T E R 2. C O N T A C T PROCESS T Y P E MODELS ON SMALL FINITE SETS 2.1. 2.2.

Introduction The Basic Contact Process on Finite Set 2.2.1. n = 1 2.2.2. n = 2 2.2.3. n = 3 2.2.4. General case

xi

7 7 7 8 9 9

xii

2.3.

Phase Transitions of IPSs

The Threshold Contact Process on Finite Set

11

CHAPTER 3. THE BASIC CONTACT PROCESS I 3.1. 3.2. 3.3.

Introduction 15 Correlation Identities 18 Harris-FKG Inequality Method 21 3.3.1. First bound 23 3.3.2. Second bound 24 3.3.3. Third bound 24 3.3.4. Fourth bound 25 3.3.5. Higher dimensions 26 3.3.6. Summary 27 3.4. Harris Lemma and Markov Extension Method . . . . 27 3.4.1. Harris lemma 28 3.4.2. First bound by the Harris lemma 29 3.4.3. Second bound by the Harris lemma 30 3.4.4. Third bound by the Harris lemma 30 3.4.5. Higher dimensions 31 3.4.6. Remark 32 3.5. Liggett and Ziezold-GriUenberger Methods 33 3.5.1. n = 1 36 3.5.2. n = 2 37 3.5.3. General case 38 3.6. Application to the Threshold Contact Process 41 3.6.1. Introduction 41 3.6.2. Liggett and Ziezold-Grillenberger methods revisited . . 42 3.6.3. General case 45 3.6.4. Lower bounds 48 3.7. Summary 50

C H A P T E R 4. THE BASIC CONTACT PROCESS II 4.1.

Introduction

53

xiii

Contents

4.2.

4.3. 4.4. 4.5. 4.6. 4.7.

4.8.

Contour Method 4.2.1. Oriented bond percolation 4.2.2. Oriented site percolation Comparison Method Holley-Liggett Method Correlation Identities I Correlation Identities II Decoupling Procedure : Nonrigorous Results 4.7.1. Renewal decoupling 4.7.2. A new decoupling Remark

53 53 56 57 59 59 62 66 66 67 69

C H A P T E R 5. T H E 0-CONTACT PROCESS 5.1. 5.2.

5.3. 5.4. 5.5.

Introduction Lower Bounds 5.2.1. Griffeath method 5.2.2. Liggett and Ziezold-Grillenberger methods Upper Bounds Coalescing Duality Gray Duality

70 75 75 78 80 84 93

C H A P T E R 6. D I F F U S I V E CONTACT PROCESSES I 6.1. 6.2.

6.3. 6.4. 6.5.

Introduction Diffusive Contact Processes on Small Finite Sets 6.2.1. n = 1 6.2.2. n = 2 6.2.3. n = 3 6.2.4. Summary 6.2.5. General case Preliminary Results Upper Bounds Proofs of Theorems 6 y e ^ a a n d a 6 . 1 . 3

100 . . . 104 104 105 105 106 107 109 119 132

xiv

Phase Transitions

of IPSs

6.5.1. Proof of Theorem 6.1.1 (2) 6.5.2. The Galton-Watson process 6.5.3. Proof of Theorem 6.1.3

132 138 140

C H A P T E R 7. DIFFUSIVE CONTACT PROCESSES II 7.1. 7.2. 7.3. 7.4.

Introduction Lower Bounds for d = 1 Lower Bounds for d = 2 Lower Bounds for d > 3

146 146 151 166

CHAPTER 8. LONG R A N G E CONTACT PROCESSES 8.1. 8.2. 8.3. 8.4. 8.5. 8.6.

Introduction Preliminary Results Upper Bounds Lower Bounds for d = 1 Lower Bounds for d = 2 Lower Bounds for d > 3

171 175 182 192 194 197

C H A P T E R 9. UNIFORM NEAREST-PARTICLE SYSTEMS 9.1. 9.2.

9.3.

Introduction 199 Uniform Nearest-Particle Systems : Finite Ranges . . 199 9.2.1. Attractive nearest-particle systems with finite ranges . . 200 9.2.2. Better lower bounds 207 9.2.3. UNPS(M) 209 Uniform Nearest-Particle Systems : Infinite Ranges . . 210

Contents

9.4.

U N P S ( M ) (Finite M a n d Finite System)

xv

212

C H A P T E R 10. P A R T I C L E S Y S T E M S ON T R E E S 10.1. 10.2. 10.3. 10.4.

Introduction Preliminary Results A*e(D) Lower B o u n d s o n A?(D)

Subject I n d e x

214 219 223 224 226

1

Introduction

CHAPTER 1 INTRODUCTION - MARKOV CHAINS

1.1. I n t r o d u c t i o n In this chapter we consider briefly Markov chains on finite and infinite sets. Some examples on Markov chains are introduced. 1.2. D i s c r e t e - T i m e Markov Chains Let X = {Xt : t 6 T} be a Markov chain which takes values in some countable set S, called the state space. In this section we consider discrete-time Markov chains, i.e., T = { 0 , 1 , . . . } . The Markov chain X is called temporally homogeneous if P(Xn+1

=j\Xn

= i) = P{XX = j\X0

= i),

for any n>0,i,j€ S. From now on we will restrict attention to the temporally homogeneous case. The evolution of a Markov chain can be described by its transition probabilities : P(hj)

= P(Xn+l

=j\Xn-

i).

The | S | x | S | matrix p = (jp(i,j)) is called transition matrix where | S | is the cardinality of S (| S | may equal oo). A measure fi is said to be a stationary measure if

5>(i)p(M) = M(j),

(1-1)

for any j e S. If p. is a probability measure, we call p. a stationary distribution. If XQ has stationary distribution p, then so does X„ for all n > 1. We will now give some concrete examples. E x a m p l e 1.2.1. Let X be a Markov chain on S = {0,1} with transition matrix : _

P

~ [

1— a

fi

a

1-/3 '

Phase Transitions of IPSs

2

where 0 < a,/3 < 1 and a/3 > 0. A stationary distribution n would satisfy Eq.(l.l),/xp = /x, which is to say that

( l - a M 0 ) + M l ) = M0), a/x(o)

+ ( i - / ? M i ) = Mi)-

So we have

M= K 0 ) , M D ] = [ ^ ^ r

Let £< ) = { 0 , 1 , . . . , r } and 5 ( o o ) = { 0 , 1 , . . . } . Define the Markov chains X W and X<°°> on state spaces S(r> and .S^00' by the transition matrices p
nM

1 0 0 0 ... g 0 p 0 ... 0 g 0 p ...

0 0 00 0 0 0 0 0

0 0 0 0 0 0 0 0

q 0

o p 1 0

1 0 0 0 g 0 p 0 0 g 0 p

p(°°) =

That is, J> (r) (0,0) = 1 and p^(i,i + 1) = p,p(T\i,i - 1) = g = 1 - p for 1 ( r , r - 1) = 1. And p(°°)(0,0) = 1 and p(°°'(i,i + 1) = p,p(°°'(i,i - 1) = g = 1 - p for 1 < i. The chain represents a particle in random walk. The particle moves one unit to the right or left with respective probabilities p and g, except that 0 is an absorbing state once the particle enters, it cannot leave, and r is a reflecting state. Let /i' r ) be a stationary distribution of X(r\ so Eq.(l.l) gives /iM = [ l , 0 , . . . , Q ] . lr)

and

(r)

That is, M (°) = 1 /* («) = 0 for i f 0. Let /x<°°> be a stationary distribution of X<°°). Similarly we have / • < - > » [1,0,0,...]. This example is a trivial one. E x a m p l e 1.2.3. Random walk with reflecting states 0 and r

» =

f0 9 0

1 0 g

0 0 ... p 0 . . 0 P • .

0 0 0 0 0 0 0 0 0

0 0

0 0

0 0

q 0

0 0

. . . .

0 p 1 0

Introduction

3

ro i o o

q 0 p 0 0 q 0 p

„(°°)

By Eq.(l.l) we obtain the following stationary distributions for r > 3 : Mw

= ,l')(0),i,.">(0)

i(£y„M(0),.

-K ; - ) ' " > « » . (■^fV'W

)

where

^m-{.+ig(£ '+C;)'"'}"' For an infinite system, if p < q, then M<~>

=

^°°>(0),V°>(0),.. L

-)V°°>(o),...

9 \«

5

where ^'(0) =

^ .

If p > 9, then there is no stationary distribution. We concentrate on the behavior of pf-T^(Q) as r increases to infinity. So we take p^r'(g) : /i' r '(0). We can rewrite p*-T\q) in the following way : 9 „<3>( = §, g) = ?<%) '2'

(4) pWf»1 (9)==

92 + ? + ( l - 9 ) + ( l - 9 ) 2 ' 93 q3 + q2 + ?(1 -?) - + (!-«)! + ( l - , ) 3 ' 94 p (6) (9)= 9< l -- ?) + g ( l g< ++g g33++? 2? 2((1 9) + 9(1- -99)2) 2+ +( l( l- g- 9) 3) 3 ++ ( (l l_-- g)4' 9)4'

,<°°) (?) = iziVo, where a V 6 denotes the maximum of a and 6. Then we obtain that for q > 0, p<°%) < p(6>(g) < pW(q) < pW(q) < ,<»>(,). Moreover as r goes to oo, for q > 0,

p^(q)\p(°°\q). pW(9)\p ( o o ) (9)-

4

Phase Transitions of IPSs

p'r'((*0 > " r

PW

=

r

/< >(l) 1 0 0

/< >(2) 0 1 0

/«(3) 0 0 1

0 0

0 0

0 0

p(°°> =

a l l d

/M(r-l) 0 0 0

/(°°)(1) 1 0 0

/(°°)(3) 0 0 1

f(°°)(2) 0 1 0

That is, p< r '(0.j) = f{r)U + !) for j > 1 and p^(i,i distributions / i ' r ' of X ' r ' is as follows :

-.M

ES/

W

(

fW(r) 0 0 0

t ;

) = lforr = l,2,...,oo. /( r >(r + l ) 0 0 0

... ... ... ...

- 1) = 1 for i > 1. The stationary

^(0),(l-/ (r) (l))M (r) (0),...,|l-E/ (r) (fc)JM ,M (0)

and p>°> = / x(-)(0),(l-/(-)(l)y°

o

»(0),...,|l-E/(oo)(fc)U(oo)(0)

where pW(0)

XEi */«(*)'

and M(o°'(0) =

1 1kfl°°

><*)' For A 6 [0,r], let {£k,r : 1 < fc < r} be independent and identically distributed (i.i.d.)

with

P(tk,r = 1) = - , and •P(6,r = 0) = 1 - - . r Define Yr = 1 + fi,r + . . . + £ r , r . Let /^(k) = P(Y r = fc) for 1 < k < T + 1. So we have E(Yr) = A + 1. Therefore for r = 1 , 2 , . . . , and A e [0, r], ^(r)(0) =

1

A+l"

Introduction

S

The weak law of small numbers (for example, see Theorem 6.1 on page 117, Durrett, 1 ) implies t h a t as r goes to infinity Yr converges in distribution to a Z + 1 where Z is Poisson distribution with mean A. For k = 1 , 2 , . . . , we let f(<x /■<°°>(jfc) e><*) =: e"

-l

1 ) ! -(( f*e-- •I)!

So for A e [0,oo), 1

M(°°>(0) = A + l ' To get more interesting bounds on /u(°°> we modify the random variables £k,r m the following way : e\ A P ( ^ = l ) = ( l + ^ . r) For example, if e = 1 then

«o)=^ r '( 0 ) = 3 I A +T 1I

forAe far A

e [0,^3, [°'^TT]'

and if e = — 1 then

,^■-<1)-(0)> ( o=)1=^ ^~ +- 11

r2 for f o rAA<=e t[0, O,^].. 7" — 1

So we can see /i( r,1 )(0) is a lower bound and / i ' r , _ 1 ' ( 0 ) is an upper bound on /i'°°'(0) = 1/(A + 1). 1.3. C o n t i n u o u s - T i m e Markov Chains Most of interacting particle systems which we will study in this book are continuoustime Markov chains, so we now turn on the continuous case and take a look at them briefly. Let X = {Xt : t £ [0, oo)} be a continuous-time Markov chain which takes values in some countable set S. T h e Markov chain X is called temporally homogeneous if P(Xt = j\Xs

= i) = P ( X t _ , = j | X0 = i),

for any t,s > 0,i,j G 5 . From now on we assume that X is a temporally homogeneous Markov chain. The evolution of a Markov chain can be described by its transition proba bilities : = P(Xt =j\X0 = i). Pt{i,j) The | 5 | x | 5 | matrix pt = (j>t(i,j)) is called transition matrix. A measure /J, is said to be a stationary measure if

X>(*)MU) = M0')> ies

(i-2)

6

Phase Transitions of IPSs

for any t > 0 and j 6 S. If p is a probability measure, we call p a stationary distribution. G = (g(i,j)) is the generator of the chain. It takes over the role of the transition matrix p for discrete-time Markov chains. That is, starting from Xt = i, until t + h (i) nothing happens with probability 1 + g(i,i)h + o(/i), (ii) the chain jumps to state j(^ i) with probability g(i,j)h + o(h), as h J. 0. Then a simpler condition than Eq.(1.2) is as follows :

X ^*)» hj)

-= 0.

(1.3)

If XQ has stationary distribution u, then so does Xt for all t > 0. E x a m p l e 1.3.1. Let X be a Markov chain on S = {0,1} with generator : —a

G = . fi

a

-P.

1

where a/3 > 0. In order to find stationary distribution we solve Eq.(1.3), fiG = 0 , so

(i-«Ko) + M i ) = o, afi(0)

+ (1 - j S M l ) = 0.

Therefore

i*-= W0),/i(l)] =

r -3

« 1

L°!+/»' Q + /3J

As for the stationary distribution, it is the same as that of Example 1.1.1 for 0 < a,/3 < 1.

References 1. R.Durrett, Probability : Theory and Examples

(Wadsworth,Inc, California, 1991).

Contact Process Type Models on Small Finite Sets

7

CHAPTER 2 C O N T A C T P R O C E S S T Y P E M O D E L S O N SMALL F I N I T E S E T S

2.1. I n t r o d u c t i o n In this chapter we will introduce the contact processes on the finite set { 0 , 1 } ^ n ^. In particular, we will calculate exact stationary distributions of the contact processes for small n. Erom now on we call continuous-time Markov chain Markov process. 2.2. T h e Basic C o n t a c t P r o c e s s o n F i n i t e Set The basic contact process r) which we will study in this chapter is a continuous-time Markov process on state space S^"' = { 0 , l } ' ' 1 " } . Then for a site x e {1, . . . , n } and a time t > 0, rjt(x) takes values in {0,1}. One interpretation of the basic contact process is a simple model of the propagation of an infectious disease. The value 0 is regarded as a healthy individual while 1 is regarded as an infected one. The dynamics of the evolution is described by the following transition rate : at site x £ { 1 , . . . , n } , 0 -► 1

at rate

\{T)(X - 1) + n(x + 1)},

1 —► 0

at rate

1,

where A is a nonnegative constant which is interpreted as the infection rate. Each healthy individual becomes Infected at a rate which is proportional to the number of infected neigh bors. On the other hand each infected individual recovers at a normalized constant rate 1. The boundary condition 77(0) = 0,77(71 + 1) = 1 (or vice versa), breaks the symmetry, so we don't adopt it. In this chapter, we take 77(0) = 77(71 + 1) = 1 as the boundary condi tion, because the other boundary conditions, 77(0) = n(n + 1) = 0 and periodic boundary condition, give just trivial stationary distribution, i.e., any state is 0, as we saw Example 1.2.2 in Chapter 1. Let TJW be a basic contact process on state space 5^"' and / i ' n ' be its stationary distribution (if exists).

Phase Transitions of IPSs

8

2 . 2 . 1 . n=l First we calculate the stationary distribution when n = 1. 77'1' is a continuous-time Markov process on state space S^1' = {0,1} with generator SW(0,0)

(?
0
-2A 1

^)(1,1)J flW(l,0)

2A -1

where A > 0. So we can write down the forward equation p't = P J G ' 1 ' : P ;(0,0)

= -2AP!(0,0)+M0,1),

P;(l,l)

= 2Ap((l,0)-pt(l,l).

In fact, this case is equivalent to Example 1.3.1 for a = 2A and /? = 1, so we can get immediately the stationary distribution ^1» = [M(1)(0),^(1)], 1 2A 2A + 1' 2A + 1 In this case we pay attention to juf^(l) : the density of an infected individual of the basic contact process when n = 1. Then we take / ^ ' ( A ) = /xM(l). So we have P (1) (A) =

2A 2A+1'

2.2.2. n=2 Next we calculate the stationary distribution fj,^ when n = 2. 7j' 2 ' is a continuous-time Markov process on state space 5 ' 2 ' = {0, l } ^ 1 , 2 ^ with generator

Gm

<7<2>((0,0),(0,0)) <7<2>((0,1),(0,0)) jW((l,0),(0,0)) ff<2>((l,l),(0,0)) -2A A 1 -(2A+1) 1 0 0 1

5(

2

>((0,0),(0,1)) >((0,1),(0,1)) «7( 2 )((1,0),(0,1)) 2 ff< )((l,l),(0,l)) A 0 0 2A -(2A + 1) 2A 1 -2J

5<

2

2

fl<

>((0,0),(l,0)) >((0,l),(l,0)) 9 < >((1,0),(1,0)) ffW((l,l),(l,0))

j«((0,0),(l,l))n 2 >((0,1),(1,1)) ffW((l,0),(l,l)) <7<2>((1,1),(1,1))

2

ff< 2

5<

From /i< 2 'G (2) = 0, we have M(2) = ^ 2 » ( 0 0 ) , ^ 2 ) ( 0 1 ) , ^ ) ( 1 0 ) , / x ( 2 ' ( l l ) ] 1 A A

2A2 2A + 2A + 1' 2A + 2A + 1' 2A + 2A + 1' 2A + 2A + 1 2

2

2

2

Contact Process Type Models on Small Finite Sets

2.2.3.

9

n=3

We calculate the stationary distribution yfP*. rf3^ is a continuous-time Markov process on state space 5 ( 3 ) = {0, l}* 1 ' 2 ' 3 ) whose generator G ( 3 ) is given by ■-2A 1 1 0 1 0 0 . 00

A 0 0 -- (( 22 A A ++ 11 )) A A 00 00 -(4A - ( 4 A + 11)) 2A 2A 1 1 -(2A + 2) 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1

A 0 A A 00 00 00 0 0 -(2A + 1) A 1 -(2A + 2) 1 0 0 1 0

0 0 2A 2A 0 A 0 -(2A + 2) 1 1 -

00 ' 00 00 2A 0 ' 2A 2A -33 ..

From // 3 >G( 3 ) = 0, we have

^ 3' /i'

3 (3) ) ) 3) 3 3 = [M [M (3) (000),/i( (000),/i(3)3(001),/ (001),^ )(010),^i (33)(011),M >(011),M( 3(3) (100),/i (100),/i( (3) (101),^ (101),/i( )(110),^ )(110),/3 i»(Hl)] ' 3 '(lll)] i ( )(010),/

l"°n\22 +i °1 \ i fi6,2A(8A ° V ° \ 22 i o\ i ¥\ ° \ 22(4A H \ 1 3), ?.\ \ 22(4A M \ 1 1Y4\ 1 ?^ = — _ , ,*w » v [20A \/SAJA - f 21A i i / \+ i " D,i/\^OA + * f 9A »A + -\- 3),2A O), LA \fkA + f O), A A ^**A + T 1)(4A *./\*rt + T^ 3), O), 2 2A(8A2 + 9A + 3), 2A22(8A22 + 8A + 3), 3), A2(4A + 1)(4A + 3), 3), 8A33(2A + l ) if], ],

where Z ( 3 ) (A) = (2A + 1)(16A4 + 32A3 + 32A2 + 21A + 6). In this case as for p^(X) there are two candidates: /i' 3 >(*l*) and /i' 3 ) (l * *). From symmetry, we adopt the first one, so p(3 (3)) ( A ) = P ((A) = />

'

Then we have

2 4A 4A*(4A" (4A2 +-t-bA 6A + 3) d; 16A" 16A ++ 32A 32A33 ++ 32A 32A22 + 21A 21A ++ 66' 4

pW(X) < pW(X) < pmm (X) 0

wm < tP>m < o (M

for A > 0.

In this case we take )(A) pW(\) ^

^

f

(2) (2) == ^\*l) (l*)> (*1) === Mp,W(l*), M

M r*<

2) 22 ) (2) ) (*D ( * 1 ) ==--M//( 2 >(01) ( 0 1 ) + +M ^ ( 2 ) ((ll), H),

(2) 2) H (1*) -= /»W(10) + M (ll)-

Therefore we get

2A22 + +A A t»m 2A P (2, A (A) = = 22 ^ ) 2 A2A ++ 2 A + 1 -"

Phase Transitions of IPSs

10

2.2.4. General case In this subsection we consider of general re. To do this, we will rewrite G^n\n in the following way :

0 0 0 0

2

G< > = A

G<3> = A

IX 0 0 0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0

G^

= A

1 1 0 0 0

0 2 2 0

1 0 0 0 0 0 0 0

0 2 0 0

H:

0 0 0 1 0 0 1 0 0 0 1 1

0 0 1 0 0 0 1 0 1 0 0 2 0 0 2 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 2A + 1 0 0 0 0 0 0

0 0 4A + 1 0 0 0 0 0

0

0 0 0 2 0 2 2

0 0 0 2A + 2 0 0 0 0

0 0 0 0

0 0

2A 0

2A 0 0 0

0 0 10 10 0 1 10 0 1 0 0 0 0

= 1,2,3)

0 1

0 0 0 2A + 1 0 0 0 2A + 1 0 0 0 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 10 0 10 10 0 0 10 1 1

0 0 0 0 0 0 0 0 0 0 0 0 2A + 1 0 0 2A + 2 0 0 0 2A + 2 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 3

From the above concrete examples, a little thought gives the following simple relation between re and re + 1. Let C?W be generator of the basic contact process on state space S<"), so it is 2 n X 2" matrix. Let O(re) be an re X re zero matrix and /(re) be an re x re identity matrix. Then we have T h e o r e m 2.2.1. For n > 1,

G< n + 1 ) = AG[" + 1 ) + G(" +1 > - G<" + 1 ) ,

Contact Process Type Models on Small Finite Sets

11

where G ("+l)

0(2")

G<">

_

.0(2" G = .7(2") /7("+l) _ ' G d

0(2")' G^n) .

n)

_

n

~

i,12

R(n) 1

■"■i.ll

.0(2")

\

+

i

0(2")'

R{n)

n

d,U

0(2"-!)

0(2") -/(2"-1)]

' J(2"- 1 )

0(2 n - 1 ) 1

0(2"- J )

0(2").

■>

0(2")' Gdn)

.0(2") D(")

+

■>

R(n)

0(2"- x ) J '

0(2"-!) 2/(2n"1)J ' [0(2"- 1 ) 2/(2"-J) J '

"d
4;i=^(2 n ). 0 2'

'2A 0

0 1

In the next chapter we study the basic contact process when n --

00.

G« = 0 0

G(1> =

0 1

0' 0

?

G<;> =

2.3. T h e T h r e s h o l d Contact P r o c e s s on F i n i t e Set In this section we introduce another contact process type model, the threshold contact process. The definition of this process is as follows at site x 6 {!,.. >"}> 0->l

at rate

A[{irf> -- 1 )

l->0

at rate

1,

+ IJ(X + 1 ) } A 1 ] ,

where A > 0 and a A b is the minimum of a and 6. In this case, the only difference between the threshold contact process and the basic one is infection mechanism when both neighbors are infected. That is, the infection rate = A, in the threshold case, on the other hand, = 2A in the basic case. Let Gy be generator of the threshold contact process on 5 ' " ' . Similarly we obtain >tain Grp

2)

4 =A

A

o r + 0

o i i o 0 0 .0

0 0 0

0 0 0

1 1 0.

+

0

"0 0 1 0 1 0

.0

0 1

0 0 0 1 L

0 0 0" 0 0 0J

A 0 0 1 ■2A 0 0 . 0

J

0 o A+ l 0 A+ l 0 0 0

o0 0 » 2.

12

Phase Transitions of IPSs

-o i o o i o o oi

G< T 3)

0 0 0 = A 0 0 0 .0 -2A 0 0 0 0 0 0 . 0

0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 2A + 1 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 2A + 1 0 0 0 0 0

-o o o o o o o on

0 0 1 0 1 1 oJ

+

1 1 0 1 0 0 .0

0 0

1 0 1 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1

0 0 0 0

0 0 0 0

0 0 1

0 oJ

°1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 A+ 2 0 0 A+ 2 0 0 0 0 3J

0 0 0 0 0 0 A+ 2 0 2A + 1 0 0 0 0 0 0 0

So the stationary distributions (i£ (i == 1,2,3) are obtained as follows : ^

$\o),4\i)}, r 1 A1

= [

.A+l'A + 1

,)

,)

,

I

J,

A4 2 ) = [/4 (00),/4 (01),^ (10),4 (11)] ' 1 A A A2 2 2 .(A + l ) ' (A + 1 ) ' (A + 1)2' (A + 1)2 (3)

FT

= [|i ( T 3) (000),^ 3) (001),/i ( T 3) (010),Atl 3) (011), & (100) ^

w

(101)

»

(110) u ( 3 ) ( H I ) ]

rfif \ 1 1Y2 \m \ 2 + 9A + 6), 2A2(2A + 3), A2(2A + 1)(2A + 3), } (

zf{xP ' '

A(4A2 + 9A + 6),2A2(2A2 + 4A + 3), A2(2A + 1)(2A + 3),4A 3 (2A + i f ] ,

where

43)(A) ==

2(2A + 1)2(2A3 + 6A2 + 6A + 3).

Prom these in a similar way we get : (1

= P?W --" A 1+

PT (A) =

p$\\) Then for n > 2,

l' 2A2(A2 + 3A + 3) = (A + 1)(2A3 + 6A2 + 6A + 3 ) '

4 n+1 » == AG^+

X)

+ G^1'

U

T,d

>

13

Contact Process Type Models on Small Finite Sets

where

4n,+1) =

G(T} 0(2")" 0(2") GJ$

Mn+1) _

Gfl 0(2") [J(2") '

* >*

U

T,r

-

0 2

G*T"i 0(2")

/-.(n+l) T,d

u

( ") G^d

•C(2" -2') E - 0(2"- ) ■"■T.i.ll — 0(2"- 2 ) .0(2"- 2 )

[0(2")

+

0(2") J '

P<"> -"■T.d.ll

0(2")

0(2")

"■T,d,22 .

_0 \

T

0 ( 2 " - 2' ) 0(2"- ) 0(2"- 2 ) 0(2"- 2 )

- 1 ( 2 " -2' ) 0(2"- ) 0(2"- 2 ) 0(2"- 2 )

0(2"-")0<2"- 2 ) 0'2"- 2 ) 0<2"- 2 ).

-"T.d.ll

r/(2"-2) 2 = A/(2") - A 0(2"- 2 ) 0(2"- ) .0(2"- 2 )

0(2"- 2 ) 0(2"~ 2 ) 0(2"- 2 ) 0(2"- 2 )

0(2"- 2 ) 0(2"- 2 ) 0(2"" 2 ) 0(2"- 2 )

D (»)

= J(2").

(n)

Din) ■"■1,1,12

-"•7,^,22

J

= /(2»), 0(2"" 2 )0<2"- 2 ) 0<2"- 2 ) 0<2"- 2 ).

>

In fact we can rewrite the above result in the following way. Theorem 2.3.1. For n>2, G<"+1>

G^ n ) -A/(2") 7(2")

where XLIJI

=

r 7(2"- 2 ) 0(2"-2) 0(2"-2) .0(2""2)

n) A/(2") + A ' i4 n) 0(2") G^ -/(2").

0(2"" 2 ) 0(2"-2) 0(2"-2) 0(2"-2)

-7(2"-2) 0(2"-2) 0(2"~ 2 ) 0(2"~ 2 )

0(2")' 0(2")

J

0(2""2)0<2"~ 2 ) 0<2"~ 2 ) 0'2"-2).

From Theorem 2.3.1 we introduce a modified threshold contact process on state space 5'"^ with generator G^t is given by ^("+1) _ — "i»r

'G|S - W(2»)

I

'(2")

A/(2")

GJS-J(2").

That is, the second term of the right hand side of is omitted. Fortunately in this case we can compute stationary distribution explicitly. For example, PmT - n

( 1 )

a(2)

-ii(2)

14

Phase Transitions of IPSs

/4& = ^kooo),^(ooi),^(oio),^kon), ^(100),^(101),^(110),^(1H)] 1 [1, A, A, A2, A, A2, A2, A3]. ((A+ A + 1)3

And as for the density of this process, P^TW $TW

= =

P^TW = P%rW P^TW = P™TW

A XTT A+l.

In general, we have Theorem 2.3.2. fbr 77 = (ri(xi)r)(x2) ■ ■ .ri(xn)) e 5 ( n ) with n = 1,2,..., the stationary distributions of the modified threshold contact processes on state space S^"' are obtained by \NW A „<») 1 KlUv) =(A + l ) " ' where N(j]) is the number ofr)(xi) = 1 for i = 1,..., n. Moreover the order parameters are given by A A+l'

Aiw =

Proof. For simplicity, we omit subscript mT in this proof. First we devide //") into two parts in the same number of each part and define ^("' = [/4i >M ]■ For example, (2)

2

) (00), M ( 2>(01)], ^ 0 2 , === [M [^>(00),/x< (OI)], M

2) 4

4 === [M 2)

[ M(<2 )2 » ( 1 0 ) , ^(2) )(11)). (IO), M

(ll)]-

Then we have

„(»+!)_ * „(n) and Ml A+l^ Mo - A + 1^ +1 To prove it observe that the definition of G'" ' implies Mo

M(n+l)G(n+l) =

is equivalent to M

(n+l)(G(„)_A/(2„))

+

(2.1)

(j

M

(n+l)=0)

V^" + 1 ) +M n + 1 ) (G ( n ) -/(2")) = 0. So /i(")G = 0 gives Eq.(2.1). Once Eq.(2.1) is established, * e desired conclusion is easily obtained.

Basic Contact Process I

15

CHAPTER 3 THE BASIC CONTACT PROCESS I

3.1. Introduction The basic contact process has been studied extensively from mathematical, physical, chemical and biological aspects. In the probability theory, the basic contact process was first introduced and studied by Harris.1 From that time on, many mathematicians have studied the basic contact process by using the techniques of interacting particle systems and percolation theory. For details, for examples, see Griffeath,2 Liggett,3 Durrett,4 and Durrett.5 From the physical point of view, see Marro and Dickman.6 One interpretation of the basic contact process is, as we showed in Chapter 2, a simple model of an infectious disease. The basic contact process is a continuous-time Markov process on state space T\ £ {0,1} Z , where Zd is the d— dimensional integer lattice. There are two different types of definition of the basic contact process. One is given by the following formal generator J2c(x,r,)[f(rix)-f(v)),

n/(i?)= x€Zd

with flip rates c(x, 77) = TJ(X) + A(l - 77(1)) x

J2

77(1/),

v-\y-x\=l

where rf denotes rf{y) = rj(y) for y ^ x and rf{x) = 1 - r}{x) and |x| = | i i | + 1- \xd\The details about this definition were discussed in Chapter I of Liggett.3 The other is given by the graphical representations (see details, for example, Durrett 4 ). This definition is more intuitive, so it is easy to understand. We consider the state of process ft C Z d ,i.e.,f t = {x e Zd : rjt(x) = 1}. The points in it are thought of as being occupied (or infected). (i) If x £ it, then x becomes occupied at a rate equal to A times the number of occupied neighbors, (ii) If x e it, then x becomes vacant at rate 1.

Phase Transitions of IPSs

16

To construct the basic contact process, we introduce the following two Poisson pro cesses. For each x and y with | x - y |= 1, let {21T ,n > 1} be a Poisson process with rate A, and let {U*,n > 1} be a Poisson process with rate 1. At times Tnx , we draw an arrow from x to y to indicate that if x is occupied then y will become occupied (if it is not already). At times U%, we put a 6 at x. The effect of a S is to kill the particle at x (if it is present). We call there is a path from (x, 0) to (y, t) if there is sequence of times t0 = 0
(y,t)}.

Note that the first definition is equivalent to the second one. See Liggett, 3 pp.172-173. In this chapter we take the first definition, as you will see later, since it is useful to compute the correlation functions of the basic contact process. Let p be the collection of all probability measure on { 0 , 1 } Z . For / i E p , nS(t) £ p denotes the distribution at time t of the basic contact process for initial distribution fi where S(t) is a semigroup corresponding to the formal generator fi. Let 3 be the set of all stationary measure ( j £ p : 3 = { / i 6 p : fiS(t) = ii for all t > 0}. The basic contact process is said to be ergodic if (a)

9 = { i ' } i s a singleton,

or (b)

lim fiS(t) = v for all // 6 p . (—KX>

Let #o and b\ denote the pointmass on 77 = 0 and r\ = 1 respectively. By Theorem 2.14 in Chapter III of Liggett, 3 there is a critical value Ac £ [0,oo] so that if A < Ac, then the basic contact process is ergodic : 9 = {SQ}, and if A > Ac, then the basic contact process is not ergodic. So we have Ac = sup{A > 0 : the basic contact process is ergodic } = inf{A > 0 : the basic contact process is not ergodic }.

Basic Contact Process I

17

Bezuidenhout and Grimmett7 showed even if A = Ac, then the basic contact process is ergodic : 9 = {fio}Let

vx = lim hS{t), t-*tx>

which is called the upper invariant measure. The well-definedness of this measure is guar anteed by the attractiveness of this process. In general, the process with rates c{x,rj) is called to be attractive if any 77, f with n(x) < f (x) for any x, c(x,V)c(x,0

if n(x) = C(*) = 0, if n(x) = C(x) = 1.

Most of particle systems in this book are attractive, so the upper invariant measures have an important role in this book. On the other hand, trivial stationary measure, tf0 = lim S0S(t), (—too

is called the lower invariant measure. Moreover Bezuidenhout and Grimmett7 proved that the stationary measure of the basic contact process is a convex combination of So and v\ : for all p. £ p, Urn pS{t) = 7*o + (1 - 7)«/A where

1=1

p

v(T < <»M*?)>

and r is the hitting time of ij = 0. Define px as the density of perticle at a site x with respect to v\ : P\ = Evx(t)(x)) = v\(v{x) = !)• Note that p\ is independent of x, since i/\ is translation invariant. We take p\ as an order parameter of the basic contact process. Then the critical value Ac can be characterized by P\Ac = sup{A > 0 : px = 0} = inf{A > 0 : px > 0}. Unfortunately Ac and px are not known rigorously ! From now on, we will restrict attention to the one-dimensional case. Here we present several rigorous results on the critical values and the order parameters of the basic contact processes in one dimension. Theorem 3.1.1. (1) px = 0 for A < Ac.

Phase Transitions of IPSs

18

(2) (3) (4) (5)

px > 0 for A > Ac. p\ is a noadecreasing function of X. px is continuous for A > 0. 1.539 < Ac < 1.942.

Note that the lower bound 1.539 on Ac is a numerical result by Ziezold and Grillenberger. 8 As for upper bounds, Holley and Liggett 9 proved Ac < 2. Recently an improved upper bound 1.942 was given by Liggett. 10 Some simulations or (non-rigorous) numerical methods by using computer reported that the critical value is estimated as, Xc « 1.649. (See, for example, Brower, Furman and Moshe 11 , Konno and Karori 12 .) The above results (l)-(4) hold also for d > 2. This chapter is mainly devoted to the lower bounds on critical value of the one-dimensional basic contact process. In general, there are many methods for getting the lower bounds as follows : (i) (ii) (hi) (iv)

Harris-FKG inequality method. Markov extension method, Liggett and Ziezold-Grillenberger methods, Griffeath method. In this chapter we will study (i) - (iii). As for (iv), we will discuss details in Chapter 5. Upper bounds will be treated in the next chapter. 3.2. Correlation I d e n t i t i e s In this section we present the correlation identities on the one-dimensional contact process. Let Y be the collection of all finite subsets of Z 1 . We define px(A) = E„x I [ J r)(x)\ = vx{n : V(x) = 1 for all x € A), \xeA ) PM)

= EUi ( J J ( 1 - »?(*))) = ">(»? ■ ri(x) = 0 for all x £ A),

for any A 6 Y. We can rewrite the formal generator of the one-dimensional basic contact process as follows :

«/0rt = E ^ M * +1) + v(x - i)}(i - T?(X)) + r>(x)][f(if) - f(n)i xez The general arguments of the generator of a Markov semigroup give

jtESl(f(m)) = ESl(nf(%)). Let A = {xi,X2,---,xn}, at time t as

where X{ E Z for 1 < i < n. Define n-points correlation functions

Pt,\(A) = pt,\(xiX2 ... xn) = ESl{rit(xi)rit(x2)

■ ■ ■ T)t(xn)).

Then we have the following system of infinite correlation evolution equations.

Basic Contact Process I

19

Theorem 3.2.1. For any AeY,

ftPtAA) = -{\A\+2\b(A)}pUA)

+ \Yl XSA

£

P*A(A\{x})U{y})

J/6A.4 |y-*|=l

+X £

w

A{x)pt,x{A \ {x}) - A J^

x&A

wA(y)ptiX(AU{y}),

y£&A

where \ A \ denotes the number of elements in A, b(A) denotes the number of bonds inside A, AA = {y G Z \ A : there is an x e A so that \x-y\=l} and for x g Z, wA(x) =| {y g A :| a; — y |= 1} |g {0, l,2},i.e., denotes the number of nearest neighbors ofx inside A. Note that this theorem is satisfied generally in the d-dimensional case by translation : Z -> Z d . By Theorem 3.2.1 for A = {0},{0,1},{0,2}, and {0,1,2}, and the translationinvariance of the initial configuration 7, = 1 and evolution of this process we obtain the following differential equations. Lemma 3.2.2. (1)

l}ipt'xW

= ( 2A - 1 K A ( 0 ) - 2Apt,,(01).

(2)

^ . * ( 0 1 ) = 2XP*A°) - 2(X + ^KxCOl) + 2 V M ( 0 2 ) - 2Aft,A(012).

(3)

^-tPt,\(M) = 2Vf,A(01) - 2/9^(02) - 2AAl»(012) + 2Aft,x(03) - 2A/>t,*(013).

(4) -^pt,x(0l2) = 2XptiX(01) + 2/>t,A(02) - (4A + Z)pt,x(012) + 2Apt,A(013) - 2Apt,A(0123). Of course, the above lemma is also given by direct computations from the formal generator. For example, in the case of /(»?) = 17(0), observe that fi/M = [ M T , ( - 1 ) + 7,(1)}(1 - 7,(0)) + r,(0)j [1 - 27,(0)] = A T , ( - 1 ) + AT,(1) - 7,(0) - AT,(-1)T,(0) - A7,(0)77(l),

since j,(0) 2 = 7,(0). Hence we have part (1). Similarly parts (2)-(4) are obtained. In the stationary state, that is, in the limit as t -> 00, Theorem 3.2.1 gives

20

Phase Transitions of IPSs

Theorem 3.2.3. For any AeY, 0 = -{\A | +2\b{A)}px(A) + A Y, x€A

E

|„-*|=1

+ AJ>,I(*VA(A\{S})-A £ 16^4

PA((A\«)UW)

116A/I

wA(»)^(AU{if}).

K6AA

Therefore we have the following correlation identities for p\(-) : Lemma 3.2.4. (1)

0 = (2A - l)pA(0) - 2ApA(01).

(2)

0 = ApA(0) - (A + 1W(01) + ApA(02) - ApA(012).

(3)

0 = ApA(01) - pA(02) - ApA(012) + ApA(03) - ApA(013).

(4)

0 = 2ApA(01) + 2pA(02) - (4A + 3)pA(012) + 2ApA(013) - 2ApA(0123).

On the other hand, by a similar way or using a duality technique, (see Section 5 in Chapter III of Liggett3 or Chapter 5 in this book,) we have Theorem 3.2.5. For any AeY,

x

E E [p^A u {»» - M*)] + E N A \ W) - ?*w] = °x£A y:\y-x\=l

x€A

So we have the following correlation identities for pA(-) : Lemma 3.2.6. (1)

1 - (2A + 1)/5A(0) + 2ApA(01) = 0.

(2)

PA(0) - (A + l)pA(01) + ApA(012) = 0.

(3)

2pA(01) - (2A + 3)pA(012) + pA(02) + 2ApA(0123) = 0.

Ba3ic Contact Process I

(4)

pA(0) + ApA(012) - (2A + l)pA(02) + ApA(013) = 0.

(5)

pA(012) - (A + 2)pA(0123) + pA(013) + ApA(01234) = 0.

21

(6) pA(01) + p A (02)-(4A + 3)pA(013) + pA(03) + 2ApA(0123) + ApA(0124) + Apx(0134) = 0. (7)

pA(0) - (2A + l)pA(03) + ApA(013) + ApA(014) = 0.

3.3. Harris-FKG Inequality Method This section gives lower bounds on critical values and upper bounds on order parameter for the one-dimensional basic contact process by using the Uarris-FKG inequality. As for the Harris-FKG inequality, for example, see pages 70-83 in Liggett.3 Results in this section appeared in Konno and Katori.13 Let M+ and M~ be the collection of all increasing and decreasing functions on {0,1} Z , respectively. Theorem 3.3.1. ( Harris-FKG inequality ) For any f,g £ M+, Eu-dfa) >E„: AW».MBy the definitins of increasing and decreasing function, we have the following corollary immediately. Corollary 3.3.2. ( Harris-FKG inequality ) For any f,g £ M~,

(i)

EVxUa)>Evx{f)E,M-

For any f e M~ amd g e M+, (2)

E^Ua) < +

For any f G M (3)

EUl>(f)En(g).

and g e M~, EM)

<

E^(f)E^(g).

Applying the Harris-FKG inquality to p\{A) and p A (4), we obtain

22

Phase Transitions of IPSs

Corollary 3.3.3. ( Haxris-FKG inequality.)

For any A,B 6 7 ,

(1)

px(AUB)>px(A)p (B), Px(AUB) > px(A)pxx(B),

(2)

(A)px(B). pM UPx(AUB)>p B) > pxx(A)p

Proof. For part (1), define

m=n <*«). s(n)=n ■»(*)• x€A

x€B

Since /,
fin) = i - II (i - <*)), a(n) = i - II ^ - rt*))x€A

xeB

So the proof is complete. From Corollary 3.3.3 (1), we have the following useful lemma. L e m m a 3.3.4. 2

(1)

Px(01)>px(0)

(2)

M012) >

(3)

p>(0123) >

(4)

.

px{0)px(01).

px(0)px(0l2).

P A ( 0 1 3 ) > p A (0)p A (01).

In the same way Corollary 3.3.3 (2) gives L e m m a 3.3.5. (!)

P,(01)>pA(0)2.

(2)

P A ( 0 1 2 ) > P,(0)p,(01).

(3)

Px(0123) > p x (0)p A (012).

23

Basic Contact Process I

(4)

fc(013)

?»(013) > ^(0)p ft(0)fc(01). A (01).

(5)

PA(01234) > />,(0)p P*(01234) pA(0)pA(0123).

(6)

p,(0124) (012). PA(0124) > p,(0)p ^(0)p AA(012).

(7)

ft(0134)

p,(0134) > pxA(0)p (0)p,(013). x(013).

(8)

PA(OW) > > Px(0)Px(M)PA(014) pA(0)p,(01).

The definitions of p\,p\(0)

and p;\(0) gi ye

Lemma 3.3.6. PA = PA(0) = 1 - P A ( 0 ) .

3.3.1. First bound From Lemmas 3.2.6 (1) and 3.3.5 (1), we have 1 - (2A + l)pA(0) + 2ApA(0)2 < 0. By Lemma 3.3.6, the last inequality can be rewritten as

4>-^),». Combining the last inequality and px > 0 gives 2A - 1

» S -2JT

„ for

2A - 1

and pA = 0 So we obtain the following result.

for

„

^r^ 0 ' - g £ - < 0.

24

Phase Transitions of IPSs

( H,1) T h e o r e m 3.3.7. Let X = 1/2. Define '\C C —

A

- / ( 2 A - - 1)/2A

lo

for A > v(»,l) for A < A^>.

Then we have

(1),

AWi)

tf™

< A < Acc

and H,1) pA
(2)

for for

A ^ >> 00..

In this way we get a lower bound Ac on critical value Ac and an upper bound p\ ' on order parameter p\ simultaneously. Note that Theorem 3.3.7 can be obtained by a different way. That is, Lemmas 3.2.4 (1) and 3.3.4 (2) imply (2A - 1 ) ^ ( 0 ) - 2Xpx(0)2 > 0, so we have the same conclusion. But, in this case, Lemma 3.2.6 is more suitable than Lemma 3.2.4 as for using the Harris-FKG inequality, since the number of negative terms in Lemma 3.2.6 is always one ! So we use Lemma 3.2.6 to improve bounds instead of Lemma 3.2.4. 3.3.2. Second bound As the first bound, Lemmas 3.2.6 (1) (2) and 3.3.5 (2) imply pA[A(2A+l)pA-(2A2-l)]<0. Using similar arguments, we obtain T h e o r e m 3.3.8. Let \iH'2) (H,2) PX

= 1/V2 fa 0.707. Define f (2A2 - 1)/A(2A + 1) for A > A ( c w ' 2) , lO forA.

Then we have A W ! < A("'2> < Ac,

(1) and et\ (2)

^ (#,2) ^ (H,l) Px < p\ < P\

, for

, . „ A > 0.

3.3.3. Third bound As the first and second bounds, Lemmas 3.2.6 (1) - (4) and 3.3.5 (3) (4) imply pA[A(2A + 1)(4A + 3A + 2)Px - (8A4 + 6A3 - 6A - 3)] < 0. In a similar fashion as before, the next result is obtained.

Basic Contact Process I

25

Theorem 3.3.9. Let A<,H'3> = inf{A > A : 8A4 + 6A3 - 6A - 3 > 0} « 0.859. Define (ff,3) _ f (8A4 + 6A3 - 6A - 3)/A(2A + 1)(4A2 + 3A + 2) for A > A<"'3), l0 forA.

X

Then we have A<»'2> < A(»'3> < Ac,

(1) and

P,
(2)

for

A>0.

Note that as Lemma 3.3.5 (4) we can get pA(013) > p,(0)pA(02).

(3.1)

3

From Theorem 1.9 (c) in Chapter VI, Liggett and the definitions ~f>\{A), we have Theorem 3.3.10. Suppose that, for each n > 1, x\ < 12 < • ■ ■ < xn a*>d Vi < 2/2 < • • • < y„ with xi+1 ~Xi < yi+i - y{ (i = 1,2,..., n - 1). Then P\({xi,x2,...,xn})

({s/i,J/2,---,J/n})-

So we get PA(02) < pA(01).

Therefore we observe that estimation by Lemma 3.3.5 (4) is better than that by Eq.(3.1). Similar arguments hold in the next bound. 3.3.4. Fourth bound As the privious bounds, Lemmas 3.2.6 and 3.3.5 (5) - (8) yield PxhiWpl

+ 72(A)PA + 73(A)] < 0,

where = A2(2A + 1)(8A4 + 12A3 + 9A2 + 9A + 3), 6 5 4 3 2 72 (A) = A(32A + 120A + 186A + 182A + 148A + 75A + 15), 6 6 4 3 2 7l (A) = -2(2A + 1)(24A + 36A + 20A - 2A - 28A - 30A - 9). 7l (A)

Phase Transitions of IPSs

26

Since 71(A) > 0 for A > 0, the last inequality can be rewritten as

«(p i -pi + ) )(p»-/'Sr , )s 0 '

(3 2)

-

where (±) Px

-72(A) ± v S ( A ) 2 - 4 7 i ( A ) 7 3 ( A ) ~ 2 7 i(A)

Note that 72(A)2 - 4 7l (A)7 3 (A) > 0 for A > 0. Eq.(3.2) gives Px < p[+)

p[+) > 0,

for

and px = 0

p[+) < 0.

for

So we have the following result.

T h e o r e m 3.3.11. l e t A = inf{A > A : 73(A) < 0} « 0.961. Define (H.4)_/A+) "* " t o

fo'A>A(c"'4\ forA.

Then we have (1)

A ( ^ ) < A<"'<> < Ac,

and Px < P{H'4) < p{"-3)

(2)

for

A>0.

As far as the Harris-FKG inequality used for A C {0,1,2,3} and B = {4}, a similar discussion in the end of subsection 3.3.3 shows the above bounds are the best ones. 3.3.5. Higher

dimensions

As for the d-dimensional case corresponding to the first and second bounds, we have the following results in a similar way. (See details in Konno and Katori. 1 3 )

Basic Contact Process I

27

T h e o r e m 3 . 3 . 1 2 . d > 1. Let X(CH,1) = l/2d. Define (H,i)

((2dX - l)/2dA for A > A ( c ff4) , lo forA^A^'1'

=

X

Then we have

(1)

f"

< K,

and Px
(2) Let \{H'2) = l/y/2d(2d-l).

for A > 0 .

Define

(H,2) _ / {2d(2d - 1)A2 - l}/(2d - l)A(2dA + 1) for A > \{H'2\ 2 2) " lo forforA ' .

P

Tien we have (3)

f »

< AW> < Ac,

and

PxHFM
(4) 3.3.6.

for A>0.

Summary

In this section we obtain the some bounds by using the Harris-FKG inequality : A*/' 1 ' = 0.5 < A*/*'2' » 0.707 < A<"'3> * 0.859 < A ^ ' 4 ' » 0.961 < Ac, and j.

(H,4)

,

(H,3)

,

(H,2)

,

(#,1)

r

> .. „

Px < p\ " < p\ " < p\ ' ' < PA for A > 0. We think this method is one of the simplest one. But, as you will see later, when compared t o other methods, this is not so good. 3.4. H a r r i s L e m m a a n d M a r k o v E x t e n s i o n M e t h o d In this section we study lower bounds on critical value and upper bounds on order parameter of the one-dimensional basic contact process by using the Harris lemma. We call this method Markov extension method. The reason will be found in the last subsection 3.4.6. Related results in this section appeared in Katori and K o n n o . 1 4 - 1 6 As you will see

Phase Transitions of IPSs

28

later, this method is more powerful than the former Harris-FKG inequality method in the case of the basic contact process. 3 . 4 . 1 . Harris lemma In this subsection we introduce the Harris lemma which gives the lower (and upper bounds) on the critical value. Recall that Y is the collection of all finite subsets in Z. For any A £ Y, we let TT\{A) = 1 - EV)l I Yl (1 - n(x)) I =v\{ff. \x€A I

v(x) = 1 for some x e A),

Wx(A) = 1 - EVx I J J T)(x) I = ux(ri: rj(x) = 0 for some x E A). \x€A I Note that xx{A) =

l-px{A\

WX(A) = 1 -

px{A),

px = Px(Q) = 1 - Px(0) = TA(0) = 1 - WX(0). From the definition of irx{A) and Theorem 3.2.5, we have the same type of the correlation identities :

T h e o r e m 3 . 4 . 1 . For any A e Y, A

E

£

[ ^ U W ) - n ( A ) ]

+ ^[.5(A\{«})-n(A)]=0.

x£A y:\y-x\~\

x€A

Similarly the definition of nx(A) and Theorem 3.2.3 give

T h e o r e m 3.4.2. For any

AeY,

0 = - { | A | +2Xb(A)}Wx(A)

+ A Y,

f

£

x£A

> P \ W ) " W )

ygA/l |y-*|=i

J

+ \ £iwA(x)Wx(A\{x})-\ x€A

J2

VA(y)*x{AU{y}).

y£AA

In order to obtain lower bounds on Ac, we have to look for suitable upper bounds hx(A) on irx(A). Because px = TT*(0) < hx(0) implies that the critical value of hx(Q) gives a lower bound on Ac. To do this we use the following lemma by Harris, 17 so we call it the Harris

Basic

Contact

Process

29

I

lemma. Let Y* be the set of all [0,l]-valued measurable functions on Y. For any h e Y*, we let

n-h(A) = \J2

[*(*U{»»-*(A)] + E[*( A \ to)-*(*)]•

£

*£-^»:|y—K|=1

I€J4

Note that, of course, fi'MA) = 0. Lemma 3.4.3. Let hi eY* (i = 1,2) with (1)

hi{4>) = 0 and 0 < /i;(A) < 1 for all

(2)

A / <j>,

lim *i(il) = l, \A\-t-oo

(3)

fi*/n(A)

Then for any (4)

< 0 < il*hi{A).

AeY, M A ) < n ( A ) < hi(A).

In particular, (5)

MO) < Px < MO),

(6)

Aj < Ac < A*,

wiere /ij(0) =fe<({0})and Aj. is a critical value on hi(0), that is, X\ = sup{A > 0 : MO) = 0}, X2C = inf{A > 0 : MO) > 0}. Therefore, by choosing suitable /ij(A), we can obtain lower and upper bounds on Ac. As we will see in Chapter 4, Holley and Liggett9 gave the good upper bound by using another hi(A). 3.4.2. First bound by the Harris lemma Let \A\ be the cardinality of A. So we have

30

Phase Transitions of IPSs

T h e o r e m 3.4.4. Let A*/*'1' = 1/2. Then for A > A< M,1) , T A ( 4 ) < hf£\A)

for all

AeY,

where

•pw-i-^r. Therefore the following is obtained. Corollary 3.4.5.

AC > r

1

=L

^"(a)-5^

«-

^2'

3.4.3. Second bound by the Harris lemma Let a(j4) be the number of neighboring pairs of points in A, that is, a(A) = \{x E Z : {x,x + 1} C A}\. Then we have the following results in a similar way. T h e o r e m 3.4.6. Let A(CM,2) = 1. Tien for A > \{M'2), TT A (A) < ht*\A)

where

»i''M =

'~{d

for

nil

AeY,

a

,rr ;

_ j \
Therefore we have Corollary 3.4.7. Ac > A<M'2> = 1,

/ ^ < i - (UA-IJ — , ) = ^=T

for

A

^

3.4.4. Third bound by the Harris lemma Let i»(A)=|{ieZ:{i,i + l,i + 2}cA}|, c(A) = \{xeZ:{x,x so similarly we have

+

2}cA}\,

L

Basic Contact Process I

31

Theorem 3.4.8. Let A(CM'3) = (1 + V37)/6. Then for A > A(CM'3), TA(A) < fti3)(A) for all where

AeY,

h™{A) = 1 - a 1 (A)Wa 2 (Ar^> a3 (A)^>a 4 (Ar< A >, =

«I(A)

a»(A)= as(A) = 04(A)=

D=

A(2A + 3) + VD (2A+1)(6A2-3A-1)' -(24A4 + 16A3 + 8A2 + A + 1) + (6A - 1)(A + 1)VD 2A(A - l) 2 24A3 + 16A2 - 2A - 3) + (4A + 3)VD 8A(2A + l) 2 3 (A + 1){12A - 2A2 - A + 1 - (3A - l)y^D} 2A 2 (A-1) 4 2 16A + 4A + 4A + 1.

Therefore we have Corollary 3.4.9. Ac

> X<MV = ll^EL

„

L180i

2

p\ < _

4A(3A - A - 3 ) . . 1 + V37 ;= for A > . 12A3 - 2A2 - 8A - 1 + y/D ~~ 6

3.4.5. Higher dimensions As for d > 1, we have the following result in a similar way. See Katori and Konno 14-16 for details. Theorem 3.4.10. Let A(CM|1) = l/2d. Then for A > A(CM'2), *\{A) < h<£](A) for ail

AeY,

where --1h™(A) = Moreover Jet

A(CM'2)

\2dXJ

= l/(2d - 1). Then for A > A(CM'2), TX(A)
tor all

AeY,

32

Phase ThnuiNoiu of IPSs

where

h^(A) = :1 ft »W_1

\W

/ 2d-l \2d(2d-l)X-lJ \2d(2d-l)X-l)

\

(2d(2d-l)\-l\a(A) (2d-l) 22A )/

So we have

Corollary 3.4.11.

(!)

A. * * « * - £ . and P > < ( ^ ) v O .

(2)

««--^.-^(S*D"

3.4.6.

Remark

In this section we obtained the bounds by the Harris lemma in the one-dimensional case : f ' " = 0.5 < A = 1 < A<M'3> a 1.180 < Ac, and PA

< P\

'
<

ft

for

A > 0,

where p ^ = h<£\o) for i = 1,2,3. We should remark that Corollaries 3.4.5, 3.4.7 and 3.4.9 can be also obtained by using Lemma 3.2.6 and the following conjecture : Conjecture 3.4.12. (1)

(2)

(3)

PA(0)PA(012) > p,(01)2

ft(01)ft(0123) fr(01)ft(0123)

2 > > p,(012) PA(012)2

/>A(0)pA(013) > ^ ( 0 1 ) ^ ( 0 2 ) .

Prom this, we will extend to the following conjecture.

Basic Contact Process I

33

Conjecture 3 . 4 . 1 3 . For any A, B 6 Y, p A (A U B)px(A

n B) > p A (A)p A (B).

For example, Conjecture 3.4.13 for A = {0,1} and i? = {1,2} gives Conjecture 3.4.12 (1). If A fl B = <j>, this inequality is equivalent to the Harris-FKG inequality in Corollary 3.3.3. So it obviously refines the Harris-FKG inequality in this meaning. Now we will explain the meaning of the Markov extension method. (See more details in Katori and Konno. 1 6 ) First we choose a subset Y^ C Y and define hx on Y(n> by a suitable way. Next for the rest of sets in Y \ Y, the function h^ is constructed from a quotient of simple products of hr" s previously defined on Y ( n ' . The formular used in the latter procedure is similar to the relation found among probability measures in generalized Markov processes, so this extension procedure is called the Markov extension. For example, see Schlijper. 18 3.5. Liggett and Ziezold-Grillenberger M e t h o d s In this section we consider the best lower bound on critical value of the one-dimensional basic contact process given by Liggett, 3 and Ziezold and Grillenberger. 8 In particular, we clarify the relation between their methods. Define an initial configuration 77+ by 77(1) = 1 for x > 0 and 77(1) = 0 for x < 0. Let l(rj) be the leftmost position of an infected individual in 7 7 ^ S0). That is, l{rf) = - 0 0 if 77(1) = 1 for infinity many x < 0 and 1(TJ) = y if T](y) = 1 and 77(1) = 0 for any x < y. Let S = { 0 , 1 } Z and Si = {77 e S \ {60} : I(y) / - 0 0 } . For n = 0 , 1 , . . . , define the Markov process 77}"' on state space S\ starting from 77+, i.e., T7o = 77+, with the following formal generator : Kl)+n E <*,v)lf(nx)-f(v)] = x=Hn)

n) " ((B) /('7)=

« /0J)

+ ■ ic(l(v)-Uv)lf(r)')-m], ,i)t/W-M X) <^)ifm-f(v)]+c(i(v)-

where 77 e Si and ,, , '' '

=

f 1, for x e {l(V) - I,Z(77) + n}, ^ 77(1) otherwise.

So this process 77}"' behaves like the basic contact process on ( - 0 0 , i ^ } " ' ) + n]. On the other hand r]tn\x) = 1 on [ f ^ 7 1 ' ) + n + l,oo), i.e., all individuals are always infected on this region. Let Tk = r£"' be the moment t at which 77'"' changes for the fc-th time with T0 = 0 and £k = rk - Tk_x for k = 1,2, Note that {& : k = 1,2,...} is a sequence of i.i.d. random variables. Now we can define a mean displacement m<")(A) G [—l,n] and an edge speed a ( n ) ( A ) G (-00,00] of 77'n) in the following way :

m<»>(A) = Urn pEMijM)), K-~+ OO K

a<">(A) = Urn t—too t

-E(l(n\n))).

Phase Transitions of IPSs

34

For n — 0 , 1 , . . . , a critical value for 7?'"' can be defined by \[n) = sup{A > 0 : m ( n ) (A) > 0} = sup{A > 0 : Q (n) (A) > 0}. Let T^ 00 ' be the basic contact process starting from 77+ and X[°°' be a critical value for this process. Ziezold and Grillenberger 8 showed L e m m a 3.5.1. (1)

A c = A<°°>.

(2)

AW / A

as

t

77 /

oo.

Moreover Ziezold and Grillenberger 8 presented a useful method for computation of A| as follows. For TI = 0 , 1 , . . . and k = 1 , 2 , . . . , we let j(")

_

;(") _

;(")

where /'"' = l(ri\n'). Note that for n = 0 , {dk' : k = 1,2,...} is a sequence of i.i.d. random variables. That is,
7n ( ^ ' ) ) = K - > ( £ < a

= X&)

=

- m -

Therefore the definition of the critical value gives A<°> = 1. For n > 1, define

Xkn) = (r,(W +

n

l),...,T,(li:)+n)),

yw = (4"\4 >). n

{X\ ' : k = 1,2,...} and {Y£n' : k = 1,2,...} are Markov chains on S^ = { 0 , l } ^ ' - " > and ,?<"•*> = {-1,0,1,...,77 + 1} x S<-n\ respectively. p(.n,*) j , e tkgjp transition probebilities and / / " ' and /^"'*' be their stationary respectively. The Birkhoff's individual ergodic theorem (see Theorem 2.1 in Durrett 1 9 ) implies

& l E 4 n ' = ^c».->(4n))

a.s.

state spaces Let p("> and distributions Chapter 6 of

Basic Contact Process I

35

Observe that

£„<...> (4 B } ) =

lim

-<£(.

so m' n '(A) = £^.,.,(4 B ) )- Then Ziezold and Grillenberger8 proved Lemma 3.5.2.

™(n)(A)=

E (i,ti)es<».-)

Forn = 3,4,..., m = 1,2, ...,n£(s) £(s) £(s) R(s)

i E ^ W ^ C C M ) , (/,*))• aes(»>

1 and s = 3^2 •••«n 6 5 ( n ) , we let

= ra+l, if 5 = 0...00, = n, if s = 0 . . . 0 l , = m, if s = 0 . . . 01s m+ i ... sn, = R(3ls2 ...sn) = {2c{s) - 1}A + N(s) + 1,

where c(s) is the number of components of l's for 01si«2 ■ ■ • *nlO and N(s) is the number of l's for «is 2 • • • sn with £(0) = 2, £(1) = 1, 5(0) = 3A + 1, R(l) = A + 2, 1(00) = 3, £(01) = 2, £(10) = £(11) = 1, 5(00) = 3A + 1, 5(01) = 5(10) = 3A + 2, 5(11) = A + 3. For example, in the case of n = 3, the above definition gives £(000) = 4, £(001) = 3, £(010) = £(011) = 2, £(100) = £(101) = £(110) = £(111) = 1, 5(000) = 3A + 1, 5(001) = 3A + 2, 5(010) = 5A + 2, 5(011) = 3A + 3, 5(100) = 3A + 2, 5(101) = 5(110) = 3A + 3, 5(111) = A + 4. So Lemma 3.5.2 can be rewritten as follows. In fact, it is more useful to compute mSn\X). Lemma 3.5.3. m<">(A) =

£

W Ms) * W<*■ T5T"

For n > 1, define

dn) = w4 n ) )>---^ n ) + »))Then Q is the Markov process on state space S*n' with generator G' L , n ' which is given by $}("). Let ju*"' be a stationary distribution of £("). So a similar argument gives the next result (see pp.288-289 in Liggett3).

Phase Transitions of IPSs

36

L e m m a 3.5.4. aW(A)

= -

£

(A - £(«))/*<»>(-).

Note that Lemmas 3.5.3 and 3.5.4 imply that \in' is the positive root of

£

I(S))p<">W-A,

since p*n» is a probability measure. 3.5.1.

n=l

In this subsection we compute m^(\) and ^ ' ( A ) by Lemmas 3.5.3 and 3.5.4 and also get Ao1 .{-Xfc :fc> 1} is the Markov chain on state space {0,1} with transition probability M) _ [P ( 1 ) (0,0) -[^'(1,0)

^(0,1) pM(l,l)j

P

0

1

1 A+2

A+l A+2

So the stationary distibution of X' 1 ) is

^

A+2 A + 3'A + 3

= [^(O),^!)] =

Lemma 3.5.3 gives m' J »(A) =

( A - X ( 0 ) ) ^ #(0)

+

v(

A-X v

( 1 ) ) ^ > " fl(l) .

Noting that 1(0) = 2,

1(1) = 1,

so we have m(A) :

E(0) = 3A + 1,

iJ(l) = A + 2,

3A2 - A - 3 (A + 3)(3A + l ) '

/■(i); Similarly Q is the contimuous-time Markov process on state space {0,1} with gener

ator G^1' =

^(l.O)

«M(1,1)

-(3A + 1) 1

3A+1 -1

Therefore the stationary distibution of f O is P<1» = [M< 1 )(0),M( 1 )(1)] =

3A+1 3A + 2 ' 3 A + 2

Lemma 3.5.4 gives aW(A) = - [ ( A - £(0))pO)(0) + (A - ^ l ) ) ^ 1 ^ ! ) ] .

37

Basic Contact Process I

So a<»(A) = In any case we get

3.5.2.

3A2 - A - 3 3A + 2 '

^i+^uao.

n=2

As n = 1, in this subsection we compute m' 2 '(A) and a< 2 '(A) by Lemmas 3.5.3 and y(Z) 3.5.4 and also get A'(2) Xj^ is the Markov chain on state space {0, l } ' ' 1 , 2 ^ with transition probability : n<

2

> =

—

P 2 >((00),(00)) p< 2 >((01),(00)) p< 2 >((10),(00)) p( 2 >((ll),(00)) n x H(00) " TpoJ 0

mm i

1

A(ioy

P<

2

>((00),(01)) p< >((01),(01)) p< 2 >((10),(01)) p< 2 >((ll),(01)) 2

.R(00) X

0

fl^olj

0

p<2>((00),(10)) p<2>((01),(10)) p<2>((10),(10)) p< 2 >((ll),(10))

P<

2

>((00),(11)) p< 2 >((01),(ll)) p< 2 >((10),(ll)) p< 2 >((ll),(ll))

1 R(00)

Tim 3X

wm

»

WT) WTj i^uywhere 5(00) = 3A + 1, 5(01) = 5(10) =. 3A + 2 and 5(11) = A + 3. So the stationary distibution of X^ is ^

= [M«(00) 1 ^(01) > /,( 2 »(10),^)(11)] = ^ ^ y [ 5 ( 0 0 ) ( 7 A + 5), 5(01)(9A2 + 11A + 3), 5(10)(12A2 + HA + 2),5(11)(27A3 + 32A2 + 15A + 4)],

where £<2'(A) = 27A4 + 176A3 + 240A2 + 130A + 27. Lemma 3.5.3 gives m<2>(A) = - [ ( A - £ ( 0 0 ) )

(2) /j(2>(00) + (A - £(01)) M (01) 5(01) R(Q0)

^)(ll)j + ( A - £ ( 1 0 )Ji(10) ) ^ S r + v( A - i (y i l )") 5(11) Using £(00) = 3, £(01) = 2 and £(10) = £(11) = 1, we have ,<2>(A) = Similarly ^

27A4 + 26A3 - 18A2 - 55A - 27 27A4 + 176A3 + 240A2 + 130A + 2 7 '

is the Markov process on state space {0, l } { a , 2 } with generator :

Q(L,2)

_

-(3A + 1) 1 1 0

A -(3A + 2) 1 1

2A A -(3A + 2) 1 -

1 2A + 1 3A 2

Phase Transitions of IPSs

38

Therefore the stationary distibution of £'

is

p<2> = ^ 2 >(00),p< 2 >(01),/Z< 2) (10),/Z< 2 >(ll)] = — i — [ 7 A + 5,9A2 + 11A + 3,12A2 + 11A + 2,27A 3 + 32A2 + 15A + 4], 7< 2) (A) 2

where, ^~ZT> '(A) = 27A3 + 53A2 + 44A + 14. Lemma 3.5.4 gives a<2>(A) = - [(A - i(00))/i< 2) (00) + (A - I(01))7i< 2) (01) +(A - £(10))/I<2>(10) + (A - Z(ll))7i< 2 >(ll). So a<2»(A):

27A4 + 26A3 - 18A2 - 55A - 27 27A3 + 53A2 + 44A + 14

In any case we get A<2) = sup{A > 0 : 27A4 + 26A3 - 18A2 - 55A - 27 < 0} a 1.280.

3.5.3.

General case

As in the case of finite sets in Section 2.2.4, we consider the general case. In fact, this case is closely related to arguments in Section 2.2.4. For n = 1 , 2 , . . . , we let G'"' be generators of the basic contact process on state space 5 ' " ' . As in Theorem 2.2.1, G ' n ' is divided into three parts ; G^^AG^+G^-G^', where G'"' is infected term, G'"' is recover term and Gj1' is diagonal term. We think of a similar division of G ' L ' " ' . In fact, for n = 1,2,3, we have G< w > = A

GW

=A

+A

[0 0

2 0

+

-0 0 0 .0

1 0 0 0

1 0 0 0

°1 2

-o

o i

o-

0 0

0 0

Lo o

2 0.

1 0 0 1 o 1.

"0 1

0] "0 0 +A 0

-oooo

+

1 0 1 0

0 0

0 0

l

o

0 0 0 0 1 0 0 0

1 1 0 1

Lo l

+

r0 0 0 L0

ll 1

+

ro ii 0

1

3A + 1 0 0 0

"

"3 A + 1 0

0 A+ 2

0 0 3A + 2 0 0 3A + 2 0 0

0 0 0 A+3

39

Basic Contact Process I

G(L,3)

=

0 0 0 0 0 0 0

A

1 0 0 0 0 0 0

Lo

0 0 0 0 0 0 0

0 10 0 0 10 10 0 2 0 0 2 0 0 0 0 0 2 0 0 1 1 0 0 0 0 0 2 0 0 0 0 2

o o o o o o o

-o o +A

+1 0 0 0 0 0 0 0

0 0 0 0 0 0 .0

0 0 10 10 0 1 10 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 U + 2 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0

0 0 5A + 2 0 0 0 0 0

o0

0 0 0 0 1 1 0 0

0 0 0 0 1 1.

0 0 0 3A + 3 0 0 0 0

0 0 0 0 0 0"! 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 10 10 0 0 0 10 1 1 0

- o o o o o o o i -

0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0

+

o o o o o o o i 0 0 0

1 0 0 0 0 0 . O O O 0 0 0 0 0 3A + 3 0 0

0 0 0 0 3A + 2 0 0 0

0 0 0 0 1 0 0 0 0 0 1 0 O O O O I 0 0 0 0 0 0 3A + 3 0

0 0 0 .

0 " 0 0 0 0 0 0 A + 4.

The above example implies that as in the case of G^n\ this generator G' L,n) can be divided into the following terms : G (L,n) =

XG{L,n)

+

£,(£,„)

+

XQ(L,n)

+

Q(L,n)

_

Q(L,n)

As for the additional terms, G*f'n) is infected term at J?(4 ) a n d G\r at rj(l\ ) . Note that for any n > 1, G(L,n)

=

G(n)

an(j

G(L,n)

_

is recovery term

^n)

So we obtain the following theorem similar to Theorem 2.2.1.

Theorem 3.5.5. For n > 1, Q(L,n+l)

_ ,$j(. L ' n + 1 ) J. Q(L,n+l)

, XQ<.L,n+l)

+

g(L,n-\-l)

_ g(L,n+$ ^

Phase Transitions of IPSs

40

wiere g(L,n+l)

_

r(L,n+l)

_

"ei

.

Q(L,n+l)

_

G (i,n+1) =

-,(£

;(L,n) / ( 2 " ) I T 4 ^ " ) 5(2") G*1""'] 1.0(2") 0(2 0(2") 0(2") 1 [0(2") +

Bfe-J0(2") J '

<#">] [0(2") 0(2") L0(2") 0(2") J L^R&T ^D,( T2 .2« ) J

(t,n)

re!

0(2")
,(L,n)

^

Q

GKd

n+1) _

p(L,n)

TtfV,"' 0(2")

L n)

.0(2" 0(2")

G[ '

p(L,n) (t,n)

« i,12

„(L,n) _ ei,21 (L, )

R ei,22n

p(L,n) —

0(2") <2'2n)

0(2"-') 0(2"-') 0(2"-') 0(2"-') 0(2"-') 0(2"-') 0(2""') 7(2"-')

-7(2""') 0(2"-') J 0(2"-') 7(2"-') 7(2"-') 0(2"- 1 ) 0(2"-') 0(2"- J )

0 0

0 0

0 0

0 0

.. ..

0 0

0 0

0 0 0

0 0 1 0 0 0

0 0 1

.. .. ..

0 0 0

0 0 0

0

0

0

..

0

1

0

R(L,n) _ , [ 0 ( 2 " - ' )

0(2"-')-

■*•" " [0(2"-') 27(2"-')J '

? (Z,,n) l R)T£' d,22 = 7(2"),

^(£,1)

0 0

0

0 0

ri

i

0

0'

1 oj '

_ 3A + 1 0 0 A+2 By using the above mentioned method, Zeizold and Grillenberger8 gave the best lower bound : 1.539 < KGL1-1' =

G[i» = 0 l ] 0 lj'

a'L,l)

^

Basic Contact Process I

41

3.6. A p p l i c a t i o n t o t h e Threshold Contact P r o c e s s In this section we apply the Liggett and Ziezold-Grillenberger methods to obtain lower bounds on the critical value of the one-dimensional threshold contact process. First we review this process. 3.6.1.

Introduction

The one-dimensional threshold contact process is a continuous-time Markov process on state space 7? 6 { 0 , 1 } Z . The definition of this process is given by the following formal generator : flr/(ij)= £ c(x,T,)[f(r,*) - / ( „ ) ] , with flip rates c(x,n) = i)(x) + A(l - 77(a;))min[77(x - 1) + v(x + 1), 1], where A is a nonnegative constant and if denotes if(y) = 7/(3/) for y ^ x and rf(x) = 1 Tj(x). The one-dimensional threshold contact process is also a simple model of an infectious disease. The value 0 is regarded as a healthy individual, while 1 is regarded as an infected one. Each healthy individual at site 1 becomes infected at rate A if at least one of its neighbors on {x — X,x + 1} is infected. On the other hand each infected individual recovers at rate 1. Let v\ be the stationary distribution of this process starting from 77 = 1. The upper invariant measure v\ is well defined, since the threshold contact process is attractive. Define the density of infected individuals by p(A) = v\(r\: T](x) = 1) which is independent of x £ Z. Note that p(A) is a nondecreasing function of A. The critical value is defined as AT,c = sup{A > 0 : p(A) = 0}. More details can be found in Durrett and Griffeath 20 and Liggett. 3 As for lower bounds on the critical value, for example, Katori and Konno 2 1 showed AL,C = \

* 1-225

by using submodularity of survival probabilities, that is, Griffeath method. The same lower bound will be obtained in subsection 3.6.2 by the Liggett 3 and Ziezold-Grillenberger 8 methods. Moreover their methods can give a better lower bound, 1.337, so we have A ^ « 1.225 < A ^ » 1.337. In Section 3.6.4 we will present an another type of lower bounds based on their methods. Then the best lower bound \jc is given as the root of

Phase Transitions of IPSs

42

This value is estimated as AJ.C « 1.310. (See Proposition 3.6.9.) Note that this lower bound 1.310 is not better than the above second approximation 1.337 by the Liggett and Ziezold-Grillenberger methods. On the other hand, Liggett 22 and Katori and Konno 2 1 gave the upper bound on the critical value by the method based on the Holley-Liggett 9 argument : Xv,c = the largest root of A3 - A2 - 3A + 1 = 0 ss 2.170. By computer simulations or (non-rigorous) numerical methods, the critical value is estimated as, ATiC ss 1.742. (See Dickman and Burschka, 23 Dickman and Jensen, 24 Ferreira and Mendiratta. 2 5 ) 3.6.2.

Liggett and Ziezold-Grillenberger

methods revisited

In this section we think of the methods by Liggett and Ziezold and Grillenberger which give the best lower bound on critical value of the one-dimensional basic contact process as we saw in Section 3.5. Define an initial configuration n+ by 17(3;) = 1 for x > 0 and TJ(X) = 0 for x < 0. Let l(rf) be the leftmost position of an infected individual in 77(5^ So). That is, l(n) = —00 if r){x) = 1 for infinity many x < 0 and l(rj) = y if n(y) = 1 and 77(1) = 0 for any re < y. Let S = {0,1} Z and Si = {77 € S \ {S0} : l(r)) f - 0 0 } . For n = 0 , 1 , . . . , define the Markov process rytn' on state space Si starting from 77+, i.e., rjg = 77+, with the following formal generator :

4

n )

m=

£

c(x,n)[fm-m]

+

c(l(v)-l,ri)[f(r/)--f(r,)],

where 77 £ Si and „>rx)

=

/ 1. \ n(x)

for

* £ (KV) - 1, Ki) + n}, otherwise.

So this process 77J"' behaves like the threshold contact process on (-oo,/(77^ n ')-(-7j]. On the other hand 77((n)(i) = 1 on [l(n\n)) + n+ l,oo), i.e., all individuals are always infected on this region. Let r/t = r^ n ) be the moment t at which T]\n' changes for the fc-th time with T0 = 0 and fk = rk - rk_! for k - 1 , 2 , . . . . Note that {£* : k = 1,2,...} is a sequence of i.i.d. random variables. Now we can define a mean displacement 7n^n)(A) 6 [-l,7i] and an edge speed a l T (A) G (-00,00] of 77J"' in the following way :

mlTn)(\)=hm^E(l(r,^)), 4 n ) ( A ) = ( limi£(/(77("»)). For n = 0 , 1 , . . . , a critical value for 77'"' can be defined by 4> =

su

P i > > 0 : mip\\) n

> 0}

= sup{A > 0 : a< )(A) > 0}.

Basic Contact Process I

43

Let 77(°° be the thereshold contact process starting from 77+ and Ac for this process. As the proof of Ziezold and GriUenberger, we obtain

be a critical value

Lemma 3.6.1. (1)

A T ,c = A<°°>.

(2)

A^'/'Ac

as

n/00.

Moreover Ziezold and GriUenberger presented a useful method for computation of Ay' as foUows. For n = 1,2,... and k = 1,2,..., we let j(») _ ;(n) _ ;(n)

a

k

—'rk

'TJ.,!

where l\n) = l{r)\n)). For n > 1, define

4 n ) = w t : ) + i).---.'?('ir)+"))!

n(n) = (4 n) .4 n) )-

{X^ : k = 1,2,...} and {Y^ : k = 1,2,...} are Markov chains on state spaces S<"> = {0,1}^ "> and ,?(">*> = {-1,0,1,... ,n + 1} x S<"\ respectively. Let p™ and pf be their transition probabilities and //y a n ( l Mr be their stationary distributions respectively. Then the argument in Ziezold and GriUenberger gives Lemma 3.6.2. For n > 1,

m ( T B) (A)=

53 (j,u)6S<».*)

j ^

M(")(')l4",*>((0.-).CJ.«)).

s6S(»)

Forra= 3 , 4 , . . . , m = 1,2,.. . , n - 1 and 5 = «is 2 ■■ -s„ € S^"', we let L(s) L(s) L(s) RT(s)

=ra+ 1, if * = 0...00, = n, if s = 0 . . . 0 1 , = m, if s = 0 . . . 0 1 s m + i . . . sn, = cT(s) + A + N(s) + 1,

where CT(S) is the total infection rate of configuration, l s i . . . snl and N(s) is the number of l's for sis2 ...sn with 1(0) = 2,

£(1) = 1,

1(00) = 3,

£(01) = 2,

flT(00) = 3A + l,

JEr(O) = 2A + 1, RT(1) = A + 2, 1(10) = 1(11) = 1,

J?T(01) = .Rr(lO) = 2 A + 2>

RT(U) = \ + 3.

44

Phase JVanjitions of IPSs

For example, in the case of n = 3, the above definition gives 1(000) = 4, 1(001) = 3, 1(010) = 1(011) = 2,1(100) = £(101) = £(110) = £(111) = 1, iJ T (000) = 3A + 1, % ( 0 0 1 ) = 3A + 2, % ( 0 1 0 ) = 3A + 2, £y(011) = 2A + 3, .RT(IOO) = 3A + 2, JJr(lOl) = ftr(110) = 2A + 3, E r ( l l l ) = A + 4. So Lemma 3.6.2 can be rewritten as follows. In fact, it is more useful to compute m y ' ( A ) . L e m m a 3.6.3. For n > 1,

vv ^- X-L(s) A - L(s)

(TI)„> mip\x) =

R

£» ^

(„). (n)

.

RT(S)

Next we introduce the method by Liggett 22 This method is slightly different from the Ziezold and Grillenberger method. However both methods give the same lower bounds A 1 ^. For n > 1, define

mn)),---,vu\n)+n)).

cr =

Then Q is the Markov process on state space 5 ' " ' with generator Gy which is given by f l y ' . Let j£p be a stationary distribution of C*n'. So the argument in Liggett, 22 pp.288-289, gives the next result. L e m m a 3.6.4.

4 n) W = - J2 (A - L(s))ftp\s). ses<">

Note that Lemmas 3.6.3 and 3.6.4 imply Ay' is the positive root of

£ since p y

L(s)7#)(s)-\ = 0,

is a probability measure.

In the rest of this section we compute m ( y'(A), m^\x) and c$\X), Qy2)(A) by Lemmas 3.6.3 and 3.6.4. So we get A^j, and A^' c . We sometimes omit a subscript T. {X^ : k > 1} is the Markov chain on state space {0,1} with transition probability : p(» =

pW(o,o)

p^Ho.i)

[ o

ii

*M(1,0)

P(1'(1,1)

L A+2

A+2 J

So the stationary distibution of X ' 1 ' is

413 = [^\o),^\i)]

=

A + 2

LA + 3'A + 3

Basic Contact Process I

45

Lemma 3.6.3 gives

^-<°»€S^mfmH^m^ -w»fff?l

m^'(A) = - (ANoting that i(0) = 2,

£(1) = 1,

RT(Q) = 2A + 1 ,

so we have .(D m^'(A) =

Similarly (t ator :

£ T ( l ) = A + 2,

2A2-3 (A + 3)(2A + 1)'

is the continuous-time Markov process on state space {0,1} with gener al)

S«(0,0)

^(1,0)

^(0,1)

gW(l,l)

-(2A + 1) 2A + 1 1 -1

Therefore the stationary distibution of fW is 2A+1 /4 1) = [J41)(O),T41)(I)] = 2A + 2'2A + 2 Lemma 3.6.4 implies a « ( A ) = -[(A - I(0))4 X >(0) + (A - £(1))J#>(1)], so

41J(A) =

4\*)

2A2-3 2A + 2"

A« = f

„ 1.225.

In any case we get

(2) (2)/ As in n = 1, next we compute TOy (A) and a^,'(A) by Lemmas 3.6.3 and 3.6.4. So we have Aj.'c as follows. 12A4 + 21A3 - 3A2 - 42A - 27 ( 2),,, T v ' 12A" + 84A3 + 150A2 + 107A + 27' 12A4 + 21 A3 - 3A2 - 42A - 27 2 4-v\_>(A) '111= — J 12A3 + 33A2 + 36A + 14 In any case we get 1

V

'

\%\ = sup{A > 0 : 12A4 + 21A3 - 3A2 - 42A - 27 < 0} 1.337.

46

Phase Transitions of IPSs

3.6.3. General case From now on we restrict attention to a^ for general n. We think of a suitable division of 2" x 2" matrix G^'. In fact, for n = 1,2,3, we have

W

(jji r GT

00 1' J ° oJ0iUI" + [l1

0 — A -X[0 0

+

l °0"U+AA[ °0 1 O0 jJ + A [ O0 lj 1

0

1'

1 ]+ 1-- [[' 22AA00++11 + |.0\° 0 l1_ j

0") r00 00 0 00 -0"00 11 1L On 1 10 0 00 0 0 00 00 1 1 =A 0 00 00 01 1 ++ 1 10 0 00 00 .0 1i i1 o. 0J ..o 0 0o o1 oJ 0J Lo 0 00 1L 001 r0o o0 o 0i -IT 0 00 11 00 0 00 00 01 1_ +A + + 0 00 0 1 1 1 1- 0 11 00 0 0 .0 0 0) 1J Lo .0 0 0 lJ 1. 1.

+

0 A + 22J )' A

°

r(2) _ . T _

Gf

00 ■ 3A+ 00 00 T3A + 11 2A + 2 0 0 0 2A + 2 0 0 0 2A » L 0 0 0 A ++ 33..

0 -i ro0 0o 0o 0o o0 o0 o 0 o 0""0o i1 o 0 o 0 i 1 o0 o0 o 0 1 1 0 0 0 0 00 00 00 00 0 0 0 1 1 00 0 0 0 0 0 00 11 0 0 1 1 0 0 0 0 00 00 00 00 0 0 00 1 10 0 00 0 0 00 11 0 „(3) _ . 00 00 00 00 0 00 00 11 0 01 11 01 00 0 0 0 0 0 00 A Gf = T 0 0 0 00 " 0 00 00 00 00 01 1 11 00 ++ 1 1 00 00 00 00 0 0 0 0 0 1 1 0 0 00 11 0 0 00 0 0 0 0 00 00 00 0 0 0 0 1 1 0 0 0 0 0 1 10 0 110 0 00 0 0 c 00 00 00 00 0 0 0 01 1 0. .0 0 0 0 0J 0. L0 .0 00 00 1 1 00 11 1 1 0. .0 00 0 0 00 0 0 -0 •o 0 0 00 0 . 0 +A + 00 00 00 .0 .0

0o

oo oo i i oo oo o -oi

00 00 00 0 0 00 00 (0

00 00 00 00 00 00 00

00 00 00 00 00 00 00

ro 0

0 o 0o 0o o0

o0 o 0i 1-1

0 00 0 0 0 00 00 00 00 11 00 0 0 0 0 0 00 0 0 0 110 0 00 00 00 1 1 00 0 0 00 1 1 00 0 0 00 0 0 0 00 00 00 00 00 00 1 10 0 + 0 110 0 00 00 00 00 0 00 0 0 0 110 0 00 00 00 0 0 1 10 0 0 00 0 0 0 00 00 11 00 00 0 0 0 0 1 1 1. .0 0 0 00 00 00 0 0 0 0 00 0 0 0 0 lJ Lo

+s

11 00 11 00 00 00 1 1.

00 ■ ■ "3A 00 00 3A ++ 11 0 0 00 00 00 00 3A 3A ++ 22 0 0 00 00 00 00 00 00 00 3A 3A ++22 0 0 00 00 00 00 00 00 00 2A 2A ++ 33 00 00 00 00 00 00 00 00 3A 3A ++ 22 0 0 00 00 •' 00 00 00 00 00 2A ++ 33 2A 00 00 00 00 00 00 00 00 2A ++ 3 3 00 2A .. 0 0 0 00 00 00 0 00 AA+4. + 4 0 0

So as in the case of the basic contact process, this generator Gy can be divided into the following terms : r («) -

AU

T,i +

U

T,r +

A(j

T,ei + "T.er " - G

(n)

Basic Contact Process I

47

where G^} and Gy^ are infection term and recovery term on (jj(J((n) + 1),..., r](l\n) + n)), respectively, G^ei is infection term at 7/(Z^n) - 1), and G ^ r is recovery term at T)(l\n^). Then from the definition of G'"' we obtain the following result. Theorem 3.6.5. For n > 2,

G £ + 1 ) = A G ^ 1 ' + 4 T * + Xdp+V + dg? -

,-.("+1) ^T,d

'

where Tl

*

(n+1)

"**

' G(r"| 0(2") 1 L 0(2") 0(2") |_0(2") L0(2") 0(2") J ') 0(2") G^] G1T} J . "0(2") 0(2")] 0(2")" 0(2") _ [0(2") r[0(2") 4G
T T rr

''

)_ _

Tr>(") [G G^ T r

0(2") "

5 R{n) 4:L,22J' ■"■T,ei,22 .

0(2") 0(2")'

~[/(2") £>]' [7(2") GG(«)

T,r . __ [0(2") [0(2") dpU (n+i) ^n) rG(n+l) F<">, T erer T [0(2") 7(2"). <' '~ [0(2") 7(2")J ' U

0(2") ' 0(2")' -("+1) =- [ «^T,d$ 0(2")] 1 n 0(2") ♦ [ 4"■T,d,l\ » 0(2") ■R<"> ' ~|_0(2") 0(2") G $ ] [0(2") 4T,d,22 J i M J. '

C rT T (d J

41. 0(2"~ 0(2"- )

22

2

rO(2"" •0(2" - )) 0(2" „(n) 0(2"--22)) 0(2"- 2 ) 2 u^ T . i . l l — T,i,u ~ 0(2" o(2"--22)) 0(2"- ) 22 0(2"- 2 ) .0(2" .0(2"-- )) 0(2""

4:1,12=^(2"), 4ti2= / (2"),

"0(2""-1) __ [0(2"-!) n 1 1 T,ei,21 - [0(2 L 0 ( 2 "-" )- ) 0(2""!) )(2"" - 1 ) C n) B ((n U 1 T,ei,22 ~ [/(2") 1(2"- M

„<»> p(n) n

n

2

-7(2"~ -7(2"- ) 0(2"- 2 ) 0(2"- 2 ) 0(2"- 2 ) 0(2""

0(2"- 2 )" )l 2 0(2"0(2"" 2)) 0(2"" 2 ) ,' 2 0(2"~ 0(2"- 2)_ ).

//(2"" ( 2 " - 1 )" )'

x 0(2"0(2"- X)J )J ''

0(2"0(2-1)" 0(2"0(2" _ 1l )J ) '

0' "0 0 0 0 . . .. 0 0' "0 0 0 0 0 . . .. 0 0 („)_ 7^(") =

0 0 0 0 ..... 0 0 J 0 1 0 0 ..... 0 0 ' 0 0 0 1 ... 0 0 .00 0 0 0 ....• 0 1.

48

Phase Transitions of IPSs

r■/ (7(2"2 " - 22) 2 -22 ) 0(2"" D(") _ AA7(2") »(-) u w °( ""- 2)) A Ai -"-T.d.ll — ^T,d,ii - - ^ > ~ 00( (22" '" -2) 22 ..00((22"""-- )

2 2 0(2"" 0 ( 2 " " 22 ) 0(2"" 0 ( 2" " ) 0 ( 2 " " ) 2 2 0(2n_2) 0(2"-2) 0 ( 22 " _2 - ) 0 (22 "-2) °( " °( "~ 22) 2 J 22) " ) 0 00((22""-- )) 00((22 " - ) 0(( 2 " -~ 2)) ' 1 22 22 2 0(2"" 0 ( 2 " " ) 0(2"" 0 ( 2 - " ) 00 ( 22 "" " ) ..

"■T,d,22 ^T,i,22 = ~ " "* * ) ) '

0 G ( 1 ) -- 0 ^ ^ V [o __ fo -ei_[o

o

rW

G(l)

T

o

1' w r _ 0 0' ) 0Oj '> r^ rT< 0 ' . r ~ - [ 11 oj 0 1 1] Wi) (i) _. . [0 1] l j ,' GrT,er > e r ~_ [ o0 l1 j '

mi ) __ [ 2' 2AA++l1

r ( ur

r,
00

0 AA++22 '

The interesting thing is that G^"' and G ^ have a simple fractal structure. On the other hand, G ^ ; and G^\T have more complicated forms. For n > 1, we let 7r( n ) = _ XGip} \ r - ( " ) i+ r>(") G$£ - /->(») Cry

Then we can rewrite the above result in the following simple form. Corollary 3.6.6. For n > 2, W n + i ) _ ["GST"'' G ^ 5 - A7(2") T "[ 7(2") 7(2") where

■7(2"~2 ) •7(2"~ 00(2"" ( 2 " " 22 ) „ (n„)) _ T ( 2 " " 22) ~~ 00(2"" ..0(2"" 0 ( 2 " " 22 )

4 =

AJ(2") A7(2")

n) ) Gi G ( T n-7(2")J -7(2")

00(2"" ( 2 " " 22 ) 2 0( 2" - 2) 0 ( 2 "" - 22 ) 0 ( 2 " - 22 )

+A

- 7 ( 2 " - 22 ) 00(2"~ ( 2 " - 22) 0 ( 2 "" - 22 ) 0 ( 2 " - 22 )

[' ij(") fl<") 0(2") [0(2")

0 ( 2 " ) ""

0(2") 0(2")J' 5

0(2"-2)" 0 < 2 " - 22) 0<2"~ 0 < 2 " - 22 ) '' 0<2"2 0<2"0 ( 2 " " 2)_ ).

and ^4"+« (n+l) = _ G ^ n ++11)) +, iAG£+« . ( n + l ) , ^.(n+1) UT - Cr£T + Ar L r T e ; + G^ O r Ti+i) . er

3.6.4. Lower bounds Using the results in the previous subsection we give new type of lower bounds on the one-dimensional threshold contact process. As for this method, the threshold contact process is more applicable than the basic contact process. Because the general form of matrix G' n ) of the threshold contact process is simpler than that of the basic contact process. To obtain lower bounds, we modify G'"' in the following way. For n > 1, let Wn+1) ("+i) G

=

l (n+1) +1 g(»+ G (n+1)) +, xG AG(n+1) + G<" G" >

49

Basic Contact Process I

where G £ } - A/(2") A/(2n) 1(2") ^'-1(2") ■o 0 ... 0 1 p{n+\) _ 0 0 ... 0 1

K»+i)

(n+1) _ m,ei

G

m

u

=

m,er

.0 0 ... 0 1 '-(2A + 1) A 1 - ( A + 2). I

G (1) • = G<J> =

0 1 0 1

^m.ei Sm( l Gm,er mean that any configuration flips to all occupied configuration at rate 1. So this gives the lower bounds on the critical value. Let Cmt De a continuous-time Markov process defined by generator Ghx and a „ (A) be an edge speed of this process. Define a critical value by AW = sup{A > 0 : a%\\)

> 0}.

So the definition of Gm gives Lemma 3.6.7. (1)

(2)

^
for

A("UAW

re

^!-

for

n>l.

Note that X£,c is a lower bound on Ac . Let /Zv be the stationary distribution of Cm,tFor n = 1,2,3, we compute 7*m as follows :

^ = [^»(0),^(1)]=[^,|^ P£P = B« ) (00),pg)(01),pg)(10),/l2>(ll)] = ^y[2,3A + 1,3A + 1,2(3A2 + 3A + 1)] Pi? = BlS)(000),7l2>(001),?igJ(010),Jig>(OU), ^)(100),/4 3 )(101),^ ) (110),^'(111)] = -^3j-[3,4A + 1,4A + 1,6A2 + 4A + 1, 4A + 1,6A2 + 4A + 1,6A2 + 4A + 1,3(2A + 1)(2A2 + 2A + 1)],

Phase Transitions of IPSs

50

where Z<2> = 6(A + l ) 2 and Z ( 3 ) = 12(A + l ) 3 . Therefore Lemma 3.4.4 gives m

m

J U A Q

11

\

) -= i 1 +I 22(A ( A+ T1 )l ) - A* ' « ^» W C)f\\

i

i

i

\

+ < « -J (A)-H W = 1 + 2 (2(A AT 3 ( +A1)2 W "AA>' + 1T ) j1 3(A 131/n . 1 i

«4; W -

l

^

i

\

I

+ 2(A + 1) ' 3(A + l) 2 ' 4(A + 1)3

So we get A # c » 1.225, AJ& a 1.283 and A $ e w 1.302. In general, from the definition of this process, we obtain L e m m a 3.6.8. n a ( n ) fA) = V

m (1)

w

m W

1

7

rr-A

for n > 1.

^ ( * + l)(A + l)*

^ w=g ( f c + D ( A + D * -A - -(A+1)los to) -A-

So combining Lemma 3.6.8 with Lemmas 3.6.1 and 3.6.7, we obtain the following result. Proposition 3.6.9. Let

AJ,c = sup{A>0:-log( x A_)> x A_} fa 1.310. Tien we nave ^T,c — AT,C3.7. S u m m a r y Finally we summarize the results on lower bounds of the critical value of the basic contact process in one dimension which appeared in this chapter as follows : Method H-FKG Markov L-Z-G Griffeath

Lower Bounds on Critical 1 2 0.500 0.707 0.500 1.000 1.000 1.180 1.000 1.180

Value 3 0.859 1.180 1.280 1.280

4 0.961

1.343

-

Basic Contact Process I

51

where n(= 1,2,3,4) in the second line on this table means the n-th bound, H-FKG means the Harris-FKG method, Markov means the Markov extension method, Harris means the Harris lemma method, L-Z-G means the Liggett and Ziezold-Grillenberger methods (only in this case, ra-th bound means (n — l)-th one). The value of the fourth bound 1.343 appeared on page 7 in Zeizolod and Grillenberger. 8 Griffeath means the GrifFeath method which can be found in Section 7 of Griffeath, 26 or pages 166-167 in Liggett. 3 Note that the third bound 1.280 in Markov is given by 1.2800 » sup{A > 0 : 15A4 + 21A3 - 10A2 - 39A - 18 < 0}, which can be found on page 884 in Katori and Konno. 27 So this value is not the same as that of third bound in L-Z-G : 1.2796 « sup{A > 0 : 27A4 + 26A3 - 18A2 - 55A - 27 < 0}. The critical value on the basic contact process in one dimension is estimated as Ac sa 1.649.

References 1. T.E.Harris, Contact interactions on a lattice. Ann.Probab. 2(1974)969-988. 2. D.Griffeath, Additive and Cancellative Interactive Particle Systems (Springer Lecture Notes in Mathematics, Vol.724, 1979). 3. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York,1985). 4. R.Durrett, Lecture Notes on Particle Systems and Percolations (Wadsworth, Inc., California, 1988). 5. R.Durrett, Ten Lectures on Particle Systems (St. Flour Lecture Notes, 1993) Springer Lecture Notes in Mathematics, to appear. 6. J.Marro and R.Dickman, Nonequilibrium Phase Transitions and Critical Phenomena (Cambridge University Press, 1995). 7. C.Bezuidenhout and G.Grimmett, The critical contact process dies out. Ann.Probab. 18 (1990) 1462-1482. 8. H.Ziezold and C.Grillenberger, On the critical infection rate of the one-dimensional basic contact process: numerical result.J.Appl.Probab.25(l988)l-lb. 9. R.Holley and T.M.Liggett, The survival of contact processes. Ann. Probab.6( 1978) 198206. 10. T.M.Liggett, Improved upper bounds for the contact process critical value. Ann. Probab., to appear (1994). 11. R.C.Brower, M.A.Furman and M.Moshe, Critical exponents for the reggeon quantum spin model. Phys. Lett. B76(1978)213-219.

52

Phase Dransitiona of IPSs

. 12. N.Konno and M.Katori, Application of the CAM based on a new decoupling procedure of correlation functions in the one-dimensional contact process. J.Phys.Soc.Jpn. 50 (1990) 1581-1592. 13. N.Konno and M.Katori, Applications of the Harris-FKG inequality to upper bounds for order parameters in the contact processes. 7.Pftj/s.5oc.Jpn.60(1991)430-434. 14. M.Katori and N.Konno, An upper bound for survival probability of infected region in the contact processes. J.Phys.Soc.Jpn.60(1991)95-99. 15. M.Katori and N.Konno, Three-point Markov extension and an improved upper bound for survival probability of the one-dimensional contact process. J.Phys.Soc.Jpn. 60 (1991) 418-429. 16. M.Katori and N.Konno, Upper bounds for survival probability of the contact process. J.5*o*.F/!j/s.63(1991)115-130. 17. T.B.Harris, On a class of set-valued Markov processes. Ann. Pmbab. 4(1976)175-194. 18. A.G.Schlijper, On some variational approximations in two-dimensional classical lattice systems. J.Stat.Phys. 40(1985)1-27. 19. R.Durrett, Probability : Theory and Examples (Wadsworth,Inc, California, 1991). 20. R.Durrett and D.Griffeath, Supercritical contact process on Z . Ann. Probab.ll(1983)l15. 21. M.Katori and N.Konno, Bounds for the critical line of the ^-contact process with 1 < 6 < 2. J.Phys.A.26(1993)l-18. 22. T.M.Liggett, The periodic threshold contact process.Random Walks, Brownian Motion, and Interacting Particle Systems. (Eds. R.Durrett and H.Kesten) Birkhauser, Boston. (1991) 339-358. 23. R.Dickman and M.A.Burschka, Nonequilibrium critical poisoning in a single-species model. Phys. iett.l27A(1988)132-137. 24. R.Dickman and I.Jensen, Time-dependent perturbation theory for nonequilibrium lat tice models. Phys.Rev.Lett. 67(1991)2391-2394. 25. A.L.C.Ferreira and S.K.Mendiratta, Mean-field approximation with coherent anomaly method for a non-equilibrium model. X P A J / S . . 4 . 2 6 ( 1 9 9 3 ) L 1 4 5 - L 1 5 0 . 26. D.Griffeath, Ergodic theorems for graph interactions. Adv.Appl.Probab. 7(1975)179194. 27. M.Katori and N.Konno, Correlation ineqalities and lower bounds for the critical value Ac of contact processes. XP/»j/s.5oc.Jpn.59(1990)877-887.

Basic Contact Process II

53

CHAPTER 4 T H E B A S I C C O N T A C T P R O C E S S II

4.1. Introduction In this chapter we consider upper bounds on the critical value of the basic contact process in one dimension. Comparing with lower bonds, it is hard to get good estimates on upper bounds. In fact, the methods for this are limited. Following Liggett (1985) 1 in Open Problem 1 of Chapter VI The Contact Process, "The real challenge is to improve the upper bound Ac < 2. So far, no method is available which would accomplish this." Moreover Durrett (1988) 2 writes "The story on upper bounds is short and sweet." 4.2. Contour M e t h o d In this section we will restrict attention to discrete-time models £„ C Z. In particular, we think of the oriented bond percolation and the oriented site percolation. This section is based on Section 5a in Durrett. 2 By the aid of the contour method, we will give an upper bound on the critical value of the one-dimensional basic contact process in the next section. 4.2.1. Oriented bond percolation Let V = {(TO, n) 6 Z 2 : m + n is even }. If (m,re) £ V, there is an oriented bond from (ra,n) to (m + 1, n + 1) and from (m,re) to (m - l , n + 1). Each bond is independently open with probability p and closed with probability 1 — p. We write x ^ y if there is an open path from x to y. That is, there is a sequence xn = x,..., xn = y of points in V so that for each 1 < i < n the bond from Xj_i to Xi is

Phase Transitions of IPSs

54

open. Let C(o,o) = { z e V : ( 0 , 0 ) ^ z } = the open cluster containing (0,0), £ =

{meZ:(0,0)^(m,n)£V}.

Moreover the critical probability is denned by pc = mf{p G [0,1] : P(£

± 4> for aU n) > 0}.

Note that P(£

$ for all n) = JBm P(£

? 4>)-

For n > 0, define feud n = {(m, n) G V : m G Z } . Then the number of the paths from (0,0) to level n is 2". So P(£J ^ 0) = P ( there is an open path from (0,0) to level n ) < 2 > " . I f p < 1/2 then lim P(C° ^ 0 ) = 0.

n—*oo

Therefore we get easily the lower bound on the critical probability as follows. T h e o r e m 4.2.1. Pc >

1

On the other hand, as for the upper bound, we have T h e o r e m 4.2.2.

Pc 0, we let AN N

C

= {0,-2,-4,...,-2iV}, = {z eV

: there is a y G A so that (j/,0) —»■ z },

D = {(a,6)GR2:|a| + |6|
55

Basic Contact Process II

the segment from (0, —1) to (1,0) is oriented in the direction indicated. We call T contour associated with CN. Note that the minimum length of contour is 2JV + 4. So E( the number of contours ) = Y^ P(T = I\) r,oo

= £

£

n=2AT+4ri:|ri|=r.

p(r = r,.).

Observe that (i) if |r*s[ = n then P ( r = Tt) < (1 - p)"/ 2 , (ii) the number of contours with the length n < 3 n _ 1 . So we have E{ the number of contours ) <

3"_1(1-p)"/2 = -

£ n=2N+4

^

{3(l-p)1/2}

n=2N+4

,1{3(1-P)^}2W - 3 1- 3 ^ 1 ^ ' The following relation is important for the contour method : P(\CN\ < 00) = P{T exists )

<£p(r = ro = E( the number of contours ). On the other hand

N

oo) < £ P(\CN\ = 00)

i>(|C((__22m oo) P(|C m , 0 )| ) | = 00)

m=0

= (7V + l)P(|C (0 , 0) | = oo), = (7V + l)P(|C (0 , 0) | = oo),

where So

C(--2m,0) = {* ev •(- -2m , 0 ) - *}. C(_2m,o) = {* 6 V : (-2m,0) -> x}.

P(iq )l ==oo)00)>> j y J^P(\C P(\C0(Q,0,o)\ q n ^ d N^\l= =oo)°°) =

>

FTT { 1 _ P ( | c N | < 0 0 ) } j l — £( the number of contours )}

- N+ l l

1 > - N JV ++ 1 I

v

1(3VT

•{'■ 3

11 -- 33VV1T- P^ /J '

Phase Transitions of IPSs

56

Let pc

denote the positive root of { 3i^^ / 1i --p] -P)

■.2N+4

{'

The definition implies that if p > Pc

+ 90-

+ 9 I / 1 -P- P -- 33 == 00..

then

-P(|C(0,o)l = oo) > 0, N)

so pi

is the upper bound on pc. Then p l c N ) -* 8/9 as N -> oo, so the proof is complete.

The above proof gives the following corollary. Corollary 4.2.3. For n > 0, we let p ( c "' e (0,1) be the positive root of (

i

{30Tien we iave

2 2n — +4 -P\ \1 ' a+4

[3^/1-?}

Then we have Pc

(1) (2)

p(c"+1)

- 3 == 0. = 0.

< „(»)

Pc 0,

p(cn) -+ -

(3)

i,

-P+ + 99^0/ 1- - ^ - 3

as n - oo.

4.2.2. On'enied siie percolation Let V = {(m, TI) 6 Z 2 : m -f n is even }. The point (TO, n) G V is also called the site. Each site is independently open with probability p and closed with probability 1 — p. We write x —► y if there is an open path from x to j / . That is, there is a sequence XQ = x,... ,xn = y of points in V so that all site Xi is open and for 1 < i < n, Xi is :tj_i + (1,1) or i i _ i + (—1,1). As in the oriented bond percolation we let C(o,o) = { z e V : ( 0 , 0 ) ^ x } = the open cluster containing (0,0), -► ( m , r e ) £ V}, £ = {™ £ Z : (0,0) {meZ:(0,0)^(m,n)eV}, C

inf{p ee [0,1] :■■P(C pPc P(C° ?/ *4> for forall alln)n)>>0}. 0}. c = inf{p Similar arguments imply the following results. (See more details can be found in pp.85-86 of DurTett. 2 )

Basic Contact Process II

57

Theorem 4.2.4. (1)

Pc>\.

(2)

Pc<

| 2 » 0.988.

Moreover for n > 0, we let p l n ) g (0,1) be the positive root of

{3(l-p) 1 / 4 } 2 " +4 +9(l- P )^_ 3 = 0. Then we have Pc
(3) (4)

p ■ —

for all

n>0.

as n -» oo.

4.3. Comparison M e t h o d The comparison method is a very powerful method for showing survival of the process and giving upper bounds on critical value. So this method is applicable to many situations, for example, see Section 11a Renormalized Bond Construction in Durrett 2 and Section 4 A Comparison Theorem in Durrett. 3 Generally these upper bounds are not so good, however in some cases this method gives sharp results. For example, see Chapter 6 Diffusive Contact Processes I and Chapter 8 Long Range Contact Processes in this book. More details can be found in Durrett. 3 In this section we think of the one-dimensional basic contact process constructed from graphical representation. The key idea to comparison method by Harris 4 is the following : if A is small and A is large, then the basic contact process observed at times 0 , A , 2 A , . . . dominates a supercritical oriented site percolation. In general, to show the survival for a given complicated process ft, we embed it in a suitable relatively simple process Q so that if Ct survives then so does f(. So, roughly speaking, the flow diagram of comparison method is as follows : complicated process & —» simple process Ct -* complicated process £ ( . In this case, basic contact process & —* oriented site percolation £ n —► basic contact process £j.

Phase Transitions of IPSs

58

To do this, we define a site (TO, n) e V is open by (i) there is no death at m between time (n - 1)A and (re + 1)A, (ii) there is a birth from m -» m + 1 and TO ->TO- 1 between reA and (n + 1)A. Note that P ( no death in [0,2A] ) = e~ 2 A , P ( birth from ra -+ m + 1 in [0, A) ) = 1 - e - * A , P ( birth from m — m - 1 in [0, A) ) = 1 - e~ AA So

P ( (TO, n) is closed ) = 1 - P ( (m, re) is open ) = 1 - P ( (i) and (ii) ) < l _ e - 2 A + 2e ->A_ The definition of pc implies that if P ( (m, re) is open ) > pc then oriented site model fn percolates. So the one-dimensional basic contact process f( survives. That is, P ( (m,re) is open ) > pc -> P{£

£ <j> for all re) > 0

~> -P(f° 5 ^ for all re) > 0. Therefore the above observations yield that if P ( (m, re) is closed ) < 1 - e " 2 A + 2e~XA < 1 - p c , then P(f° ± for all n) > 0. To get upper bound we see that 1 _ e -2A is equivalent to

+ 2e -AA

< l

_

p c

-M- -*y -2A _

A> _2A

2

where e > p c . So we obtain the upper bound on the basic contact process using the critical probability on the oriented site percolation as follows. T h e o r e m 4.3.1. Let pc be the critical probability on the oriented site percolation and Xc be the critical value on the one-dimensional basic contact process. Then

-M^) -2A

xc< wiere A > 0 with e

2A

> pc.

Basic Contact Process II

59

From e _ I > 1 - x for x > 0 and Theorem 4.2.4 (2), we see that e~ 2 A - Pc > 1 - 2A - pc

-2A. >± - 81 So for 0 < A < 1/162, Theorem 4.3.1 gives Aa<

1 , / i - m M A 6 V 162 )

In particular, if we take 1 A =

l62

7 X

8'

then T h e o r e m 4.3.2. Ac < 1328. 4.4. Holley-Liggett M e t h o d As we mentioned in the previous chapter, HoUey and Liggett 5 gave a good upper bound on the critical value : Ac < A<**> = 2, and a lower bound on the order parameter :

1 1 P P^A12 > ^++ V ]/l-]x 4 ~ 2A

fMA for A >2 2,

^'

by using the Harris lemma, (see Section 3.4.1). In this chapter we omit this original argu ment about the basic contact process in one dimension. See Holley and Liggett, 5 Chapter VI in Liggett. 1 We will apply this method to the 0-contact process in the next chapter. Recently Liggett 6 gave an improved upper bound Xc < 1.942 by an extension of the Holley-Liggett method. 4.5. Correlation Identities I In Sections 3.2 and 3.4 we gave some correlation identities. See Theorems 3.2.3, 3.2.5, 3.4.1 and 3.4.2. This section is devoted to other types of correlation identities. So using them and assuming some correlation inequalities, we obtain easily good upper bound on the critical value given by Holley and Liggett. 5

Phase Transitions of IPSs

60

For it > 2 and re; > 0 (i = 1 , . . . k,) we let 7(rei, n 2 , . . . , nk) be the probability of having l's at 7H + l,7i! + re2 + 2 , . . . , r a i + re2 + ••• + n fc _i + fc - 1 and 0's at all other sites in [1, n\ + re2 H h re^ + it - 1] with respect to the upper invariant measure vx. That is, J(ni,re2,...,refc) = k-\ n, EVjt (j[ { j j [1 - T,(TU + ■ ■ • + m-X + ft + * - 1)] X rfai + "' + TH + i)} x { f l [ l - 7 ? ( 7 l l + --- + n fc _ 1 + g*; +

fc-l)]}).

For example, 7(0,0) = 1^(7, 7,(1) = 1) = pX, 7(1,2) = ^ ( 7 , 7,(1) = 0,7,(2) = 1,7,(3) = 7,(4) = 0), 7(1,2,3) = ^ ( T , 7,(1) = 0,7,(2) = 1,7,(3) = 7,(4) = 0,7,(5) = 1,7,(6) = 7,(7) = 7,(8) = 0). Using Theorem 3.2.5 for A = { 1 , 2 , . . . ,re}',we have A [ pXA({0,1,.. ( { 0 , l , . .■ . , r e } ),n})-p - ^ ( { lx({l,.. , . . . , r-,"})] e})]+A [ p A (A{({l,...,n,n l,...,re,re + + l})l } ) --? p AA(({l,...,n)}] {l,...,7l)}] A[P +A[p n

..,fc-l,fc + l,...,n})-ft({l,.....n})]= 0.

. . , « } ) ] = 0.

+ J2[M{h---,k-l,k+l,...,n})-px({l,.

So the definition of 7(711,712) gives the following correlation identities.

l e m m a 4.5.1. For n > 1, L e m m a 4.5.1. For n > 1,

n

-2AJ(0, «)

+ n£>(*- 1 , 7 1 - - * ) == 0. -2A7(0, re) + J ^ 7(fc - 1,re- it) = 0. ■

1

fc=i For example, when re = 1, this lemma gives - 2 A 7 ( 0 , l ) + 7(0,0) = 0.

(4.1)

From the definition of 7(711,712), we have J(0,l)

= px-px({0,l})

and

7(0,0) = px.

Hence from Eq.(4.1) and the last relations we obtain ( 2 A - l ) p A - 2 A p A ( { 0 , l } ) = 0. The last equation is equivalent to Lemma 3.2.4 (1). On the other hand we expect the following conjecture.

61

Basic Contact Process II

Conjecture 4 . 5 . 2 . For k > 2 and ni,U2,...,nk

> 0,

7(0,0)7(71!,ra 2 ) < 7 ( n i , 0 ) 7 ( 0 , n 2 ) .

For example, 7(0,0)7(1,1) < 7(0, l ) 2 is equivalent to p A (0)p A (012) Ac, define -jv J(ni,m,...,nh)=

\

7(ni,n2,...,rait) — .

Note that the definition of A<j gives 7(0,0) = p\ > 0 for A > A,.. So we can rewrite the above conjecture as follows. Conjecture 4 . 5 . 3 . For k > 2 and n i , n 2 , . . . , n t > 0, 7 ( n i , n 2 ) < 7(ni,0)7(0,re 2 ).

Let OO

¥>(«) =

^ 7 ( B , 0 ) /

+ 1

.

n=l

Lemma 4.5.1 can be rewritten as n

-2A7(0,7 1 ) + J ] 7 ( f c - l , 7 i - f e ) = 0.

(4.2)

Observe that Y^, [ - 2A7(0, n)un+1\

= -2\[
(4.3)

n=l

Assume Conjecture 4.5.3. So we have oo

£

oo

n

£

7(fc - l , n -

fcK+1

TI

< J ] ^

7

(

f c

- 1.0)7(0, n - * K + 1 -

(4-4)

Phase Transitions of IPSs

62

Similarly Eq.(4.4) yields OO

71

£^7(fc-l,n-fcK

+ 1

=
(4.5)

Combining Eqs.(4.2),(4.3) and (4.5) gives
(4.6)

Note that if A > Ac, then 00

_

*) = E^0)

1

1

=^ . J ( 0 , 0 ) = -p.'

(4-7)

From Eqs.(4.6) and (4.7) we have 2\p\ - 2Xpx + 1 < 0.

(4.8)

So the continuity and monotonicity of p\ implies

»*i + i/Fs

for 2

^-

(4 9)

-

Note that this lower bound on p\ is nothing but the Holley-Liggett 5 one. So if we assume Conjecture 4.5.3, then Eq.(4.9) gives a good upper bound on the critical value:

Ac < AHL) = 2As you will see in Section 4.7.1, this argument is closely related to renewal measure. To clarify this relation we introduce another type of correlation identitiies which are slightly different from J ( « i , % , . . . , * * * ) . 4.6. Correlation Identities II For k > 1 and ti{ > 0 (i = 1 , . . .k) we let K{ni,n%,... ,»*) be the probability of having l's at 0, ni + 1, m + ra2 + 2 , . . . , nx + n 2 + 1- nk + k and 0's at all other sites in [0, rei + n2 + • • ■ + 7ifc + k] with respect to the upper invariant measure v\. That is, K(m,n2,...,nk)

=

E„ \V{0) f[ I f[ [1 - J/(nj + • • • + n ^ j + q{ + i - 1)] x T,(m + • • • + nt + i) I V For example, K(0) = n ( r / ij(0) = r?(l) = 1), K(l)

- « * ( i7:7) , 77(0)= 1,77(1) = 0,77(2) = 1),

K(0,0)=vx{V

7:?}7/(0) = 77(1) = 7,(2) = 1),

K{1,2) =vx{t,n-n : 7,(0)

= 1,7,(1) = 0,7,(2) = 1,7,(3) = 7,(4) = 0,7,(5) = 1).

Basic Contact Process II

63

L e m m a 4 . 6 . 1 . For any A > 0, oo

(1)

E *M --

= /»A,

m=o CO

K ,n"2)) === ir( £jC( K{n ), Bl 2 ai x), E fa>

(2)

n 2 =0 OO

J ^ ijf(ni,n K(m,n22)-). ,n 3 ) == A"(ni,n ,«»; 2 ,n3)

(3)

E *(»!•

n 3 =0 n 3 =0

In generai, oo CO

(4)

^ E

"1+1=0

K(n1,...,nk,n ■ k+i) ,n*,n*+i) == ^(«i»-

lt...,nk). if ( nK(n i l - ..,»*)•

Proof. Part (1). Observe that K(n) = vx{r) : 7/(0) = 1,7/(1) = ■ ■ ■ = r,(n) = 0,7/(n + 1) = 1) = v\(t) : 7/(0) = 1,7/(1) = ••■ = 7/(n) = 0) - vxiv ■ n(0) = 1,7/(1) = •• • = n(n + 1) = 0) = J ( 0 , n ) - J ( 0 , 7 i + l). So JV-l

Eii-(7l) = p>1-J(0,iV),

(4.10)

71=0

since 7(0,0) = p\. If 0 < A < Ac, then i>A = *o, so J(0,N) = 0 for any N. Hence in this case we have the desired conclusion. On the other hand, if A > Ac, then Theorem 1.10 (d) in Chapter VI of Liggett 1 gives lim ux(v ■ VW = ■ ■ ■ = V(N) = 0) = 0.

iV-»oo

(4.11) (4.11)

Noting that

/(0,N) < i/xfa :nW = ■■■ = •»W = 0), so combining the last inequality with Eq.(4.11) implies Mm J(0,N) N—*oo

= 0.

(4.12)

64

Phase

Transitions

of IPSs

Using Eqs.(4.10) and (4.12) we obtain JJV V--11

co oo

== Jim Y. K(n) = Px.

5>(n) n=0

" " ^ n=0 n=0

n=0

For parts (2),(3) and (4), the proofs are similar to part (1), so we omit them. For A > Ac, we let

-=?,

# ( % , . . .,nk)-

v

K(ni,...,n K(n!,...,n k) k) ^ .

So Lemma 4.6.1 can be rewritten as follows. L e m m a 4.6.2. Assume A > Ac. CO

(i)

£E ri r( rme i)) ==' 1i,, ni=0

(2)

^ ¥ ( r e i , n,»s)(ni), 2 ) == ZKim),

oo OO

n 2 =0 Tl 2 =0 oo

7 E ^ ,%) === *
OO

(3)

"n 3 = =0 3 0

1

2

1( " 2 ) -

3

2

In general, CO OO

(4)

^E

¥

"* + l = 0

K(ni,...,nk,n = k+i)=~E(ni,...,n K(ni,..■ ,nk).). ("i'-- • ,n*,n/t+i) k

From the above Lemma 4.6.2 (1) we consider {jK"(n) : n > 0} as probability den sity function. As in the proof of the last lemma, we have the following relation between J(ni,...,nk) and K(ni,... ,nk). L e m m a 4.6.3. For any A > 0, CO 00

(1)

.j 7( m ( n,io, 0))==

£

K( ), = E *o»o, Pl

Pi=m

oo OO

(2)

/J(ni, ( » i , nIH) , )= =

oo 00

^(Pi.ft). = E E *&*. £

E

Pi—nj P 2 = n 3 P i = n i p 3 =n 3

P2),

65

Basic Contact Process It

OO

(3)

J(ni,n2,n3)=

CO

OO

X ! X ! Kfa>Ps*Pa)-

£

In general, OO

(4)

OO

2Z ■■■ E

J(ni,...,nk)=

Pl=nl

K(Pl,...,pk).

P*="ft

So Lemma 4.6.3 can be rewritten as follows. Lemma 4.6.4. Assume A > Ac. CO

(i)

7(m,o)= X) *(*)■

(2)

J(ni,na)= X! £ *"(Pi>^)-

CO

OO

P l = n i P2 = 7l2

CO

(3)

CO

J(nj,n2,ti3)= 5 3 £

OO

£

*"(Pi'P2>P3)-

P l = « l P2 = "2 P 3 = " 3

In general, OO

(4)

OO

7 ( « i , . . . , » * ) = X ) '" XI ^(Pi'--->?*)pi=ni

pt=ni,

From the definition of S2 for the basic contact process in one dimension we have the following lemma. Lemma 4.6.5. For n,m,l > 1, OO

(1)

-pA + 2A£ff(») = 0,

(2)

-2(A + l)K(n) + 2\K(n + 1) + X) #(» - *>n - 0 = °.

66

Phase

Transitions

of IPSs

(3) -(4A + 3)K(n, m) + XK{n + 1, m) + XK(n, m + 1) m

n

+ ^ i T ( p - l , n - p , m ) + ^ i i r ( n , g - l , m - g ) = 0, p=l

, =1

(4) -2(3A + 2)K[n, m, I) + XK(n + 1, m, /) + XK(n,ro,I + 1) n

m

I

+ 5 3 - ^ ( P - l . « ~ P » m > l ) + YlK(n,q-l,mp=l

q,l) + £ } K ( n , m , r - 1,1 -r)

g=l

= 0,

r=l

(5) -4(A + l)K(n,0,l)

+ Aif(n + 1,0,/) + XK(n,0,l

+ l)

+ AA'(n + l,Z) + AA'(n ) / + l) n

/

+ £ j r ( p - l , » - ] » , 0 , J ) + ] £ j r ( M , f - U - r ) = 0. p=l

r=l

4.7. Decoupling Procedures : Nonrigorous Results In this section we introduce two decoupling procedures of the correlation identities K(ni,n2,.. .,nk) to get nonrigorous upper bounds on Ac and lower bounds on p\. Results in this section appeared in Konno and Katori. 4.7.1. Renewal decoupling We present the following decoupling procesdure based on the renewal measure : for A > Ac, K^(nun2,...,nk)=^K^\ This decoupling gives the measurement of correlation functions A ' ( 1 ' ( n 1 , n 2 , . . . ,nk) renewal measure with probability density /M(ra) on { 0 , 1 , . . . } . That is,

(4.13)

by a

k

KM(nun2,.

..,nk)

= p™ J ] fW(nt),

(4.14)

i=i

where Er=o(« + l)/(1)(n) The explicit form of / ^ ' ( n ) is given by fW{n)

= F(n + l)-F(n

+ 2)

with

F(n + 1 ) «

j 2 n } \ \ for n > 0. n!(n + 1)! (2A) n

Basic Contact Process II

67

From Eq.(4.14), Lemma 4.6.5 (1) and (2), we have

^1)(0)=^11),

(4.15)

K"(l)=^P™.

(4.16)

Define the generating function by oo

$<1>(u) = £/< 1 >(rcK+ 1 . n=0

Multiply Lemma 4.6.5 (2) by u n + 1 , sum over n > 1 and use Eqs.(4.14), (4.15), (4.16) to get u$W{u)2 + 2[A - (A + l)u]$ (1) (u) - u[(2A - 1) - 2Xu] = 0.

(4.17)

$ (1) (1) = 1,

(4.18)

£ # « « U = ^J-

(4-19)

Note that

Differentiating Eq.(4.17) twice with respect to u, setting « = 1 and using Eqs.(4.18) and (4.19), we obtain a quadratic equation of p^ ' : 2A(p)2-2Ap + l = 0.

(4.20)

We choose the following solution which has a physical meaning,i.e., pA —» 1 as A — ► oo :

^-l+VTs

forA 2

^-

(4 2i)

-

Define the critical value Ac ' by A*1' = inf{A > 0 : p^ > 0}, so we have A « = 2Note that the above p\' and \y' are equal to the bounds by Holley-Liggett method. 4.7.2. A new decoupling In a more general setting, we introduce the following new decoupling procedure to obtain improved approximations for Ac and p\.

68

Phase Transitions of IPSs Suppose A > Xc. For m = 0 , 1 , . . . , define

K^(kuk2,...,k2m+1)

w

1

(2)

^ _

m

w

-]

=\ P y \f {h) n / (fc2S,fc25+i)+n f (hs-i, k„)fV(k2m+1)\, KW(kuh,.~Mm) = M 2 ) i n / < 2 ) { ^ - I , M + / ( 2 ) ( ^ ) n /(2)(*2S,*2S+i)/(2>(*2m), where {/ ( 2 ) ( f c )>/ ( 2 ) (™> m ): fc,ra,ro < 0} is a set of nonnegative integer with oo

E/ (2) ( fc ) = 1oo

£/(2)(m,«) = /(2)W

and

/<2>(m,rc) =

fm(n,m),

m=l

1

„(2)_ PA

^

=-

•E^ i)/(22)»(»)' E~=1 (» (n + i)/< (»)'

For example,

# (2) (M = ^ 2 7 ( 2 ) (M, tf(2)(fci,M = \P^[im{kiM)

+ / (2) (*i)/ (2) (M],

ir(2)(fci,fc2,fc3) = 5Pi 2) [/ (2) (fci)/ (2) (*2,* 3 ) + / (2) (fci,fc 3 )/ (2) (fc 3 )]. Set

$<2>(«) =

OO

n=0

In a similar way, we have ai(A)(A 2 ) ) 3 + «2(A)(A 2 ) ) 2 + « 3 (A)A 2 ) + a 4 (A) = 0, where ai(A)

= -2(64A 4 + 72A3 + 14A2 - 5A - 2),

a 2 (A) = 256A4 + 208A3 + 4A2 + 2A + 5, a 3 (A) = -2(4A + 3)(16A3 - 4A2 + 9A + 4), a 4 (A) = 4(4A+l)(4A + 3). The cubic equation of p\

has the only one real root : A(c2) « 1.790.

Basic Contact Process II

69

4.8. Remark We should remark that an upper bound on the critical value of the basic contact process can be obtained by a slightly modified version of the contour method without using the comparison method. The estimated values are between the Holley-Liggett bound=2 and the Harris bound=1328. This method was first introduced by Bramson and Gray8 in order to show the survival of the branching annihilating random walk. This approach was also used in Katori and Konno9 for the pair annihilation model proposed by Dickman.10 The good point of this method is that it does not need the attractiveness of the process, so it would be more applicable. In fact the branching annihilating random walk and the pair annihilation model are not attractive.

References 1. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York,1985). 2. R.Durrett, Lecture Notes on Particle Systems and Percolations (Wadsworth, Inc., California, 1988). 3. R.Durrett, Ten Lectures on Particle Systems (St. Flour Lecture Notes, 1993) Springer Lecture Notes in Mathematics, to appear. 4. T.E.Harris, Contact interactions on a lattice. Ann.Probab. 2(1974)969-988. 5. R.Holley and T.M.Liggett, The survival of contact processes. Ann. Probab.6( 1978)198206. 6. T.M.Liggett, Improved upper bounds for the contact process critical value. Ann. Probab., to appear (1994). 7. N.Konno and M.Katori, Correlation identities for nearest-particle systems and their applications to the one-dimensional contact process. Mod.Phys.Lett.B 2 (1991) 151159. 8. M.Bramson and L.Gray, The survival of branching annihilating random walk. Z. Warhsch. verw. Gebiete 68 (1985) 447-460. 9. M.Katori and N.Konno, Phase transitions in contact process and its related process. Formation, Dynamics and Statistics of Patterns. (Eds. K.Kawasaki and M.Suzuki) World Scientific, Singapole. (1993) 23-72. 10. R.Dickman, Universality and diffusion in nonequilibrium critical phenomena. Phys. Rev. B40 (1989) 7005-7010.

70

Phase Transitions of IPSs

CHAPTER 5 THE 0-CONTACT PROCESS

5.1. Introduction The 8-contact process Tjtte is a continuous-time Markov process on state space {0,1} . The formal generator is given by

n/fo) =

xez xezdd

c{9: x,r,)[f(rix)- -/(•?)],

with flip rates c(9 :x,T,) = r,{x) + A(l - r]{x))[ri(x - 1) + r,(x + 1) - (2 - 9)r,{x - l)r)(x + 1)], where A and 0 are nonnegative parameters and rf denotes rf(y) rix(x) = 1 - T](x). That is, 1 -► 0

at rate

1,

001 - > O i l

at rate

A,

100 -» 110

at rate

A,

101 -> 111

at rate

0A

= n(y) for y ^ x and

This process is also called the generalized contact process in Jensen and Dickman. 1 The first three sections in this chapter we will restrict attention to a family of the 0-contact processes for i1<9<2 < e<2 which was first introduced by Durrett and Griffeath. 2 This restriction comes from the existence of a coalescing duality for the 0-contact process. After that we will discuss a general case, 9 > 1, by using the Gray duality. 3 The process for 9 = 2 is nothing but the basic contact process. When 9 = 1, this process is the threshold contact process.

u-Contact

Process

71

When 6 > 1, the ^-contact process is attractive. So the upper invariant measure v\ $ exists for each A > 0 and 0 > 1 : vx,e = lim

SiS(t),

where S\ is the pointmass on i) = 1 and 5(t) is the semigroup given by SI. As usual the order parameter and the critical value are denned by px(0) = ux,e{v ■ V(x) = 1), Ac(0) = sup{A > 0 : px{9) = 0}, = inf{A > 0 : px(6) > 0}. Here we present another definition for the ^-contact process. The state (t,e C Z at time t is given by the sites being occupied ( or infected ) : 6,e = {xeZ:

T]tie(x) = 1}.

The system evolves as follows : (i) If x e £t,«, then x becomes vacant at rate 1, (ii) If x £ £t,t, then x becomes occupied at the rate f(Nx) of occupied neighbors,

Nx=\^n{x-l,x where f(N)

which depends on the number

+ l}\,

is given by (■ 0, f(N)=\X, {OX,

if N = 0, if JV = 1, if JV = 2.

Let ffe denote the ^-contact process starting from A C Z. Here we clarify the relation between the original processes and their coalescing dual ones. (We will explain the coalescing duality in Seciton 5.4.) Let £fe denote the coalescing dual processes for the 0-contact processes. The evolution of these processes are described as follows : { i } dies

at rate 1,

{x} branches to {x — l,x}

at rate (6 — 1)A,

{x} branches to {x,x + 1}

at rate {6 - 1)A,

{x} branches to {x — \,x,x

+ 1}

at rate (2 - 6)X,

and two particles which try to occupy the same site coalesce to one. The duality equation (which will be discussed in Section 5.4,) i.e., P(ft>fl^)

= P ( ^ n i ^ )

(5.1)

implies that if we let f j s denote the process starting from fo,« = Z, then as t —> oo, £]g decreases to a limit £ ^ B. The limit is a stationary distribution for the ^-contact process and P(0 e £ c , 8 ) = P(8J / 4>forall t > 0). (5.2)

Phase Transitions of IPSs

72

Similarly J°(0 e & , « ) = P(.8f = ^ for all t > 0).

(5.3)

Then p.\(0) and Ac(#) can be rewritten as follows : Px{9)

= p ( o e &,,,),

Ac(0) = inf{A > 0 : P ( 0 G &,,«) > 0}. Now we think of the relations between the following critical values : Ac(0) = inf{A > 0 P(0 e &,,,) > 0}, \,{9)

= inf{A > 0 i>($ ( S / for all t > 0) > 0},

\c(6)

= inf{A > 0 P(0 6 g o , . ) > 0},

Xf(0) = inf{A > 0 P ( | ? e / # f o r a l l t > 0 ) > 0 } , where / stands for survival starting from finite sets. So Eqs.(5.2) and (5.3) yield Xc(0) = Xf(9)

and

Xf(6) = Xc(9).

(5.4)

2

In fact Durrett and GrifFeath proved that the above four critical values are identical as follows. To do this we introduce the following edge processes. Let rt{9) and 7t(9) be the right edge processes for ^-contact process and its coalescing dual respectively, starting from ( - o o , 0 ] . That is,

rt(9) = ^

^ ' ° \

P

f t (0) = s u p g - ° ° ' 0 1 , with sup
««,*(»)

t—+oo

ax(9) = Mm

at,x(9)

< — :.;■

i

'

where atA(9)

=

E(rt(9)),

3(,A(0) =

E(r,(0)).

Then the standard argument yields X,(9) = inf{A > 0 : Urn ^

^ > 0} = inf{A > 0 : ax(8) > 0},

X}(9) = inf{A > 0 : lim 3t,xW > 0} = inf{A > 0 : 5A(6>) > 0}. This means that edge speeds characterize the critical values for survival probabilities. So to prove A/(0) = A/(#), it suffices to show 0 ( , J ( « ) = 5(,A(«)

forallt,
(5.5)

8-Contact Process

73

That is, ax(8) = ax(8). Using the duality equation, and translation invariance and symmetry of dual processes we see that oo

oo

<*t,x(6) = E kP(r,(9) = k) - J ] kP(rt(8) = -k) oo

=

J2lP{Tt(0)>k)-P(rt{e)<-k)} k=i oo

= E [ p (f'r' 0 1 n [fe.oo) ±4)- i>(^°°'01 n (-*,«,) = 4,)} = E h ? , ^ n (-00,0] / 4) - J»(gi;*-) n (-00,0] = *)]

= E h£r'01 n i*>°°) / * > - -p^r' 01 n (-fc'°°)=*)\ oo

= E i w ) ^fc) - p(f«w * -*)i =5'.* wTherefore we have Theorem 5.1.1. For tie 0-contact process with 1 < 8 < 2, (i) (2)

«*(•) =

SAW,

A0(«) = Xf(8) = \c(9) = \s{8).

So from now on as for critical values we consider just \c{8). Similar results about discrete-time version of the 0-contact processes can be found in Chapter 5 of Durrett.4 Note that this relation holds for nearest neighbor additive systems. (See page 9 in Durrett and Griffeath.2) Moreover Durrett and Griffieath2 showed Theorem 5.1.2. In the 0-contact process, (1)

(2)

ifl<8<2,

then aKie){8) = 0,

it 1 < 8 < 8 < 2, tien Xc(8) > Xc{8).

74

Phase Transitions of IPSs

One more interesting thing is the relation between the density of particles P(0 £ f ^ j ) and the survival probability P(£° g / <^> for all i > 0). The former is sometimes called the steady state concentration of particles by physicists. When 0 = 2 (i.e., the basic contact process), the process is self dual w.r.t. coalescing duality. So ^ n B ^ )

= P ( ^ n i ^ ) .

(5.6)

Hence two order parameters coincide :

p(o € & ,*)-When 9^2,

-- m%^ 4> for all t > 0).

unfortunately the exact relations are not known. Let

ti£ = { # , * <$> for all * > 0}, so the survival probability can be rewritten as

Px(d£) where we add subscript A in order to emphasize this role. Applying the results of Bezuidenhout and Gray 5 about the second-order phase transition for P;\(fioo ) with respect to A to the ^-contact processes, we have T h e o r e m 5.1.3. For the 8-conta.ct process with 1 < 0 < 2,

i W n & ) = °This result is a generalization for the basic contact process. Similarly from Theorems 5.1.2 (1) and 5.1.3 we expect Conjecture 5.1.4. For the 8-contact process with 1 < 0 < 2, Pxc(e)(0) = 0.

Here we assume that there are critical exponents /3(9) and ~J3(0) so that if A J. A c (0), then

p,(0)~(A-Ac(0))«e), ^(^)~(A-AC(
The critical exponents on the basic contact process are estimated as 13(2) = P(2) ss 0.277.

8-Contact

Process

75

See, for example, Brower, Furman and Moshe, 6 Konno and Katori. 7 By using Pade approximant analysis of series, Jensen and Dickman 1 derived the critical exponents for the 0-contact processes over a wide range of 8 values and supported /?(
That is, the exponents for the steady state consentration of particles are independent of 8. In this meaning a family of 0-contact processes belongs to a single universality class. The rest of this chapter is organized as follows. In Section 5.2 we give lower bounds on Ac(0) for 1 < 8 < 2 by using the Griffeath method and the Liggett and Ziezold-Grillenberger methods. Section 5.3 is devoted to upper bounds derived from the Holley-Liggett method for 1 < 8 < 2. In Section 5.4 we introduce the coalesing duality and discuss about some examples. The Gray duality is presented in Section 5.5. So using it, we will give lower bounds on A c (0) for 8 > 1. 5.2. Lower B o u n d s In this section lower bounds are obtained by the Griffeath method and the Liggett and Ziezold-Grillenberger ones. 5.2.1. Griffeath

method

Our main result in this subsection is the following theorem which appeared in Katori and Konno. 8 In fact we will remove the restriction 8 < 2 in Section 5.5 by using the Gray duality. T h e o r e m 5.2.1. Let x

^ = 2(8TT)[e-1

+ Ve2 + m

+ 13

\-

Then we have x

A L W < c(0)

for

1 < 0 < 2.

This is a simple extension of the lower bound on the basic contact process, Az,(2) = (1 + V^7)/6 « 1.180, which was given by the following methods : (i) Markov extension method ( Corollary 3.4.9 ), (ii) Liggett and Ziezold-Grillenberger methods ( Section 3.5.1 ), (iii) Griffeath method.

76

Phase Transitions of IPSs

The last one (iii) can be found in Section 7 in Griffeath9 and pp.166-167 in Liggett.10 The method presented here is the Griffeath method. In the next subsection we apply the method (ii) to getting lower bounds. Let Y be the collection of finite subsets in Z. The duality equation implies P(&j

n A f 0 = P(lt,e + 4>forall t > 0),

P(tl,e nA?) = Pltf, ± 4> for all t > 0). The definitions give "\Av ■ r&x) = 1 for some x e A) = P(&,» n A ^ <£). As in Section 3.4.1 we let 7r(A) = irx,e{A) = vx,$Cl : rj(x) = 1 for some x £ A). Moreover following the notation in Liggett10, define for all t > 0), so (7(A) = 7r(A).

To prove Theorem 5.2.1, we use the following correlation identities which obtained as in Theorem 3.4.5. Theorem 5.2.2. For any A £ Y, («-!)*£

E

| ^ U W ) - ^ ) ] + ( 2 - « ) A ^ [ ^ U { I - 1 , I + 1})-^)]

+

^2[a(A\{x})-a(A)]=0. xeA

Prom now on for example we sometimes abbreviate
-(2A + l)
(2)

(7(0) - (A + 1)<J(01) + A
(3)

2cr(01) - (2A + 3)(7(012) +CT(02)+ 2A
9-Contact Process

(4)

77

(7(0) + A
When 1 < 9 < 2, the following lemma comes from Theorem 5.2.2 for all A C {0,1,2}. L e m m a 5.2.4. (1)

- ( 0 A + l)
(2)

(3)

2(7(01) - (2A + 3)
(4)

So we can easily check this lemma is an extention for Lemma 5.2.3. Noting that Proposition 5.9 on page 165 of Liggett 10 is also valid for the 0-contact process with 1 < 9 < 2, we have the following useful lemma. More general case will be proved in Lemma 5.4.3 (2). L e m m a 5.2.5. cr(A) is submodular in t i e sense that
(2)

CT(0123)

< 2(7(012) - (7(01). (7(013) < CT(02) + (7(01) - <7(0).

P r o o f o f T h e o r e m 5.2.1. By Lemmas 5.2.4 (3) and 5.2.6 (1), we get (7(02) > - ( 2 A - 3)(7(012) + 2(A - 1)(7(01). Lemmas 5.2.4 (4) and 5.2.6 (2) give (A + 1)CT(02) < (3 - 0)A(7(O12) + (20 - 3)A(r(01) - [(9 - 1)A - 1](7(0).

78

Phase Transitions of IPSs

So

{2A2 + (2 - 9)X - 3}
+ (20 - 3)A + 2}
{(9 - 1)A - l}<j(0) > 0. Lemma 5.2.4 (1) and (2) yield

ex + (3 " ex + (2 -# < 0 ) . 0A2 + (3 - 0)A +1

(7(01) =

(7(012) =

X{9X ++(2(2-e)} - 9)} x{ex

7(0).

So the last two equalities imply that the last inequality is equivalent to {(9 + 1)A2 - (9 - 1)A - 3} 0. Note that XL(9) is given by the positve root of {9 + 1)A2 - (9 - 1)A - 3. So if A < XL(9) then
methods

In this subsection we will give better lower bounds than X^(9) given by the previ ous subsection by the Liggett and Ziezold-Grillenberger methods. For n = 3 , 4 , . . . , m = 1 , 2 , . . . , n — 1 and s = S1S2 ■■■«„ £ 5 ' " ' , we let L(s) = n + 1, if s = 0 . . . 0 0 , L(s) = n, if a = 0 . . . 01, L(s) = m, if s = 0 . . . 0 1 s m + i . . . sn, R(s) = c(s) + A + N{s) + 1, where c(s) is the total infection rate of configuration, L s i . . . s „ l and N(s) is the number of l's for S1S2 ...sn with £(0) = 2,

1(1) = 1 ,

£(O) = ( 0 + 1 ) A + 1,

£(00) = 3,

£(01) = 2,

£(10) = £(11) = 1,

£(00) = 3 A + 1 ,

£ ( l ) = A + 2,

£(01) = £(10) = (f?+l)A + 2,

£ ( l l ) = A + 3.

For example, in the case of n = 3, the above definition gives £(000) = 4, £(001) = 3, £(010) = £(011) = 2, £(100) = £(101) = £(110) = £(111) = 1, £(000) = 3A + 1, £(001) = 3A + 2, £(010) = (26* + 1)A + 2, £(011) = (9 + 1)A + 3, £(100) = 3A + 2, £(101) = £(110) = (9 + 1)A + 3, £(111) = A + 4. First we compute m<1)(A) and a<1)(A), so we will get X£K {X^ chain on state space {0,1} with transition probability : Pm

W = P (O,O)

j>M(l,0)

^'(o.i) pM(l,l)

0 1

1 X±x

X+2

A+2

: k > 1} is the Markov

O-Contact Process

79

So the stationary distibution of A"'1' is

n™ = H1)(o),/*«1)(i)] = [

1 A + 2, A + 3'A + 3J

Then we have

„,>w._[(,_iW)^ + (i_i(1))^,. Noting that i(0) = 2,

1(1) = 1, mmfA) l J

~

5(0) = (5+l)A + l,

#(l) = A + 2,

(5 + l ) A 2 - ( 5 - l ) - 3 (A + 3){(5 + l ) A + l } '

Similarly Q is the continuous-time Markov process on state space {0,1} with gener ator : -{(5 + l)A + l} {(5 + l)A + l} G<1' = s«(o,o)
(5+l)A + l. (5 + l)A + 2'(5+l)A + 2J

Then we have Q(D(A)

= - [(A - L(0))^(0) a*1'(A)

+ (A - i(l))M(1)(l)],

(5+l)A 2 -(5-l)-3 {(5+l)A + 2} ■

In any case we get

xi ){e) =

°

WVT) le~1 + v^ 2 + 1 0 *+ 1 3 ] ■

Note that Xh '(6) is equal to Xr,(6) in Theorem 5.2.1. As in n = 1, furthermore we compute m(2'(A) and cS2\X) as follows. m

(2)

( }

a<2>(A)

_ ~

3(5 + 1)2A4 - (252 - 115 - 12)A3 - (52 + 125 - 10)A2 - (135 + 29)A - 27 3(5 + l)2A4 + 4(452 + 115 + 6)A3 + 5(52 + 155 + 14)A2 + (235 + 84)A + 27' 3(5 + 1)2A4 - (252 - 115 - 12)A3 - (52 + 125 - 10)A2 - (135 + 29)A - 27 3(5 + 1)2A3 + (52 + 175 + 15)A2 + 4(25 + 7)A + 14

Therefore we get A(5) = sup{A > 0 : 3(5 + l) 2 A 4 -(25 2 - 115 - 12)A3 - (52 + 125 - 10)A2 - (135 + 29)A - 27 < 0}. Then we have

Phase Transitions of IPSs

80

Theorem 5.2.7. Assume that 1 < 0 < 2. Xc(0) > A<2>(0) > \W(6) = XL(0). 5.3. Upper Bounds In this section we get upper bounds by using the Holley-Liggett method as in the basic contact process, i.e., 0 = 2. Liggett11 reported upper bounds on the critical values for a family of contact processes which is called the periodic threshold contact processes. This family contains the threshold contact process (9 = 1) as a special case and his upper bound A(/(l) ss 2.170 is the largest root of the cubic equation A3 - A2 — 3A + 1 = 0. His argument is also based on the Holley-Liggett argument, however it can not be extended for 1 < 0 < 2 directly. Moreover the explicit form of /(n) is not given. Comparing with his result we can also present upper bound on critical value Ac(0) and lower bound PL,X(0) on the order parameter

p\(0) = nAn ■ vix) = i). whenever 1 < 0 < 2. Note thatCT(0)= o({0}) = p\{0). As in Section 3.4.1 the one of key ideas is based on the Harris lemma. Here we rewrite this as the version of 0-contact processes. Let y* be the set of all [0,l]-valued measurable functions on Y. For any h G Y*, we let

Sl'h(A) = (0-l)\-£

£

[h(AU{y})-h(A)]

x€Ay:]y-x\

+ (2 - 0)X Y, [KA U {x - 1, x + 1}) - h(A)] + £ [h(A \ {x}) - h(A)]. xeA xeA Note that Theorem 5.2.2 gives ftV(A) = 0. Lemma 5.3.1. Let ht G Y* (i = 1,2) with (1) (2)

hi{4>) = 0 and 0 < h{(A) < 1

for alJ

Mm hi(A) = l, \A\-HX

(3)

n*/n(A) < o < n*h2(A).

Then for any A e Y, (4)

h2(A) < o-(A) < hx(A).

A ^ <$>,

0-Contact

In

Process

81

particular,

(5)

h2{0) < Px(6) < ^ ( 0 ) ,

(6)

Ac < Xc(9) < A2,

where hi(0) = hi({0}) and X'C is a critical value on i;(0). That is, Xl = sup{A > 0 : /ti(0) = 0}, A2 = i n f { A > 0 : / i 2 ( 0 ) > 0 } . Therefore, by choosing suitable h{{A), we can obtain lower and upper bounds on Xc. As for upper bounds we obtain the following main result. 8 T h e o r e m 5.3.2. Let Xu(9) be the greatest root of the cubic equation OX3 - (30 - 2)X2 - 3(2 - 9)X + (2 - 9) = 0. Tien we have K(0) < Xu(9)

for

\<9<2.

When 9 = 2, this cubic equation is 2A2(A — 2) = 0, so the above thorem gives the Holley-Liggett upper bound, Ac = 2. IS 9 = 1 then this cubic equation is nothing but A3 - A2 - 3A + 1 = 0 which was given by Liggett. In this way upper bounds {Xu(9) : 1 < 9 < 2} contains both upper bounds as special cases. It is interesting to note that f(n) is expressed by Gauss's hypergeometric series in the form F(—(n — 2),— (n — l),2;z(9)) for 1 < 9 < 2. Particularly, if 9 —* 2, then z(9) —► 1, so the Gauss's hypergeometric series are reduced to a combination of the gamma functions (i.e. factorials), as a result we have the same form given by Holley and Liggett. Following the Holley-Liggett argument, we choose h(A) in the next way. (i) Choose h of the form h(A) = H(T] : r)(x) = 1

for some x e A),

for some renewal measure fj. on { 0 , 1 } Z with n(
82

Phase Transitions of IPSs

L e m m a 5.3.3. Let 1 < 9 < 2. Suppose that there are functions f(n) satisfy

..} which on {1,2,. {1

2A/(3) + / ( l ) 2 - ( 2 + 0A)/(2) = O,

(1)

n-l

(2)

2A/(« + 1) + £

/(*)/(!» - fc) - 2(1 + X)f{n) = 0

(n > 3),

k=i CO

(3)

£/(») = !,

(4)

^n/(n)
(5)

0 < f(n) < 1

for

any n > 1,

and (6) Tien for any non-empty

/(rc) / ( n + 1) ., ' > ^ *• / ( n + l ) - / ( n + 2) sets A 6 Y,

for

any n > 1. * -

fi(rj : rj(x) = 1

for some x 6 A) > 0,

where // is the renewai measure corresponding to / . Let

*x») = £/(*)• From the choice (ii), .F(ra) should satisfy the following identities : 0AF(2) + ( 2 - 0 ) A F ( 3 ) = F ( l ) 2 , n J2 F(k)F(n + 1 - k) = 1\F(n + 1) fc=i And Lemma 5.3.3 (3) and (4) can be rewritten as

m=i,

for n > 2.

(5.7) (5.8)

(5-9)

9-Contact Process

and

83

E jr '( n ) <00 '

(5.10)

respectively. So it is easy to check that Lemma 5.3.3 (l)-(4) are equivalent to Eqs.(5.7)(5.10). We introduce the following generating function to solve Eqs.(5.7)-(5.9). oo

YlF(n)un.

Note that Eq.(5.9) is equivalent to d<j>(u)

= 1.

(5.11)

+ (fl-l),

(5.12)

du Let

s =~

y = (2-8) + 9\.

(5.13)

So 4>(u) satisfies 2y

2y

1-x

w-if^+T+V+i+r =°-

(5.14)

Therefore the unique solution of Eq.(5.14) with Eq.(5.11) is given by L „1 + J! 1 - 1 /„1 + X V W 1 - 2——u - 77T 2—I— u ) V y 4(1 + c) V y ) From now on we assume that 0 < x < 1 and y > 0. Let

(U):

1+x

1

u±

2/ l + x±y/2{l

(5.15)

+ x)'

so u- < 0 < u+. In fact the function (5.15) is real analytic only when u_ 1, i-e-, y>1 + x +

A/2(1

+ x),

(5.16)

then we can obtain a real solution F(n) with Eqs.(5.7)-(5.9) by expanding Eq.(5.15) in a power series in u, which satisfies

g>( n ) = X>/(n) = T -J-

„l + x 1-x2 1-./1-2—— U ;—

(5.17)

Phase Transitions of IPSs

84

Here we present useful formula to expand Eq.(5.15) in a power series in u. If 0 < x, s < 1 then

f

with « = !

»d

-s-

1~*

OO

4(l + i )

.7 = 1S n=l

^ = 2 ^ x ( l | ^ )

n / 2

^ "

2 )

^ , e - ^ )

(n>2)

where

eiv :=

/1 + a V 2 "W 22

and ?j(re, z) is Gauss's hypergeometric series of the form »(n,z) = F ( - ( n - 2 ) , - ( n - l ) , 2 ; z )

(n > 2).

Let

iu(l,x) = 1

tu(n, x) = f i ±+ £ j)

2 v, iv e*<"e , '^- 2 >* ) «(n, «(n,ee~ 22iv ))

(n > 2). 2).

Applying the above formula to Eq.(5.15), we have L e m m a 5.3.4. If 0 < x < 1 and 3/ > 1 + x + ^/2(1 + x), then the unique solution of (1) and (2) with (3) and (4) in Lemma 5.3.3 is given by f(n) = F(n)-F(u where

F(n) =

1

■«;(n.

+ l)

,s).

Note that u+ > 1, i.e., y > 1 + x + -y/2(l + x) is equivalent to U + ^ - j

[ex3 - (30 - 2)A2 - 3(2 - 8)\ + (2 - 9)] > 0.

(5.18)

So if A > 0 and 1 < 9 < 2 then Eq.(5.18) gives the following cubic equation in Theorem 5.3.2 : 0A3 - (39 - 2)A2 - 3(2 - 9)\ + (2 - 9) > 0. Moreover we can show that the above f(n) satisfies Lemma 5.3.3 (5) and (6). However the proofs are messy, so we omit them. Details can be found in Katori and Konno. 8

O-C'ontact Process

85

5.4. Coalescing Duality Let Y = {A C Zd :| A |< 00} and

R(n, A) = IIC 1 - »K*))

for

ail ^ e y,

that is, 5(?7, A) = 1 if 77(0:) = 0 for all x e A, = 0 otherwise. Note that the product over empty set is 1. Suppose

am --= nmm+a./(•»)

= 5>(.,!,)[/(!».) - /(1?)] + D £ > ( X ) ( I - iK»)M*. »)[/(•»-*) - /fo)l. x

x,y

where C(*,IJ) = c[(l - T?(X)) + (2i/(x) - 1 ) $ ] p(x,A)5(»?,A)], /ley

c>0,

p(x,A)>0,

D > 0, p ( x , j 0 > 0 ,

V p ( x , A ) = l, ^ p ( x , 3 / ) = l,

sup J ] p ( x , A ) | A | < 00, p(x,7/) = p(i/,x),

»7x(j/) = i?(2/) if V / i , i7x(i) = 1 - 17(1) ™d f*y(u) = v(y) if « = *, >?*»(») = T?(X) if u = y, r]Xy(u) = 77(74) otherwise. We compute fi applied to H{TJ,A) as a function of 77. So fltffa, A) = c ] T ( l - 77(X))[5(T ?X ,A) - IFfoA)] + c^(27,(x) - 1) J2 p(x,B)H(r,,B)[H(r,x,A) x + i? ^

- 5(T?,A)]

B£Y IJ(*)(1 - T?(X)MX, »)[ff (17,,,, A) - #(77, A)]

x,y

= Hi + H2 + -ff3 ■ H x e A, then JTfo., A) - fffo, A) = T?(X) x j J ] (1 ~ »»(»)) I - (1 - »?W) x J n (1 - 77(2,)) 1 l»€A\{*} J Uex\{*} J = (2T,(S) -1)5(7?, A \{x}). If i € 4 , then J(77 X ,A)-J7(77,A) = 0. So for A g y , 5(7,,, A) - 5(77, A) = lA(x)(277(x) - 1)5(7,, A \ {*}),

(5.19)

86

Phase Transitions of IPSs

where 1,4 denotes the indicator function of A, Using Eq.(5.19), we have - V(x))[H(r,x,A) -

Hi = c^(l

H(TI,A)]

X

= e £ ( l - 7?(x))(27j(x) - l)H(r,,A\ xeA =

{x})

-C^(1-I?(X))5(T,)A\{X}) T£A

= -C£>(7fcA).

B)H{n,B)[H(n ff22 = c£(277(x) c£(2i,(*)-- 1) K^B)H{ ~ H{V,A)] V,B)[B(T,Zx.,,A) ,A)-H{r,,A)] H i) J2 E «*' X

= =

Bey

cC ^£ (( 22 r7, ?( (xI)) -ix€A €A

B l)22 E 2 H«. )*(l> B)E^'V, -1) *:,B)H(r,,B)H(

A\{x}) A \ {x})

Bey BeY

c = E E *:,B)H(v,B)H( ,A\{x}). = * E E P(x,B)H(r,,B)H( ,A\ {X}). V

V

xSAB^Y

= H(T],AUB), we have Noting if (77, A)fl(7/, B) = ff (T;, A U .B), we have Noting H(T),A)H(T),B)

* =cx£AB€Y E E R,*■S )B%^H( A^\ (WA \) U< xB»)u 5 )

H H2

C = - EEEE

E E

,B)H(V,B). Rx,B)H( V,B).

BgV ie.4 F:(A\{x})uF=B :B BgV i £ / l F:(A\{i})uP=

Let q(A,B) =

-E

E

p(x ,B).

x€A F:(A\{x})UF= x£A F:(A\{x})UF=B :B

Note that

E 9(^.5) = c. BeY

Therefore we have Hi + H2 = Y, q(A,B)[H(V,B)

- H(r,,A)].

BeY

That is, q(A,B) can be interpreted as the transition rates for a continuous-time Markov process on state space Y which evolves as follows : (i) each x £ A is removed from A at rate c and replaced by the set F with probability p(x,F), and (ii) If a particle tries to move at a site which is already occupied, the two particles coalesce.

$-Contact Process

87

In this sence, this duality is called the coalescing duality. Moreover the symmetry of P{x,y) gives ^

= YEP(I.»)[%».^)-%4)1

= f E E ><*.»)[*(*,. A) - J?(i?, A)] + f E E * , V ) [ % » , A ) - fffo.A)] D 2

r)p(*. *)[(i -«?(?)] -(1-7,(0;))] n 11 (i-»/M) (!-!?(«)) EE*.*-*))-^-^))]

^EE**-*))-^*))]

II (i-^w)

= f E E «<*. »)[*(* ^ u {»»\ <■}) - *(* w YEEP(^)I%(AUW)\W)-%4

+

a?^-^ y€A

So the symmetry of p(x,y) again yields ^3

=

f ^P(x,3,)[5(7j, A IS ) - H{r,,A)} x,y

= ^ E E K*. r)i*fo. A*») - # (^ ^)i = / > £ U ( « ) ( 1 - 1 A ( » ) M * , » ) [ 5 ( I J , A „ ) - 5(7?, A)],

where

' (A U {»}) \ {x} f(AUAxy = < (A Ul{^})\{2/} IA

iixeA,y$A, ifseA.s^A, otherwise.

Therefore we obtain

QH (i,, A) = 52q(A,B)[H{r,,B)-H{T,,A)] +D^Ato)(i-u(v))pto,y)[Bto^^)~B(%A)]. (V,A)]. x,y

Let

uv(t, A) = P"totto) = 1

for all x € A)

= S(t)H{-,A)to)

Phase Transitions of IPSs

88

So Theorem 2.9 (c) of Chapter I in Liggett 10 implies that jtun(t,A)

=

S{t)SlH(;A)(r1)

= ^(A,B)[S(t)H(;B){V)

-

S(t)H(-,A){V)}

B

+ DY,1A(X)(1

- lA(y))p(x,y)[S(t)H(;Axy)(r,)

-

S(t)H(-,A)(n)]

x.y

= Y/q(A,B)[u,,(t,B)-uv(t,A)} B

+ D ]T 1*00(1 - u(y))p(x, »)[«,(*, AXy)

"TJ(*>

A)].

x,y

Since Markov process At is nonexplosive, the unique solution to these differential equations with initial condition E{r), A) is EA(H(i),

At)) = PA{n{x)

= 0

for all x € At).

So we have the following duality theorem. T h e o r e m 5.4.1. For any r\ e {0,1} Z " and (1)

S(t)H{-,A)(V)

AeY, EA(H(n,At)).

=

P"{r]t(x) = 0 for all x 6 A) = PA(n(x)

(2)

Pvijit{x)

(3)

= 1 for some x e A) = i "

4

= 0 for all x e

At).

^ ) = 1 for some i e A t ).

This theorem can be rewritten as follows. Corollary 5.4.2. For any A C 1d and B 6 F,

(i) (2)

P ( t f n * = fl = P(8*nA = fl, p(6 4 n£ ? ^) = P(£fnA^<0.

When D = 0, a more general case can be found in Theorem 4.13 of Chapter III in Liggett. 10 And this theorem is an extension of Theorem 1.1 of Chapter VIII in Liggett. 10 Assume that the process rjt is attractive. In this setting, we let p(A) = 4 lhn ) P''- 1 (7 ?( (a;) = 1 for all x G A), p{A) = ]imaP"=1(r]t(x) <j(A) = for any AeY.

Then we have

l-p(A),

= 0 for all x £ A),

89

O-Conlact Process

Lemma 5.4.3. For any A,B e Y, (1)

p(AUB) + p(Ar\B)>p(A)

+ p(B),

(2)

Proof. For A, B £ Y, V(x) e {0,1} gives

| l - II (l-7?(x))l II (l-^(x))jl- n (1-,W) >0. \

xeA\B

J xeAnB

From the definition H(n, A) = YlxeA(l

y

xeB\A

)

~ ^(x)), the last inequality can be rewritten as

(1 - H(V,A\B))H(r,,An

B)(l-

H(r,,B\A))>0,

So H(TI,

A U B) + B(rj, A n B) > H(T), A) + H(r], B).

Then we have EAUB(H(V, (A UB)t) + EAnB(H(V,

(A n B)t)) > EA(H(r,, At)) + EB(H(r,, Bt)).

Note that Theorem 5.4.1 (2) gives p(A) = lim P" = 1 (%(i) = 0 for all x e A) t—toa pA

= Urn r ( % = M , ) ) . (—► oo

So paxt (1) comes from the last inequality. Part (2) is equivalent to part (1) by the definition : a{A) = 1 - p{A). Prom now on we will give some examples. Example 5.4.1. Basic contact processes. Here D = 0 and c(x, rj) = T)(x) + (1 - 7/(x))A x

J^

i)(y).

y:|v-i|=l

In this case we have c = 2<JA + 1, ( 1 / ( 2 ^ + 1), if A = , p(x, A) = I \/(2d\ + 1), if A = {x, y} with | y - x j= 1, (0, otherwise.

90

Phase

Transitions

of IPSs

To see this, observe that c[l - 77(a) + (2*7(3!) - 1) J2

p(x,A)H(r,,A)]

A€Y

= (2dA + 1)[1 - 7j(x) + (2T?(X) - 1)

p(x,)H{x,)+

= ( 2 d A + l ) ( l - r / ( x ) ) + (27 7 ( a ; )-l) 1 + A

£

Y,

P(*»{*.S»-ff («»»{»»»})

(l-r,(x))(l-r,(y))

y:|t/-i|=l

= 77(x) + (1 - V(x))

5Z ^

= C

(*' "?)■

2/:|j,-a;|=l

Of course, we can confirm it by the following direct computation. nJff(7?)J4) = ^ T)(x) + (1 - 7?(x))A X

£

7/(2/) [jr(i h f J 4)-iT(ij l A)]

3/:|y—ar|=l

= ^i/(x)(2iK*)-l)J(i7,i4\{x}) + 52(1 - IKX))(2IK«) - 1)A r€A

J2

IK»)2T(I;,A\{*})

j/:|y—x)=l

= 52ti-(i-'?w)] n a-»K«)) -52
5 ] [i-(i-7,(2,))] !/:|a-r|=l

n

(!-•»(«))

uSA\{x}

= 52[-ff(^A\M)-ff(77,A) igyl

+ A X)

{%^{i})-%^)}].

y:|»-a:|=l

Let £ = {x G Zrf T?(X) = 1}. The definition of the basic contact process implies that for * 6flCZ«), £ —► f \ {x} at rate 1, £ —► £ U {2/} at rate A for | y - x \= 1. On the other hand cxp(x,) = 1, cxp(x,{x,j/}) = A impliy that for x 6 A(e Y), A^A\{x} A^ Al){y}

for | ! / - x | = 1

at rate 1, at rate A for | y ~ x | = 1.

B-Contact Process

91

So this coalescing dual process is exactly same as the basic contact process, restricted to configurations with finitely many ones. In this meaning, the basic contact process on Zd is coalescing self dual or self dual w.r.t. H{n,A) = X[xiA{l - n(x)). However it is not self dual w.r.t. H(r),A) = Y[xeArj(x). For example, see Theorem 3.2.3. Example 5.4.2. 9-contact processes. Here d = \,D = 0, and = 0 A+ 1 ,

C :

if A =■4>, I 1/(0A+1), if A =:{X- 1, X} or {i, x + 1} I(0-1)A/(0A + 1), P~(x, A) = ) (2-0)A/(0A + l ) , if A =:{X- 1, x,x + l}, otherwise.

u

That is, at rate 1, at rate (9 --1)A, at rate (2 --0)A.

(A\{x) A - I A U {x - 1} or A U {a; + 1} | A U { I - 1 , I + 1}

From this we see that if 8 jt 2 then the 0-contact process is not self dual w.r.t. H(r], A) = TIX£A{1 — v(x))- As i n t n e basic contact process, the above relation is closely related to Theorem 5.2.2 : For any A e Y,

('"!)*£

[o-(Au{y})-o-(A)]+(2-8)\J2{°(AU{x-- l , s + l})-
£

x€Ay:\y—x\=l

x€A

(A\{x})-

-a(A)\- = 0.

xeA Example 5.4.3. Exclusion processes. In this example, we consider only the stirring part; 1

«./(•») == ^E»K*)(i-- ^(s/M , !0[/(»fc,y)"- /(•»)] where D > Q,p(x,y) > 0 , £ „ p ( a , s ) = 1 and p{x,y) = p{y,x). The evolution of this process is as follows : (i) a particle at x waits an exponential time with stirring rate D and then chooses a y with probability p(x, y) (ii) if y is vacant at that time it moves to y, while if y is occupied, it remains at x. Then we have

= "£*

0 .#(»?, A) =

A ( * ) ( 1 "-

1A(»)M* ,»)[#(»?, A** ) • -■ff(i;,A)].

So

A^(AU{j,})\{x}

at rate Dp(x,y).

92

Phase Transitions of IPSs

IWi -

Hencei in this meaning the exclusion process is self dual w.r.t. H(V,A) = "?(*))• Next we consider another choice of H(n, A) ■= IXrgjl tfi®) *• m t n e P r o o ; f of Thorem 1.1 of Chapter III in Liggett. 10 Similarly we get

njfft, A) =^ E i i ( # -

* ) - - 5 ( 7 , A)].

- ^A{y))p(x ,y)[S{n,Ax

= n,

Therefore the exclusion process is self dual also w.r.t. 2T(ij,A) =

M5A "?(*)■

E x a m p l e 5.4.4. Diffusive• contact processes. In this example we combine the basic contact process with the exclusion process. We call this process the diffusive contact process which will be studied in Chapters 6 and 7. The observations of Examples 5.4.1 and 5.4.3 imply that the diffusive contact ]process is self dual w.r.t. E{n,A) = r U ^ U - •/(*))■ That is, at rate 1, at rate A for \yat rate Dp(x.,y).

[A\{x] A-> { AU{y} {{AU{y})\{x}

■ X

1=1,

Note that the diffusive contact process is not self dual w.r.t. H{n,A)--

= IL<EA V(X)-

E x a m p l e 5.4.5. Diffusive 8-contact processes. As in the last example, we combine the ^-contact process with the exclusion process. So for x 6 A,

A-»

(A\{x) 1 Au{x-l) or AU{x | AU{xl,x + l} l(AU{2/})\W

+ l}

at at at at

rate rate rate rate

1, ( » - 1)A, ( 2 - 9)\, Dp{ x,y),

where 1 < 9 < 2,A > 0, and D > 0. When 9 ^ 2, the diffusive 0-contact process is not self dual w.r.t. H(n,A) = T] ^ ( 1 - ^ ) ) E x a m p l e 5.4.6. Long range contact processes. The flip rates are given by c(x,Tl) = 1)(x) + (l -

T {x))

> vd(M)\

£

/:1<|!/-*I <M

v{y),

where M = 1., z , . . . , 1>d(M) := | { *

and |z|

=11*\\P

ez d

is the V norm with 1 < p <

:1< CO.

1*1< ^ } | ,

This process is self dual w.r.t. % A )

6-Contact

Process

93

I l i g / i ( l - ^(z))- To see this, as in Example 5.4.1, observe that
=

E[^) + (1-^))^)X

E

i(y)

[H(T,X,A)-H(T,,A)]

y:l<\y~x\<M

=xeA5> =

^2v{x){2v(x)-l)H(V,A\{x})

xeA

+ Ed--»(«))(*»(*)-D^ = E [ i - ( i - ^ ) ) ]i

E

IJ(»)J(I»,A\{«})

y:l<|y-i|<M

n (i-^w)

E u-a-»K»»] n (i-^w) - » - * » « BBS:l/:l<|j,-j;|
+

dw)

£

{%AU{»})-%1)}].

y:l<|!/-i|<M

So, for s £ A(e Y), x} A _ / 1A4\ {11/,A IAU{,}

at rate 1, at rate \/vd(M)

for 1 < | y — x \< M.

Note that when M = 1 and p = 1, this process is equivalent to the basic contact process. 5.5. Gray D u a l i t y In this section we consider a more general duality which was introduced and studied by Gray. 3 The treatment in this section is based on Katori's paper, 12 so details can be found there. Let Y = { the collection of subsets in Y}. For A £ Y, we let

H(n,A)=U ( i - I N 1 ) ) ASA V

^A

For example, in the case of A = {A} = {{xi,X2,..

'

■, xn}}, n

H(V,i)=i - n n(x)=i - n x£A

In another case of A = {A} = {{xi},{xi},..

H(V,A)--

»K*.-)-

t=l

.,{xn}}, n

- »/(*)) =

-»?(« .'))•

Phase Transitions of IPSs

94

Note that H(r,,<j>)=l, H(r,,{}) = 0. In this chapter we will restrict attention to the 0-contact process with 0 > 1. Then for {x} 6 A = { { i i } , {x2},...,{x„}}, f A U { { * - 1 } } U { { * + 1}} A->

{ I U { { I - 1 , I + 1}}

U\{W)

at rate A, at rate (9 — 1)A, at rate 1.

Roughly speaking the second mechanism is equivalent to A^A\J{{x-l,x

+ l}}

= A U {{x - 1}} + A U {{x + 1}} - A U {{a: -- 1}} U {{* + 1}}. So A-- (8 - 1)A = (2 -
A^

f A U { { X - 1 } } U { { * + 1}} 2 } , . . • i {^n}}* then this dual process is equivalent to the coalescing dual process . As in a similar computation we have the following theorem which is analogous to Theorem 5.2.2. From now on, define "\ = v\,e for short. T h e o r e m 5.5.1. For A e Y,

A J2 [S(AU{{*--

1}} U {{x + 1}}) --5(A)]

{x}e2

+ (0-1)A J2 [5(1 U{{*--l,x + !}})-?(!)] {T}£A

+ £[< (A \

{{*}})--

5(A)] = 0,

{X}SA

where

a{A) = l-Evx ( n ( l-IN*))) From this we have some identities which will be used for getting lower bounds with 6 > 1.

6-Contact Process

95

(i) A = {{x}}. In this case {«}-<

f {{{i { * --! }1}, , {{x}, * } , {{x* ++ 1}} {{x},{{x-l,x+l}}

u

! So we have

{{x},{{x<j>

l , i + l}}

at at at at at

rate A, rate (9 - 1)A, rate 1. rate (9 - 1)A, rate 1.

A[5({{-l},{0},{l}})-5({{0}})]+(e-l)A[5({{0},{-l,l}})-5({{0}})] So we have + [5W-5({{0}})]=0. A[5({{-l},{0},{l}})-5({{0}})]+(e-l)A[5({{0},{-l,l}})-5({{0}})] For simplicity we write 5(0,1) = 5({{0},{1}}), 1}}), and so on. + [ * W5(0,-11) - 5 ( { { 0 }=} )5({0},{-l, ]=0. For simplicity we write 5(0,1) = 5({{0},{1}}), CT(0,-11) = 5({0},{-l, 1}}), and so on. -(9X+ 1)5(0) + (9 - 1)A5(0,-11) + A5(-l,0,l) = 0. So -{6X + 1)5(0) + (0 - 1)A5(0,-11) + A5(-l,0,1) = 0. The definition gives The definition gives 55(0,-11) (0,-ll) = 5 ( 0 , l )- - 55(-l,0,1) (-l,0,l) = 55 (( -- ll ,, 00 )) + + 5(0,1) = = 25(0,1)-5(0,1,2). 25(0,1)-5(0,1,2). Combining Combining last last two two equalities, equalities, we we have have -(e\ + 1)5(0) + 2(9 - 1)A5(0,1) + (2 - 0)A5(O,1,2) -(9X 0)A5(0,1,2) = 0. (ii) A = {{a;},{a: + l}}. From Theorem 5.5.1 and 5(0,-11,1) = 5(0,1), 5(0) - (A + 1)5(0,1) + A5(0,1,2) = 0. (iii) A = {{x},{x + l},{x + 2}}. From Theorem 5.5.1 and 5(0,-11,1,2) = 5(0,1,02,2) = 5(0,1,2,13) = 5(0,1,2), we have 25(0,1) - (2A + 3)5(0,1,2) + 5(0,2) + 2A5(0,1,2,3) = 0. (iv) A = {{x}, {x + 2}}. From Theorem 5.5.1, 5(0) + (9 - 1)5(0,2,13) - (9X + 1)5(0,2) + A5(0,1,2,3) = 0. By 5(0,2,13) = 5(0,1,2) + 5 ( 0 , 1 , 3 ) - 5(0,1,2,3), we have 5(0) + (9 - 1)5(0,1,2) - (9X + 1)5(0,2) + (2 - 0)A5(0,1,2,3) + (9 - 1)A5(0,1,3) = 0. Therefore we have the following lemma.

96

Phase Transitions of IPSs

L e m m a 5.5.2. (1)

-(9X + 1)5(0) + 2(9 - 1)A5(0,1) + (2 - 0)A5(0,1,2) = 0.

(1')

-(9X + 1)5(0) + (9 - 1)A5(0, - 1 1 ) + A5(0,1,2) = 0.

(2)

5(0) - (A + 1)5(0,1) + A5(0,1,2) = 0.

(3)

25(0,1) - (2A + 3)5(0,1,2) + 5(0,2) + 2A5(0,1,2,3) = 0.

(4) 5(0) + (9 - 1)5(0,1,2) - (9X + 1)5(0,2) + (2 - 9)A5(0,1,2,3)

+ (9 - 1)A5(0,1,3) = 0.

As in Lemma 5.4.3 we have the following useful result. L e m m a 5.5.3. For any

A,B£Y, 5(A U B) + o(A n B) < 5 ( 1 ) + 5 ( 5 ) .

Proof. For A e Y,

c(A) = l-E„x i n * ( M ) l . \AeA

/

where k(V,A)

=

l-^\T,(x). x€A

For A,B EY,

r)(x) 6 {0,1} gives

j l - II k(V,A)\ Y[j(v,B)\l[

A£A\B

) BgJnB

I ] fc(77,C)l>0. (

C€B\A

j

The last inequality can be rewritten as

II k(t,,A) + H k(r,,A)>Y[k(r,,A)+Y[k(r,,A). AZAUB

AzAnB

AeA

Note that

E(n,l)= Y[k(r,,A). AeA

A€B

97

9-Contact Process

So H(r),AUB) + H{V,AnS)>

H(V,A) + H(n,B).

Then we have EIuS(H(T,,(AUB)t)

+ E^nS(H{r,,(AnS)t))

> E*(H(V,At))

+

ES(H(r,,Bt)).

So the desired conclusion comes from 5(A) =

\-EVx{E{V,A)).

The following result can be obtained from Lemma 5.5.3 immediately. Corollary 5.5.4. For any A = {{x1:} .. .,{xn}} (1) For any A, B

a(AUB)

and B = {{J/J},...,{y m }},

+ a(Ar\B)
+ a(B).

eY,

(2)

5({A U B}) + a{{A n B}) < a({A}) + *({B}).

Using Lemma 5.5.2 for A = {{0},{1},{2}} and B = {{1},{2},{3}}, and A = {{0}, {—1,1}} and B = {{1},{3}} respectively, we have Lemma 5.5.5. (1)

5(0,1,2,3) < 25(0,1,2)-5(0,l)

(2)

5(0,2,13) < 5(0, -11) + 5(0,2) - 5(0).

As in Theorem 5.2.1, we obtain the following main result in this section using Lemmas 5.5.2 amd 5.5.5. In this theorem, we can remove a restriction 9 < 2 in Theorem 5.2.1. Theorem 5.5.6. Let

A

^=2WI)[ (? - 1+ ^ 2 + 10<' + 1 3 ]-

Tien we have ALW

< Ac(0)

for

9 > 1.

Phase Transitions of IPSs

98

Proof. By Lemmas 5.5.2 (3) and 5.5.5 (1), we get 5(0,2) > - ( 2 A - 3 ) 5 ( 0 , 1 , 2 ) + 2(A-- 1 ) 5 ( 0 , 1 ) . Lemmas 5.5.2 (4) and 5.5.5 (2) give (A + 1)5(0,2) < 2A5(0,1,2) + {9 -- 1 ) A 5 ( 0 , - 1 1 ) - - A 5 ( 0 , l ) --[(
iF(o).

{2A2 + (2 - 9)\ - 3}5(0,1,2) + {-2A 2 + (29 - 3)A + 2}5(0,1) - {(9 - 1)A - 1}5(0) + (6 - 1)A[5(0,1,2) - 25(0,1) + 5 ( 0 , - 1 1 ) ] > 0.

Note that 5(0,:1.2)- - 2 5 ( 0 , 1 ) + 5(0, - 1 1 ) = = 0. Hence

{2A2 + (2 - 9)X - 3}5(0,1,2) + {-2A 2 + (29 - 3)A + 2}5(0,1) - {{9 - 1)A - 1}5(0) + (9 - 1)A > 0.

The rest of proof is the same as that of Theorem 5.2.1. That is, Lemma 5.5.2 (1) and (2) yield 9\ + (3 5(0 : ,1) = 9\ + (2 0A2 + (3 -B)X + 1 5(0). 5(0,1, .2) = \{9X + (2 - 9)} So the last two equalities imply that the last inequality is equivalent to {(9 + 1)A2 -- ( • - - 1 ) A -- 3}5(0)

>o.

Note that \L{9) is given by the positve root of (9 + 1)A 2 - -(#• - 1 ) A - 3 . So if A < *L(0) then 5(0) < 0. Therefore the definition of \c(9) gives \L(9) < Ac for 9 > 1, since K{6) = inf{A > 0 : 5(0) > 0}.

W

References 1. I.Jensen and R.Dickman, Series analysis of the generalized contact process. Physica 4 . 2 0 3 (1994) 175-188. 2. R.Durrett and D.Griffeath , Supercritical contact process on Z. Ann, Probab.ll( 1983)115. 3. L.Gray, Duality for genera 1 attractive spin systems with applications in one dimension. Ann.Probab. 14 (1986) 371-396. 4. R.Durrett, Lecture Notes on Particle Systems and Percolations (Wadsworth, Inc., California, 1988).

9-Contact Process

99

5. C.Bezuidenhout and L.Gray, Critical attractive spin systems. Ann. Probab., to appear (1994). 6. R.C.Brower, M.A.Furman and M.Moshe, Critical exponents for the reggeon quantum spin model. Phys. Lett. 576(1978)213-219. 7. N.Konno and M.Katori, Application of the CAM based on a new decoupling procedure of correlation functions in the one-dimensional contact process. J.Phys.Soc.Jpn. 59 (1990) 1581-1592. 8. M.Katori and N.Konno, Bounds for the critical line of the 0-contact process with 1 < 9 < 2. J.Phys.A.26(1993)1-18. 9. D.Griffeath, Ergodic theorems for graph interactions. Adv.Appl.Probab. 7(1975)179194. 10. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985). 11. T.M.Liggett, The periodic threshold contact process.Random Walks, Brownian Motion, and Interacting Particle Systems. (Eds. R.Durrett and H.Kesten) Birkhauser, Boston. (1991) 339-358. 12. M.Katori, Reformulation of Gray's duality for attractive spin systems and its applica tions. J.Phys.A. 27(1994)3191-3211.

Phase Transitions of IPSs

100

CHAPTER 6 DIFFUSIVE CONTACT PROCESSES I

6.1. Introduction The diffusive contact process is a continuous-time Markov process on {0,1} . The formal generator is given by

a/fa) = £

r/(x) + (1 - i7(i))A x

]T

JJ(J0

1/(17.)-/(»»)]

!/:|y-j;|=l

+ 2dC £

IJ(B)(1 - iKlOM*. lOl/O?...) -

M l

where A > 0, D > 0, p{x,y)>0,

^ p ( i , ! / ) = l,

?(&>») = ?(»,&),

y

Vx(y) = »?(») if 3/ ^ *, V*(*) = ! - v(*). ^ d W 1 ) = ' / W i W » ) = 7 /( I )>'?x»( u ) = "?(«) otherwise. The first term of right hand side is equal to the basic contact process and the second one is the exclusion process (or stirring system) on Zd. As for the exclusion process, more details can be found in Chapter VIII of Liggett's textbook 1 and Example 5.4.3 of Chapter 5 in this book. In this chapter we consider only the nearest neighbor stirring mechanism : P(x,y)=

2^

for

| * - y 1=1.

So for the stirring mechanism, the following (i) and (ii) are equivalent : (i) for each i , j £ Z ' with | x — y \= 1, we exchange the values at i and y at rate D (ii) a particle at x waits an exponential time with stirring rate 2dD and then chooses y, i.e., one of 2d nearest neighbors randomly with probability l/2
Diffusive Contact Processes I

101

This process is attractive, so the upper invariant measure is well denned : for X,D > 0, V\,D = t—t-oo lim SiS(t), where Si is the pointmass on 77 = 1 and S(t) is the semigroup given by SI. As we saw in Example 5.4.4, the diffusive contact process is coalescing self dual. So for x £ f with f C Z d ,

£ -* £ \ ix}

at rat

f -y f U {y}

at rate A for | y - x |= 1,

£ -* (£ U {y}) \ {x}

e 1,

at rate D for | jy - x |= 1.

Let fj D denote the diffusive contact process starting from the origin. Define the survival event and survival probability by ftoo = {£?,D # <>/ f o r a l l t > 0 } , PkiOoo) = P(&iD f t for aU t > 0). Let fJ D be the diffusive contact process starting from 77 = 1. Then the order parameter and critical value are defined as follows : px{D) = U^D{T) : r]{x) = 1)

= Px(Slco), \C(D) = inf{A > 0 : px(D) > 0} = inf{A > 0 : P(0 £ ^ , D ) > 0} = inf{A > 0 : Pi(floo) > 0}. De Masi, Ferrari and Lebowitz2 and Durrett and Neuhauser3 studied mean fields the orem on more general interacting particle systems with the stirring mechanisms. In our setting, as for critical values and order parameters, the following results were obtaind by Durrett and Neuhauser.3 Theorem 6.1.1. Let d > 1. (1)

Xc D

(2)

Px{D) -+

M

^ ^2d cy J\ u x

-D->0°-

-i

as

P-+00.

102

Phase Transitions of IPSs

Roughly speaking, the above results can be obtained as follows. As in Lemma 3.2.4 (1), we have the same correlation identity even for D > 0 starting from n = 1 : (2
(6.1)

where e\ = ( 1 , 0 , . . .0) and = V\,D(V : n(xi) = 1 for all i e { l , . . . , n } ) .

p(xix2.-.xn) As D —► oo, we assume that

p{0ei) = KO) 2 .

(6-2)

That is, the adjacent sites are independent. This assumption means that mean field theory is correct for the diffusive contact process with rapid stirring limits. In other words, this limit has a product measure as stationary measure. Combining Eq.(6.1) with Eq.(6.2), we have ,„. 2dA-l

" W = /*»)=-MAAnother interpretation of the mean field theory is given in the following. To do this, we introduce ZfD which is a continuous-time branching random walk on Z d at stirring rate D starting from A. The dynamics of this process is as follows : (i) each particle dies at rate 1, and gives birth at rate 2dA, (ii) a particle born at x is sent to a site y chosen at random from 2d nearest neighbor sites, (iii) each particle jumps to a randomly chosen one of 2d nearest neighbor sites at rate IdD. We call this process the diffusive branching process. As in the diffusive contact process, we define the survival event and survival probability of the diffusive branching process by fioo = {Zt,D j=<j> for all t > 0}, P>(ft«,) = P(ZlD where Z\ that

D

?4>

for all t > 0),

denotes the diffusive branching process starting from the origin. It is well known 2d\ - 1

^n~) = I T ' inf{A > 0 : P - ^ c o ) > 0} = i . So in this meaning, as D —* oo, the diffusive contact process approaches to the diffusive braching process. Similar phenomena occur in the long range contact process as the range goes to infinity. Details about this can be found in Chapter 8 Long Range Contact Processes in this book. Moreover Thoerem 6.1.1 is analogous to results for the basic contact process with large dimension : 2dAc(d) —► 1 as d —> oo, , ., 2<2A - 1 M d PxW 2dX * ° ~* ° ° ' where Ac(d) and p\(d) are the critical value and the order parameter on the basic contact process on d dimension. As for these resurlts, for example, see Schonmann and Vares. 4 In this chapter, we present the following results which refine Theorem 6.1.1.

Diffusive Contact Processes I

T h e o r e m 6.1.2.

103

13

f(C/DW, C/D ' , d=l, x { C(logD)/D, — ^lc(logD)/D, d = 2, (C/D, {C/D, d>3, where ss means that ifC is small (large) then right hand side is a lower (upper) bound for large D. UD)-

T h e o r e m 6.1.3. Let

13

C/D ' , fCfDW, d=l, X~-=lc(logD)/D, d = 2, C(logD)/D, M {C/D, d > 3. C/D, There is a constant Sd(C) with Sd(C) -» 0 as C -* oo, so that if D is large, then 1

P,(D)>(l-Sd(C))2-^. Theorem 6.1.2 resembles a result of Theorem 1 in Bramson Durrett and Swindle 5 for the long range contact process which will[ be be studied studied inin Chapter 8 : 23

\C(M)

- l a

23 (C/M C/M ' ,' , C ( l o g M ) / M2,2 , C(logM)/M

d

[ C/M ,

d=l, d= 2,

d>3.

However Theorem 6.1.2 can not be obtained by an obvious translation : Md —► D. Remark that recently Katori 6 gave another proof of upper bounds for d > 3 in Theorem 6.1.2 by comparison with the binary contact path processes with stirring mechanisms. As for the binary contact path process, see Chapter IX in Liggett. 1 From the physical point of view, Jensen and Dickman 7 studied the one-dimesional diffusive contact process by using the series expansion and Monte Carlo simuilations. Their results suggest the stirring mechanism does not change the critical behavior of the diffusive contact process. T h a t is, if we assume that there is a critical exponent /3(D) so that as A I XJD). D D px(D)~(\-\c(D))MK{D)f< \ \ then P(D) = fl « 0.277

for any D > 0,

where j3 is the critical exponent on the basic contact process. So their results supported that /3(D) is independent of D > 0, and a family of the diffusive contact processes belongs to the directed percolation universality class as well as the ^-contact processes. In the language of statistical physics, Theorem 6.1.3 means that a crossover occurs from contact process to branching process. When C is large, the survival probability of the diffusive contact process is the almost same as that of the diffusive branching process for large D. As for crossover, for example, see page 98 in Durrett 8 and Amit. 9 Here we present the following interesting conjecture which was suggested by Jensen and Dickman. 7

104

Phase Transitions of IPSs

Conjecture 6.1.4. If 0 < D < D,

then XC{D) >

\C(D).

This conjecture is not trivial. Roughly speaking, particles are more spread out, the process is more survival. From the biological point of view, Matsuda et a/.10 studied the diffusive contact process as a model of population dynamics of interacting species of organisms, where the stirring rate D means the migration rate of each individual. This chapter is organized as follows. Section 6.2 is devoted to some computations on critical values and order parameters on finite sets in one dimension. Section 6.3 is devoted to preliminary results. In Section 6.4, we will present the proofs of Theorem 6.1.1 (1) and upper bounds for any d > 1 in Theorem 6.1.2. The proofs of Theorems 6.1.1 (2) and 6.1.3 will be given in Section 6.5. Note that the proofs of lower bounds in Theorem 6.1.2 will be found in the next chapter. Most of the results in Chapters 6 and 7 appeared in Konno. 1 1 6.2. Diffusive Contact Processes on Small Finite Sets In this section we study the diffusive basic contact process 7 7 ^ with infection rate A and stirring rate D on state space 5' n * = {0, l}^1'•••■"} for small n. Then for a site x e { 1 , . . . , « } and a time t > 0, r]t(x) takes values in {0,1}. The value 0 is regarded as a healthy individual while 1 is regarded as an infected one. The dynamics of the evolution is described by the following transition rate : at site x £ { 1 , . . . , n}, 0^1

at rate

\{q(x

1 —> 0

at rate

1,

- 1) + T?(I + 1)},

where A > 0. Moreover for n > 2, we exchange r)(x) and 77(1 + 1) with x e { 1 , . . . , n — 1} at rate D. The boundary condition is 77(0) = 77(71 + 1) = 1 as in Chapter 2. Let ftp be stationary distribution of the diffusive contact process 7 7 ^ .

6.2.1. n=l First we calculate the stationary distribution when n = 1. 77J p is a continuous-time Markov process on state space SW = {0,1} with generator G$

V1)(0,0)

5 ( 1 ) (0,1)1 _ [-2A

y«(i,o) ^>(i,i)j - [ 1

2A

-1

where A > 0. In fact, this case is independent of stirring rate D and equivalent to the basic contact process on {0,1} (see Section 2.2.1.) So we can get immediately the stationary distribution u ( 1 ) = IMS >>(O),M8 ,,(I)I, v® L

2 A + 1 ' 2 A + 1J'

Diffusive Contact Processes I

105

The density of infected individuals p* (D) = ^ ' ( l ) is taken as an order parameter of the diffusive basic contact process when re = 1. So we have pW{D) Hx K

'

=

- " 2A + 1

6.2.2. n=2 Next we calculate the stationary distribution /xg' when n = 2. r^2^ is a continuous-time Markov process on state space S^ = {0, l}^1'2} with generator -2A A A 01 1 -(2A + 1 + D) D 2A Gg» = 1 D _(2A + 1 + I>) 2A 0 1 1 - 2 . So we have

/#> = D«g ) (oo),Mg ) (oi),/ig , (io),^ , (ii)] 1 A A 2A2 2 2 2 L2A + 2A + 1' 2A + 2A + 1' 2A + 2A + 1' 2A + 2A + 1 In this case the order parameter is denned by 2

/£> = /#{*!) = Mg»0*). where ^2)(*i) = ^ ) ( o i ) + ^ ) ( n ) , M ( D'(1*) = ^ ( 1 0 )

+

^(H).

Therefore we get

2A2 + A ft'W 2A 2 '+2A+l' In fact, this results are equivalent to the case of D = 0. See Section 2.2.2. „(2>f

=

6.2.3. n=3 We calculate the stationary distribution ir(3) D'. As a result of fact the n = 3 case is very different from the previous re = 1,2 cases. Because the stationary distributions for re = 1,2 are independent of D, however this one strongly depends on stiiring rate D.rf^j^is a continuous-time Markov process on state space S^ = {0, l}*1'2-3) whose generator G^' is given by [9(0,0) 1 1 0 1 0 0 . 0

A 5(1,1) D 1 0 1 0 0

0 D S(2,2) 1 D 0 1 0

0 A 2A •7(3,3) 0 D 0 1

A 0 D 0 5(4,4) 1 1 0

0 A 0 D A 9(5,5) D 1

0 0 2A 0 A D 5(6,6) 1

0 0 0 2A 0 2A 2A 5(7,7)

106

Phase

Transitions

of

IPSs

where ff(0,0)

= -2A, S (1,1) = -(2A + 1 + D), g(2,2) = -(4A + 1 + 2D), g(3,3) = -(2A + 2 + D),
So we have

^ g , = tMg ) (000) > /ig ) (001),/ig ) (010) > /ig , (011),/*g ) (100),^g ) (101), / ig , (110),/*g ) (lll)] 1 ~ Z<3>(A,D) [20A2 + 21A + 6 + 3D{16A + 9 + 9D}, A[2(8A2 + 9A + 3) + 3D{12A + 7 + 6D}], 2A[A(4A + 3) + 3D{4A + 2 + 3D}], A2[(4A + 1)(4A + 3) + 8D{5A + 3 + 3D}], A[2(8A2 + 9A + 3) + 3D{12A + 7 + 6D}], 2A2[(8A2 + 8A + 3) + D{20A + 9 + 12D}], A2[(4A + 1)(4A + 3) + 8D{5A + 3 + 3D}], 4A3[2(2A + l) 2 + D{20A + 11 + 12D}]], where Z(3)(A, D) = (2A + 1)(16A4 + 32A3 + 32A2 + 21A + 6) + D{J(A) + J(A)D}, /(A) = 80A4 + 164A3 + 162A2 + 102A + 77, 7(A) = 3(16A3 + 24A2 + 18A + 9). (3)/

\

(3\

The order parameter p\ (D) = fj,D (*1*) is given by P ){D)

^

=

z(3)(A,D)

where

[4A2(2A + 1 ) ( 4 A

' + 6A + 3 ) + 2AD{if(A) + L(X)D}]

K(X) = 2(20A3 + 31A2 + 18A + 3), 1(A) = 3(8A2 + 8A + 3).

6.2.4. Summary For n = 1,2,3, we have the following results. Proposition 6.2.1. (1) (2)

p[3)(D)
forA,D>0.

and lim p^(D) = 1 for i = 1,2,3. A—t-oo

Diffusive Contact Processes I

107

»Wc ti>(D) = J» X>(0), PWD) = P\»(0)

(3)

P(x\D) < pf{D>)

(4)

for D < D'.

In particular, 3 „<3> p\"(0) < #,(' (h« > ) .

(5) with Pl3)(0) = 0) Hx

= K

'

4A2(4A2+6A + 3) 16A + 32A 3 +32A 2 +21A + 6' 6A(8A2+8A + 3) 3(16A3 + 24A2 + 18A + 9)' 4

6.2.5. General case In this subsecton we think of the general case. To do this we summarize the stirring -.(n) n term DGT> - DG-.(») d J of G%> for n = 1,2,3. <#> = Ml) =

2

G< > =

0 0 0 0

0i 0 0 OJ

0 0 0 1 1 0 0 0

G<3> =

0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0

0 0

J

rO 0 0 0 i

0 0

Lo

1 0 0

0 0 1 0 0 0

0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 10 0 10 0 0 0 0 0 1 0 0 1 0 0 0 0 10 0 0 0 0 0 0 0J

oooooooo0 10 0 0 0 0 0 0 0 2 0 0 0 0 0 ,(3) _ 0 0 0 1 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 10 . 0 0 0 0 0 0 0 0 . Prom the above concrete examples, a little thought gives the following simple relation between n and n + 1. Let 0(n) be an n X n zero matrix and I(n) be an n x n identity matrix.

Phase Transitions of IPSs

108

Lemma 6.2.2. For n > 1, G(n+1)

G<"+!) _^ s

,(n+l) __ —

Gds

0(2") .0(2") 0(2")

G<">

bds ^ds

0(2")

.0(2") 0(2") M R<

0(2") R'(")

+

(n)

+

# [0(2-)

•"-3,12

0(2") 0(2")

R?(n) s[22

0(2"- x )

0(2"- 1 )]

/(2"- )

0(2""^ J '

1

0(2""!) R 0(2"- 2 ) '0(2"" R («) _ 0(2"eis.ll — '/(2"- 1 ) R' (») _ 0 ( 2 " - ' ) 0 Gl1' 0 (n)

<=£? =

i

/(2""1)] 0(2"-!)] ' 0(2"- x )l

/(2"- 1 )] '

0(2"-!)' 0(2"-!)] ' 0' 0

GQ is generator of the diffusive contact process on state space 5'"'. Combining The orem 2.2.1 and Lemma 6.2.2 implies Theorem 6.2.3. For n > 1, G(Dn+1) = AG - Gdn+1) + £Gl" +1 > where

DGdns+1\

-,(«) 0(2") R{n) Gj"' ■"■i.ii ■"^12 ' L0(2") G\n) 0(2") 0(2")J ' G(rn) 0(2")' )' G(«+D 7(2") G(rn) . i (n) -.(«) 0(2") i?d,ll 0(2") G(n+1) = (n) n + 0(2") p ( " ) t 0(2" 0(2"-!) - / ( 2 " - 1 ) D(") 0(2""!) 0(2"->) /(2"" 1 ) 0 ( 2 " - ' )) "' ?(«) 0(2"-!) 2/(2"-!) . [0(2"->) 0(2"-!) D(") -"•d.ll [0(2"-!) 2/(2"- 1 )J' R (") _ 7(2"),

G(n+1)

)J' )J'

Diffusive Contact Processes I

109

_ [ GiB) 0(2")] [0(2") R% ■ 1.0(2") G<"> J T [ £<»\ 0(2") J '

G (n + i)

G (n+D =

GS'

0(2")]

C41

0(2")

L0(2") Gg> J + [0(2") 4») M ' r-d) - [° 2 1 r>(i) - [° °1 rW - \2X ° ' [o o j ' w ~ [ I oj' Urf _ [o l j ' G(i)

6.3.

_

G(i)

_ [0

0'

Preliminary Results

From now on we change the space from Zd to ZdD = Zdj\[D with D > 0. Note that this change does not have any effect on critical values and order parameters denned in Section 6.1. First we introduce some notation. For z e Rd define || z || p = {\zx\T -\ + \zd\'')1/p with 1 0, let « ? ( * ; r ) = {» € Zfe :|J 9 - x || p < r } , Q?(u;r)

=

fl°(*;r)

= Q°(x;r)

\ {x},

{ye1ld:\\y-u\\p
When x is the origin, we omit x = 0. For example, Qp(r) = Qp (0;r). Let 77;^ denote the basic contact process at stirring rate D with state space { 0 , 1 } Z " at each site x E Zfj, 77<,£>(x) takes the value 0 or 1. They evolve as follows: (i) there are flip rates c0(x,T}) = 1,

ei(x,77) = A K n O f ( z ; - ^ = ) | D , so that if 77(,D(X) f i then P(r)t+s,D{x) = i | Vt,D) ~ «={(*> r)tjD)s, as s -» 0 where / ( s ) ~ 5(3) means f(s)/q(s) —► 1. (ii) for each i . j e Z j with || x - j / | | i = 1/-/D, we exchange the values at x and j / at rate D. This process is the diffusive contact process on state space ZdD. From now on we call it DCPior short. Let £, n be the set-valued process for nt n on Zdn :

ft,D = {x £ Zl, : % ) D (*) = 1}. Sometimes we identify r)t,D with £t,D- Let ^ D be the DCP starting from § £ D = A and = £?D(B) l £t?£>n -S lo where AcZdD and B C R d . Moreover we introduce Z t ^ D , a diffusive

Phase Transitions of IPSs

110

branching process on Z£, at stirring rate D starting from A. We call this process DBP for short. The dynamics of the DBP is as follows: (i) each particle dies at rate 1, and gives birth at rate 2 ^ <j> for all t > 0} where $D denotes the DCP starting from $ D = {0} ( i.e., {*D when A = {0}). We define the critical value AC(Z>) = inf{A > 0 : P(iloo) > 0} and the survival probability ( order parameter ) p\(D) = P(fi <x) ). It is well known that the limits l/2 for all t > 0}, so l/2d = inf{A > 0 : PCiT^) > 0} and (2dX - l)/2dX = P(Qoo) where Z?D denotes the DBP starting from Z% D = {0} (i.e., Z*D when A = {0}). Our objective is to give the proofs of the following theorems. T h e o r e m 6.1.1. (1)

(2)

XC(D) -+ ^

px(D) -

as D -» oo.

^ ~ ±

as D - oo.

T h e o r e m 6.1.2. For x > 0, let ( 1/x1'3,
{ l/x,

d= 1, d = 2,

d > 3.

Then we have

U D ) - 55 * C
T h e o r e m 6.1.3. Let 2dA - 1 = C(l-6d(C))™±=^.

Diffusive Contact Processes I

111

Prom now on we present several preliminary results. Let S*'" (resp. S*'Q) be a continuous-time simple random walk starting from x in which a particle jumps to a ran domly chosen neighbor at rate 2du on Zd (resp. ZdD). Define pAf,x,y)

= P(Sr

= y)

and

p^(t;x,y)

= P(S^

J»„(«;*,B) = $ > „ ( * ; * , » ) USB

and

p f (*;*,£) = £ j,gB

and

p f ( 0 = ?f(*;0,0).

= y).

Moreover for B C R.d put p°{t;x,y).

For simplicity we define P„(0=P.(*;0,0) Note that P„(t) = P ? « For n = ( m , . . . , n d ) £ Z d , l e t 5 ° = [Ui,m + 1)X ■■•X [l»4,»d + l ) , S i = K - l , w i + l ) x - - - X [ n i - l . j i d + l). Let g f (*; 5 ) = (| B \D pD{t)) let

m A 1 where a A Aft6 denotes den tes the minimum of a and i>. For x > 0,

(

,T/X, /i, ld(x) = 3 . d= = 22,, d>3.

| logs The following facts are very useful for our proofs.|, 1, The following facts are very useful for our proofs. L e m m a 6 . 3 . 1 . For v > 0,2? > 0,i > 0,x,y £ Z d , u , u € Z j , and A C R d , (1)

(2)

(3)

p^(i;ij,i;) = p 1 (i/i;-v / I?u 1 'v / Z?i;),

P.(*J»,»)^A(*).

P?(f,*,y)
*?(*;«, A) <«?(*; A).

For A > 0 and t > 0, there is a C > 0 so that if D is large, then (4)

&„(.+,***)
+ t)*fi,

112

Phase Transitions of IPSs

fp^D{s)ds
(5)

Proof. (1) Definitions give p?(t;u,v) = P(S°t$ = v - u) = P(S°'" = VD(v - u)) =

P^VD^VDV)

= P , (p^\ I/<;\/DU,V^«). =

(2) Here we will give two proofs. Let p*-n\x, y) denote the probability of an ra-step transition from x to v on Zd with M)(x v) - I 111/2* V \x^)-\^ \

if||*-»||i=l, otherwise.

Then it suffices to show that Lemma 6.3.2. Let —,p(n)(x,y)

Pt(x,y) = e"' V n=0

be the transition probability for continuous-time simple random walk on Z t > 0 and x,y e Zd, ft(*.»)
Then for any

Proof of Lemma 6.3.2. Using the inversion inversion formular for characteristic functions, 3ing tne ic p(n)(z,y)

(2T)"

4

e -«'<*-!'M[0(0)]»d0 l

(

where

m = J2 Pm(o,xy*-B = \ (x>s* fc ) , x€Z d

\Jfe=l

12

see, for example, page 13 in Lawler. gives

For simplicity, we assume d — 1. So the last equality

pt(x,y) =£-* r i

= — / Tix-y

/

r

e

^re-^-y^±^rde

-t(l-!/)6

coae

cos(x — - y)0( y)ee-'^-

'>de

= 0 then cos(i - y)9 = 1, so pt(x, y) < pt(x, x) = p ( (0,0).

Diffusive Contact Processes I

113

Next another proof is given as follows. Erom Markov property and the Cauchy-Schwarz inequality, we have Pt(x ,y) ==

J^P^fa,z)pt/2{z V

,y) 1/2 \v 1/2

1/i

>f(?

*(? s

\Y,pt/t(z

=

pt{x,xfl2pi(y,y)1/2

2

\Y,Pt/2(z>M*iV?

)

= P,(0,0).

(3) To prove this, from (2) we see that

= E*f(*s«,y)

pf(t;s,vl) =

y£A

= [x;rf(*5-.»)ji A l VveA \VSA

y

?(')) A l = (] A | D pf '(f)) A 1

= ??(M). (4) Note that V So we see that

<:

m< Wi-

c,(*;#) == (l*i[»*?«(*)) =

Al

(1 * ; l„ J W W ) A i

< (C£>"/ 2 P I ((> + ^ ) < ) ) A 1 <

w^ri)"

< C <

c (1 + tyl* '

Phase Transitions of IPSs

114 (5) Observe that

D j\i((X J\f+Dv(s)ds ,(s)ds: = J\ 1((\>,((A + D)s)ds

Jn

Jn

=

A\+D)t

1

/

p,(u)du

A + 2Dt Wo ^ du 1i_ r /.■an* U'U - 5DJ / 00 (i + uyn uyn

;

'

d

For A,B C K ,x,y

€ Z D, and fc = 1 , 2 , . . . , define

mf (<; A, A, B) B) = m£(<; = M{$ £(£&>(£)*), D{B)»), "»?(*) = ^(ff.u(Zi,)*),

mff (t; (*; A, A, B) B) *= = £(Z&,(B)*), m £(Z&(5)*), mf (*) = £ ( ^ ( Z j ) * ) .

In particular, we omit the subscript when k = 1. Moreover,

JD(<;x,a;y,6) = £ ( ^ i D ( a ) ^ D ( 6 ) ) ,

ID(t)=

°(t;x,i v,x,b),

/°(*5*,ai»,4) = £(ZflD(a)Z» (6)), /D

aezj

w= E £A*;».«»?«.*). aezj,

S>~a

where 6 ~ a denotes 6 is a nearest neighbor of a on Z £ , that is, b e <2f (a; 1/-/D). Note that mj~'(<),mj?(i),/ D (2) and / (i) are independent of i . On the other hand, £fD and Z*D can be constructed on the same probability space with

£&(*) < « & W So the above definitions imply that for A,B c R^x,?/,!!,!) 6 Z ^ f c > 1 and t > 0, we have m%(t; A, B) < m<m%(t;A,B), m°(t\A,B) f (<; A , B ) , IIDD(t;x,a;y,b)
l°(t;

(t;x,a;y,b), x,a;y,&),

(i) < mDD(t) < mD(t), D

/ID (t) (i!) < < 7 Dr(t). («).

Moreover noting that

| p ? + D (*;*.») = (A + 0 ) £ *f+0 (*;*•» + 2 ) - 2 < A + ^)pf+D (*;*.»). zz~0 ~0

we have

Diffusive Contact Processes I

115

L e m m a 6.3.3. (1)

5RDc (i; X .») = e (2t ,(*; X ,y). ™ (t;x,y) = eW-V p?+D(t;x,y).

(2)

^ D( f ; ! ,B) =-. e^dx- i)tpD t ,(«! a;, 5 ) . m (t;x,B) = eW-V p?+D(t;x,B).

(3)

l)« mDD(t) = e (2
Hence we have immediately L e m m a 6.3.4.

o<

-l)t 0 <m mDD(t) {t) < mDD{t) (t) = = ee(2*X("*-i><.

So this result allows us to conclude that that 1/24 l/2d is the the trivial lower bound on on X ACfl{D) (K) : Lemma 6.3.5. Xe(D) >

1 2d'

The following results will be used to analyze various behavior of the DBP. Let Y denote the set of all real-valued bounded measurable functions on Z^,. In particular, for a £ ZD,Sa £ Y is defined by Sa(a) = 1 and Sa(b) = 0 if 6 ^ a. For each f £Y, let

= J2 z?.D(y)f(y)vez%

L e m m a 6.3.6. For f,g G Y, x,a,b e Z £ , B C R d , D > 0, \,t > 0, p > 1, r > 0, and K — 1,2,...,

(1)

£(,/> fc ) = £()

+ A E f*) E E (2)

/ ' ^ « ZU,DA >)E(< ZlD,f>')E« Z*s%*,f >*-*),

E(< ZlDJ>< ZlD,g >) = E{< Zfojg >) + A E E fd*E«

Z?_BtD,6v >){E(< ZlD,f >)E(< ZJ**tg >)

Phase Transitions of tPSs

116

+E{)E{)}, / > ) } ,

(3)

m°(t;x,B)

=

mD{t;x,B)

k-\ A

+

IT()]C]C

TD(t;x,a;x,b)

(4)

/

ds

mD{t-s;x,y)mf(s;y,B)m%_j(s;y

+ z,B),

= 6a(b)E(Z?D(a)) + \Yi Y] / ds mP mD(t-s;x,y) '~"vez"D x {mD(s; y, a)mD(s; y + z,b) + mD(s; y, b)mD(s; y + z, a)},

EE' D j2127D(t'x'a'x^^e(2d^1)tp^D(t'^AnB^

(s)

+MAe 2<M*-i). +MAe 2<M*-i).

| p D D f t , , A) jf g?+D(a; S)ds + pf +D ft*,!>) jf «f+B(a; A)d. j , | p D D f t , , A ) j f g f +D (a; S)ds + pf +D (*;*,B) jf' ,f + D (,; A)ds} ,

E E E E

(6)

(6)

»eZi a

7D(t;x,a;x,b) < 4dXe^2dx-^ \\Q?(r)\D ft & » « * • Jo b€Q?(a;r) ^ f t x . a ^ . f c J ^ M A e ' ^ - ^ I ^ W L / V ^ W ^ -

eZ J D 6€Qj> (o . r)

Proof. (1) To begin, the branching property and strong Markov property of the DBP imply £(exp{-}) = e - < 2 ^ + 1 ) ^ ( e x p { - ^ / ( 5 ^ ) } ) 11 __ +

-(2dA+l)t ££ -(2dA+l)t

2<2A + 1

(,W+1) ) 3 x + f*» *X + / * E 2J *?(»!*. ??(■!*» »>" *>■*

A E ^ x p { - » < Zl < 2Z^^D,f, / >}). >}). Z(VStD _SJ,W >})E(ex >})£(eXp{-0 P{-9 < z~0 2~0

Differentiating k times w.r.t. 0 and letting 6 —> 0, £(< Z * D , / >*) = e - ( ' «e-W+V*E(f(SZ£) + ' ) « B ( / ( 5 ^ ) *k)) (W1) ^ ( t _ i ; S i V ) e - rf('-^,*(»«+i).A' 'A^(^£(<^,/>')£(*^). vezi i=i u / 2~o

+ /** E /0 •

Diffusive Contact Processes I

117

So we have the desired result. (2) Applying part (1) for k = 2, so we have

E(*) = E() +2AE E /

dsE

(< Z?-s,D,Sy »E(< ZlD,f »E(< Zftj

»•

Hence we see that E() = \ {E{< ZlD,} + g>2) - E(< ZfiD,f = \ {E(< Z?iD,(f + gf >) - E(< Z£D,f

+A E E

> 2 ) - E(< ZlD,g > 2 )} >) - E(< Z^g2

>)}

fdsE{< ZU,D,h >) X {E{< ZlD,f + g>)E(< Z%j',f + g>)

- E« ZlD,f»E(< Zffsf >) - E(< ZlD,g »E(< Z^D\g >)}. So the proof is complete. (3) This is the special case of (1) for f — lB where 1B is indicator function of set B;= 1 on B , = 0on Bc. (4) This is the special case of (2) for f = Sa and g = St. an< (5) Combining Lemma 6.3.3, the Markov property of Sfb ^ Lemma 6.3.1 (3) implies

A

EEE E /

oeX 66B z~0 j , e z £

< e 2(2dA-i)t A £ £ <2dXe^dx-^pf+D

ds

™D(t-s;x,y)™D(s;y,a)mD(s\y

+ z,b)

J

°

fds

P?+D(t-s;X,y)p?+D(s;y,A)p?+D(s;y

+ z,B)

(t; x,A) f 9f+D (s; B)ds. Jo

The rest of the second term is also obtained by a similar argument. So part (5) follows from part (4). (6) Using Lemmas 6.3.1 (2), 6.3.3, the Markov property of St£+D, and part (4), the desired conclusion can be obtained in the similar way. In the DBP, it follows from Lemma 6.3.3 that

Phase Transitions of IPSs

118

L e m m a 6.3.7. jmD{t) '« = == (2dA (2dX -dt

V)mP{t). l)mD{t).

So we are interested in the term ID{i) which appears in Lemma 6.3.8 (1) but not in the last differential equation. We call this term the interference term. Lemma 6.3.8 (1) will be used for the proofs of lower bounds in Theorem 6.1.2. In particular, Lemma 6.3.8 (2) will be useful to have more detailed lower bounds for d > 3 in Theorem 6.1.2. L e m m a 6.3.8. For t > 0, 2dx - 1l)m XI (t), = ((2dX jmD{fi = ~ ) mD{t) (*) - - XI °Ct), 5»°« D

(i) (2)

&>m) << (2dX - l)ml)m(t){t)) --

^mD(t) dt

D D

D

2 4<2A + i++l +( 2( 2i -di-)l )f C l ))((tt -- »» ))mr oDD( a()s<) fdas_. 4 d A 2 ff* ee --2V( A JO Jo

Proof. By the definition of the DCP, we have

jtn* ^

= -P(x -PCx e€ tf rfiDD)) + A5] P(v e fa) )-xJ2PCx,ye^ e6 ilo) - A £ P(x,y e %D & ,)) &>)= y~X

y~x

+D E P{y P(y ee & etlD, )) -- 2d£>P( 2dDP(* &,). a; ee fa). y^x

Summing over a; gives the last two terms cancel, so (1) is obtained. Let e\ = ( 1 , 0 , . . . ,0). Similarly we obtain j-P( +ei ei£e$l tf^,) {P(x 6 £ ° D ) + ( s ++e i eele& & , J) ) )} } --2{X 2(A + + l)P(x,x l)P{x,x fP(x,xx,x + =A*{P{xe8j,) + PP(x D) =

+ A{ + M ^E

P (* + ,yeffn; P{x + eeuiye8j,)+

y~x y^x+e! y^x+e,

- AA

E

+ & t,n)) +d £ e1Et°

P{*,VG&J>)} EE n*,vs&j>)} + j~x+ei y~x+ei v^x yjtx

P (s,i + P(x,x + eeu1y, j /ee ^Q£ D) )

Yl

y~{x,x+e!} y~{x,i+ei]

+ £{ E E +!>{

> iP(* (i + + ei,yee?,r>) ei,ye$,D)++

y~x y^x+e! y^x+ei

EE y~x+ei y^x

^nx,yetl . > £ ^D))}}

--2D{2d-l)P(x,x 2D(2d - 1} PCx,x ++ ei ee£ ?e, t,D) ,), ei D {x,y} means ao ~~ ii, ,a a/ t/ /j o r i ai ~~ jj/,a where aa ~~ {s,3/} , a // i i. . Therefore, we have
/ D ( t ) = 4dAm D (i) - 2{A + 1 + (2d - l)D}ID{t)

+ fi(t),

119

Diffusive Contact Processes I

where

*w = 2 E E

E

{ A p ( s *$J» X + z >y e «?.D)+ D p i*+^e«?,D)}-

Noting ID{0) = 0, and #(<) > 0 for any t > 0, we can easily get (2) from (1) and the last equation. We need the following lemma to estimate the difference between the DCP and the DBP. Part (1) will be used in the proof of Theorem 6.1.3. Note that in part (2) a time t is fixed, so we apply this part to the proof of Theorem 6.1.1 (2). Lemma 6.3.9. Let A?i£) = Zf^Zf,)

(1)

-

g,D{ZdD).

E(AlD) < S d W ^ - 1 " J* Qf , £ » * • ) *■>

(2)

lim E(AlD) D—*-00

= 0.

'

Proof. The definition of Af D gives ~E(Af, D ) = XID(t),

E(AlD)

= 0.

So we have = \ f ID(s)ds. Jo Part (1) follows from the above equation and Lemma 6.3.6 (6) for p = r = 1. For part (2), using part (1) and Lemma 6.3.1 (5), if D is large, then E(AlD)

E(AlD)

< 8d s AV(« > - 1 )«( <

Jo

l\°+D{u)du

Cid(Dt)/D.

With t fixed, -)d{Dt)ID -> 0 as D -» oo, so part (2) follows immediately. 6.4. Upper Bounds In this section we prove the upper bounds for any d > 1 in Theorem 6.1.2 by using comparison method which was studied in Section 4.3. That is, DCP £t,D —* oriented site percolation Cn —* DCP ft,D-

Phase Transitions of IPSs

120

To do this, we need some notation. Let Ki,K2,K3, will be chosen later. Define

and K^ be positive constants which

s = U\ - 1,1 = K2/e and Ij = [2j - l,2j + 1] x [-l,l] r f _ 1 , where ;' £ Z 1 . Let £t
6 B).

Proof. Let P?+D(f,x,y) = P(Sxt;n+D = y). From the definition we have 4 ? f U (*; x, y) = (A + D) £ \z..\yi+zA<2VL)P°+D (t;x,y + z)-2d(X

+ D)f°+D (t;«,,).

z~0

So (2dA - 1)£(Z?, C «) +

e ( 2 " A - 1 ) t |pf + D {f,x,y)

= (2dX -

l)E(%j,(y))

+ e^-V< [(A + D) £ lWyi+il|<2VE)Pf+D(*; «,» + *) z~0

-2d(A+ £)?£„ (*;*,„)]. On the other hand we have d A W - t a K i ^ ^ f r + *)) - E(Z*t,D(y)) dt E{X,D{y)) = E z~0 +

D

1

E

( Z : | V ) + , 1 | < 2 V T ) £ ( < P ( 2 / + »)) - 1dDE{Z°t,D{y))

z~0

=

(2d\~i)E<X,D(y))

+ (A + D) J2 \,:\vl+zl\<2VL)E(KD(y z~0

Therefore the desired conclusion is obtained.

+ ')) ~ 2
Diffusive Contact Processes I

121

Let Bf be a Brownian motion with mean 0 and covariance tl starting from x on Rd. The Lemma 6.4.2 is obtained by the local central limit theorem. As for this theorem, see Durrett, 1 3 Section 5 in Chapter 2. For convenience to readers, we present the local central limit theorem for simple version, that is, discrete-time in one dimension as follows. Set h > 0. We let X i , . . . , A " n be i.i.d. denned on hZ = {hz : z 6 Z} with EX-y = 0 and E(Xf) = a2 e (0, oo), that is P(X1 e hZ) = 1. Define S ° ; f = Xj + ■ • ■ + Xn. Let W?-°* be a random variable which have the normal distribution with mean 0 and variance a2. Then Local Central Limit T h e o r e m . As n —> oo,

p($g E VSt-i,iD - P(w?y e i[-i.iD: From this theorem we have the following result. L e m m a 6.4.2. Let A C Rd. If x/y/i -» x* and i/(t) -> i/ as t —> oo, then Km P ( 5 ^ ( " e VtA) = P ( 5 j ; D £ 4 ) , wiere A/*A = {v/ix : x 6 A] C R d . Proof. For simplicity we assume that A = [—1,1] and d = 1. From the local central limit theorem and the definition and property of Brownian motion, we have

P(S*£(/) e Vt[-i,i}) - P(B?-" e

VB[-I,I])

R0,i/

= P(^e[-i,i])

=4£K11)

= P(B^e[-i,i]). The last equality comes from the famous scaling relation for Brownian motion (for example, see page 334 in Durrett 1 3 : The Brawnian Scaling Relation.) That is, for any a > 0, {^L : t

> 0} 4 {B0/ : t > 0},

where = means that the two families of random variables have the same distributions. Note that Lemma 6.3.3 implies

E(ZltD(VLA)) = eK*P(,Sl$D 6 VIA). So using Lemma 6.4.1, we have the following corollary, since »M = * + D = £-±r + D->v=-

+

D

as^O.

finite-dimensional

122

Phase Transitions of IPSs

Corollary 6.4.3. Let A C Rd. For any D and K2 > 0, if L = K2/s and x/-JZ -* x* as e -* 0, then Um o B(Zf iB (vTA)) = eK*P(B$dD+1)/2iD E A). On the other hand, we have the following lemma. Lemma 6.4.4. Let L = K2/e. There is an £0 > 0 so that ife E (0,eo),K2 > 0 and j = ±1, tien inf P(BZ*DD E sfllj) > p/2 > 0. Proof. Let Bt denote a modified Brownian motion that starts at x and is killed when it leaves (—2,2) x R d _ 1 . As in the Corollary 6.4.3, we obtain that for any D and K2 > 0, if L = K2je and x/\/L —> x* as e — ► 0, then for j = ± 1 , l i m P ( ^ + D e VLI}) = P(EfldD+1)/2dD

E I,).

Note that if e E (0,e0) and D > 1, then 1 < (A + D)/D < {2d + 1 + e0)/2rf. Define P = j ^ f -P(B^D+l)/2dZ3 6 ^') > 0, £>>1°

so the proof is complete. Let V^D(.B) =
E{viD(VLh)) = Y, P&°

6 VZ/i) > f I A J,

On the other hand, using Lemma 6.3.6 (3) for k = 2 and I? = Z£>, we have ^ ( ^ D ( Z D ) 2 ) = E{ZlD{Zi)) + UX f

E{Z^D{Zi))E(Z*D(Zi)fdS,

Jo BO

E(Z?,D(Z%)) = e " gives E{ZlD{ZdDf) = e " + 4dAe" / ' e"ds Jo _ c 2 . f f 2 ( £ + l)

e + 2 c -««1

(6.3)

123

Diffusive Contact Processes I

Hence

e~**LE{ZlyD{VLhf) < e-icLE{ZlD{ZdDf) < ?i£±i> -

^—e^'.

Therefore if 0 < e < 1, and we write var(Y) for the variance of Y, then

vw(V£iD(VZ/i)) < X) ^(^,o(^^) 2 ) <EB(V^(VLii)») <EBCVZ,^(VLi'1)») xPA

<| A |D <\A

e~ieLE{ZlA-ZIlxf)

,„ { *±i)_i±i«-. 1

<^]AI Using Eq.(6.3) and Chebyshev's inequality now it follows that if | A j D > Ki/e, then P(VUVLII)<^-)

P\A 64

- £P2 I A \D ■ Taking K\ and K2 so that 64/(/>2-fiTi) < 6/2 and exp(-fiT2V>/4 > 3, we see that A 3 ^1 _<
g E

~

4 4

4 4 '

{^(VI/x)<^} c\zU^h)<e^-^\ {VUVLI )<^]. ={nD(v^o<^}= So

p{zUsLh)

1

< *?L) 1 - P(AC) - P{BC), we get .-ft", > 3-^,

P(ZLA^W

j = ±1) > 1 - S.

That is, we have Proposition 6.4.5. For any S > 0, there are e0 £ (0, l),D0 > 0, and .6"; > 0, i = 1,2 so that if e e (0,£0),.D >D0,L = K2/e and A C VX/o witi [ A |„> -ffi/e, then P ( Z £ o ( V T j » > MTi/e,

j = ±l) > 1 - * .

The following lemma will be used for the proof of Proposition 6.4.8 . Lemma 6.4.6. There are £0 £ (0,1), -Do > 0, and K{ > 0, i = 1,2 so that if £ e (0, £o), £ = K2/e,D > D0 and A C %/iio with I A | D < 2iTi/£, then E{ZiD{B\f)

<

Proof. To begin, note that E(Z£>D(B\)2) ioT A = B = B\ and t = L implies *?(Z;s,J?i) = £

£

CeV'-^l/e).

= mf (£; A,Bj) bY definition. Lemma 6.3.6 (5)

/ D (£;x,a;*,6)

o6BJ 6£BJ K 2 /e

<**'pf + 0 (tf»/e;*,*i)

l + 2(l + £ ) e ^ / Jo

gf +D (s;i?i)d

(6.4)

By Lemma 6.3.1 (3) and (4), we have P?+D{K2/e;x,Bln)Bl)ds
(6.5) (6.6)

(6.7)

125

Diffusive Contact Processes I

By Eqs.(6.5), (6.7) and | A \D< 2.8'i/e, we obtain l m?(L; A,B\) = £ m ? (?(L;x,B m?(£;A,fli) £ ; x , f l in)+ )+ XGA

D J2 £ ™ *(£;« ™D<,^K (L;x,Bl)m (L;y,Bl) *(J-,i,*i)

x,yeA x,yeA x?V dd 22 ydd(l/e) (l/e) Dee // y

+ {\A\ {\A Dl De^K2> s u p p f ^ / c ; * , ^ ) }} 22

dd 1
so the proof is complete. Next we prove the following lemma for a general process UAD on 7idD starting from A. For t fixed, we can thin UtAD by U£D(Bl) = U£D(B\) A K3jd{l/e) for any n 6 Zd. Note that UAD is not uniquely determined in general, however it can be if we impose some rule. To estimate the difference between UAD(T/LIJ) and UAD(V~LIj), we need the following result. It will be also used for the proof of Lemma 6.5.9. Define D(t;A,B;U) D(t; A.fl:iF) == £ ngJ3

A

U D(Bl)l(UA(BJ)>K (1/e)). D(Bi)>K3yd ^^(■^BJ^OJJJ, 3 7<.(l/e»-

Lemma 6.4.7. For j = ± 1 , _E(U£D(Bl) {Bl)22)) PCDCi; A,VLIfi V) >K i/e) d 2 el PWiA^V) > Kl/s) -< KC1KK3Jjg_ 1Ml/e). Proof. Note that

-*m*'-

|VT/ |=(2^ =={2s/I)d =-- 2 ^ f ) " = c ^. 1 SLIj J 1 d/2 ■ Z

e

So the definition gives E(D{L; A, VLIf,U)) ==: ££ E(D{L;A,VZlj;U))

A E{V : UAD{B\) > > KJf37d(l/e)) D{Bl) 3ld(l/e)) E(Ut1 &(*D D(K) ■■ D

nevTfi

1 Vi/,-1

W^i) 1 ).

Therefore the desired result follows by using Markov's inequality : if X > 0 and t,r > 0, then P(X > t) < E(Xr)/tr Note that D(L;A,>/LI D(L; A,VLIj; i;Z) Z) = E nevTi,

,{yfLBl)-Z {zAL,D{SLBl)-t LiD{s/ZBl)) LiD{yfLB\)}

> ztA&h) zfoiVZij) --- Z£D(SLIJ) > zUWlh) >ztD(Viij)- - ZL
Phase Transitions of IPSs

126

By Lemmas 6.4.6 and 6.4.7, we have PG8$JD{>/LI,)

- f ^ C v / Z l j ) > Ki/s) < P(D(L;A,VLIf,Z) E(ZJiD{Bjf)

> Ki/e)

-lld{l/e)

< C/K3So if K3 is large, the last inequality and Proposition 6.4.5 gives the following result. Proposition 6.4.8. For any S > 0, there are e0 G (0, l),Do > 0, and JiTj > 0,i = 1,2,3 so that ife 6 (0,eo),L = K2/e,D > D0 and A C VIlo with Kife <\A\D< 2ifi/e, then > 2Ki/e,i

P(%tr,(y/Zlj)

= ±1) > 1 - 26.

The following lemma is essential to the estimate on the difference between the DCP and the DBP in order to obtain the upper bounds. Lemma 6.4.9. There are e0 e (0,1), D0 > 0,andKt > 0,i = 1,3 so that ife e (O.eo),^ > D0 and A C \/~LIa with \A\D< 2Ki/e and \ A n B\ \D < 2i$_37
nf2et

£ E(ZtD(a)Z*D(b))

< ^—

r ft

/ pf+J> (s)ds + 7 <(l/e){pf + „ (*) + D-

d/2

} ■

n 6 Z i b~a

Proof. For simplicity, define Yla b~a for p= r = 1,

=

Saez d Y^b~a ■ Using Lemmas 6.3.3 and 6.3.6 (6)

J2 E(ZfiD(a)ZfiD(b)) = J2 JLlD^x'a'x'b)+ +

E£ E£

?Si'(*;!E,a)mD(t;y,6)

z#y 2

2

<8d Ae " | A |„

Joi

+^E E*

fP°+D{s)ds

X+C

it; X;

■

«

)

yeA

The first term of the right hand side is bounded above by Ce2ct £

'* Jo

P

,(s)ds,

pf +0 (*;»,*)•

127

Diffusive Contact Processes I

since | A \D< 2Ki/e. n G Z d , we have

To estimate t h e second term, by | A n Bln \D< 2K3jd(l/e)

DO*?**)- E

E p?+D(*;v,6)

me(2Z)<1 yeAns;, < E l^nB^|D

v€>*

for any

<2K3li{lle)

£

sup

Pf+D(t;y,b)

sup pf+J) (*;„,&).

mG(2Z)'< " e B ™

Note that for any b e Z% there is the unique nb e (2Z)d so that b e B\b. Define Q(n;r) = {x £ (2Z)'' :|| a; — n ||i< r}. Then there is an r{d) (depends only on the dimensionality d) so that

™PP?+D(f,V,b)
E

(6.8)

yeB

me{2Z)*\Q(ny,r(d))

™

For example, in d — 1, we can pick r ( l ) = 2;Q(ra;r(l)) = {n - 2,n,n + 2}. So the local central limit theorem implies the above inequality. On the other hand, by Lemma 6.3.1 (2), we have

E

supO*;j,,&)
y€B

meQ(nb-,r(d))

(6-9>

™

Therefore combining Eqs.(6.8) and (6.9) gives E

^P?+D(f,y,b)
+ D-^}.

(6.10)

) 5 C7d(l/e){P?+D(0 + ^ * / 2 } -

(6-H)

mg(2Z)" ! , e B -

So by Eq.(6.10) we have

E*0'^

6

On t h e other hand

E £ !£„(*;*.«) = M I ^ I » -

(6-12)

By Eqs.(6.11), (6.12) and | A \D< 2iTi/e, we obtained the desired result. For x > 0, let (x3, d=l, lfid(x)= I x I log* I, d = 2, [x, d>3. Note that if d ^ 2 then x = l/ a n d i f d = 2 t l l e n s ~ l/(p2(ip2(x)) and i ~ ^ 2 (1/V2(*)) as a: -> oo. Let a ( d ) = (d/2) A 1. The following lemma will be used for t h e proof of Lemma 6.4.11.

128

Phase Transitions of IPSs

L e m m a 6.4.10. For any K4 > 1, tiere is an e0 so that HE e (0,£o) and D > then MD/e)/D
K4
Proof. To begin, note that (6"13)

*7<(«0 = W f ' ) * and ipd(xy) = &(aO£d(y) for x, y > 0,

(i) d £ 2. By D > K^d{l/e)

W^A) <

mpd(D)/K4.

Using Eq.(6.13), the last inequality and xa^-2tpd{x)"^

= 1 for any x > 0,

{d)

ld(D/e)/D<E/K?

.

(ii) d = 2. By Eq.(6.13), fofjcy) = m ( « 0 + * & ( » ) ■y2(D/e)/D = eih(l/e)/D

for

* , » > 1, and

+

fr(D)/D2

= ^(D),

Vt{D).

Using Z) > K4 K4»(l/e)) ~ £ as e —> 0 implies WiD) < 4e/K4 when .fi^ is large. So the proof is complete. To estimate the difference between the DCP and the DBP, we introduce the following notation. Let At
= ~ZttD(x) - (t
\D(X)-

L e m m a 6.4.11. For any S > 0, t i e r e are £ 0 e (0,1) and Ki > 0,i = 1,2,3,4 so that if E e {0,e0),L = K2/e,D > Ki # ! / £ ) < ,5/2.

Proof. The definitions give jtE(EtD)

< (2dA -

1)E(K^)

+ A E E E hz-.^+^^VL^iTtMy^loiy +*)) xEA

< (2dX -

y

z~0

l)E(K?iD)

+ A E E W+*ltf v O ^ ^ f o ) ^ * + *))•

129

Diffusive Contact Processes I

So < sE(K*D) + A £

jE(KtlD)

P(a, b e f£ D ).

(6.14)

S(^D(a)^0(6)).

(6.15)

Noting that if 0 < e < 1, then A = (e + l)/2d < 1. So A^

P(a,6 e fjo) < £

Therefore, using Lemma 6.4.9, Eqs.(6.14) and (6.15), we have the following differential inequality : ~f(t)<^f(t)

/(o) = o,

+ g(t),

(6.16)

where fit) =

E(A£D),

+ 7d(l/e){pf+D (0 + JD-d/2}]-

»(*) = ^ H fp^oW* So it follows from Eq.(6.16) that E(KAL,D)

> 1 and D/e is large, then (LVD^D{t)dt
(6.17)

(6.18)

JO

So using Lemma 6.4.10 and Eq.(6.18) gives

From Eq.(6.19) it follows that

iLp°,D{i)dt Jo

^ {[P°+o{s)d*)ids) * ~ CIK"W'

(6.19)

(6 20)

'

By Eq.(6.13), if D > Jf 4 ?rf(l/e) and D > 1, then d2

ld(l/e)/eD

< 1/Kfd).

'

2cl

(6.21)

Therefore by Eqs.(6.19)-(6.21), sup 0 < 4 < L (e ) < e 2Ka and sup 0<E < 1 (e7 d (l/£)) < 1, we get

LL^dt=jJoUotp"°{s)ds)dt + %il/s) £

I P?+Dit)dt Jo

+ e^d72-7d(l/e) + 2Dd/2'ld' ;i/«) <

c

eK**> ■

[s)ds j dt

Phase Transitions of IPSs

130

Then using Eq.(6.17) we can obtain E(A£tD) CjKl

< C/e K^d).

So P(ALiD

> Kt/e)

<

. Picking A"4 the last quantity less than 6/2 proves Lemma 6.4.11. By definitions, we observe

= E

{ZL,D(BI)--ilABn))

n€(2Z)''

>

E

n6(2Z) J n£(2Z)

tfloVil) -hM)}

A > >%Z^VLJj) D(VZI)). L,D(SLIJ) ---? £ ^(va».

So combining Proposition 6.4.8, Lemma 6.4.11 and the last inequality gives the following result. P r o p o s i t i o n 6.4.12. For any 6 > 0, there are e 0 e (0,1) and Kt > 0,i = 1,2,3,4 so that He e ( 0 , e o ) , I = K2/e,D > Ktipd(l/e) and A C -/LI0 with Kx)e < | A | D < 2*Ti/£ and \AUB\\D< 2K3jd(l/e) for any n e Zd, then

> K*/e> 3 = ±1) > 1 ~ 3«-

PtLA^i)

Proposition 6.4.12 shows that if A satisfies (1) Kxje < | A | D < 2ii r 1 /e, and (2) | A n JB^ | D < 2K3-yd(l/£) for all n e Zd, then with probability greater than 1 - 36, there are sets A' C ttoriVLIj for j = ± 1 with (1') | A> \D= tfr/e, and (2') | A>C\B\ \D< K3ld{l/e) for all n € Z d . To use comparison method, we construct a general case. To do this, we let 6 Z 2 : m + n is even },

V = {(m,n)

Im = [2m - 1,2m + 1] x [ - l , ! ] ^ 1 ,

VZlm = [(2m - l)VZ,(2m + 1)SZ] X [-VI, Hm = ((2m - 2)VZ,(2m

+ 2)vT) x

VL]^1,

d

K ~\

where m e Z. For construction of the re-th stage, for (m,ra) £ V, sets A( mi „) C VTlm given by

(i) (2)

f

I ^(m,n) n B j | D < 2iiT3Td(l/£) for all k G Z* ,

are

Diffusive Contact Processes I

131

where B\ = [ ni -1,71! + l ) x ---X [nd- l,nd + l), and

( y/S, d=l, ld{x)= 3.

A site (m,n) 6 F is called open if a modified DCP ttj) starts with A ( m i n ) occupied at time nL and is not allowed to give birth and is killed outside Hm survives up to time (re + 1)£ and there are sets A^mn+1) C ?|^>'n+1) n VZlm+j for j = ±1, which have I A(,m,n+1) \D= ~t

^ ) (2')

I A(mn+1) nBl\D<

K3ld(l/e)

for all k e Z"

If (m,re)is not open, we set A

(m,n+1) = *■

To continue the construction on level n + 1, define ^(m-l,n+l) = J4(m-2,n+l) ^ ( m + l , n + l ) = ^(m.n+l)

U

U

^(m.n+l)'

^(m+2,n+l)-

To use comparison thorem, for I , J £ V , we write x —> y if there is an open path from i to y. That is, there is a sequence XQ = x,.. .,xn = y of points in V so that all sites xi is open and for 1 < i < n, Xi is aii_i + (1,1) or 2j_|_i + (—1,1). Let C° = {m € Z : (0,0) - . (m,re) 6 V}Hence the comparison method guarantees that if the DCP starts with A(0|o) satisfying (1) and (2) occupied and there is an infinite open path on V starting as (0,0) : P(C / for all n ) > 0, then the DCP survives : P(£,D

^ <^> for all n ) > 0.

That is, the comparison method gives the following flow chart : DCP i*D -> modified DCP (f:D -» oriented site percolation C° ->■ DCP £? D . We should remark that IT; and Hj are disjoint for |i - j \ > 2, so given what happened at level k < n — 1, the fates of sites (m,n) on a fixed level re are independent. That is,

132

Phase Transitions of IPSs

{ (m, n) is open }, (m, n) £ V, are independent. However they are not necessarily identically distributed, since initial configuration ^4(m,n) is n ° t identical in general. Let p (m,») =

p ( ( m , n ) j S open ),

so Proposition 6.4.12 gives p(°'°> > 1 - 3S, moreover the above observation implies that for any (m, n) £ V, j / m ' " ) > 1 - 36. Therefore comparing with the oriented site percolation, the result that upper bound on the critical probability for this model is 80/81 (see Theorem 4.2.4 (2)) yields that if

^—>>l-3*>g, that is, 3(5 < 1/81, then the DCP survives. So the argument above has shown that if D > C(pd(l/e) and C is large, then the DCP survives. Note that e > C C£><j(l/e) when D is large. Therefore we can prove the upper bounds on \C(D) given in Theorem 6.1.2. Moreover, combining this result and the trivial lower bound, l/2d, (see Lemma 6.3.5) gives the proof of Theorem 6.1.1 (1) :

lim XC(D) = 1 . 6.5. Proofs of T h e o r e m s 6.1.1 (2) and 6.1.3 In this section, first we will show Theorem 6.1.1 (2). Next the proof of Theorem 6.1.3 is given. Note that our proofs are close to those of BDS, so we will omit some details. 6.5.1. Proof of Theorem 6.1.1 (2) In the proof of Theorem 6.1.1 (2), A > l/2d is fixed, then we pick e < 2dA - 1. So the parameter A of the DCP is strictly larger than (e + l)/2d. The following lemma is well known results for branching procesess. L e m m a 6.5.1. (1) (2)

e- ( 2 d A - 1 "Z ( 0 i D (Z^) P(W > 0) = P{Z°tiD{ZdD)

W >0

a.s.

fora,;i)

as

t —» oo,

=

-2dAIn our setting, Z^D{Z D) is independent of D, so W is. Using Lamma 6.5.1 (2), we can get the following trivial upper bound on p\{D). d

133

Diffusive Contact Processes I

Theorem 6.5.2. For A > l/2d, 2dX-l Next we consider the following lemma. Lemma 6.5.3. If D > 1, then Urn £ | e-^x-^Z°
- e - P « - 1 ) « Z » D ( z £ ) P ( S j ^ t D 6 Vtlo) | 2 = 0.

Proof. To begin, we observe that if D > 1 and t > 1, then

^ II <'D + D lli< c \ / ^ r £

C V A T I V ^ < cVt. So for any 77 > 0 there is a K > 1 so that if Z? > 1 and t > 1, then

P(l * S » ll,> *<^ + 1» < a ^ l

S

; * L j <§ <„

(6-22,

On the other hand, I P(S°tLx+S e Vtlo -y)-

P(S°t£+D e Vtlo) I

< I P(S*°it£ e V^/o - 2/) - P(P ( °i- r)( A +D )/D € I„ - J/*) I + I P ( 5 ( U x X + 0 ) / c 6 4 - if) - P(B°ix+D)/D e /„) I + I P(Bl+D)/D e /o) - P(S°t£+D e Vt/o) I =Ji+72+/3, where 0 < r < r 0 < 1. Lemma 6.4.2 implies J3 -* 0 as t —» 00. By Lemma 6.4.2, if 3//V* —>■ y* as * — ► 00, and « = r(t)t with r(i) —> r, then Ji -» 0 as t —> 00. For part J2, note that sup£»i J2 —* 0 as j/*,r —> 0. So for any 77 > 0, we take an To so that if *" < '"Oi || ym ||i< K-V°i t n e n ^2 — \/^ - B-emark that if y/V -* y" as t — ► 00,a < r 0 t, and II 1/ ||i< ^ ( V " + 2), then || j/* ||j< Ky/¥o~. Therefore for any 77 > 0, there are t0,T0 and K > 0 so that if t > t0,u < r0t and || y \^< K(y/u + 2), then

I P(SlXu£ e Vtfo - ») - P ( 5 ^ + D e x/t/o) |< 2^77, where D > 1. Then we have

£ I e-i"*-»>*3?iD(v&y - e-(^-^^D{zdD)P{s°t^D ^-(MA-D.yD +2/'*! Jo

D ( f . 0>

Vi/o)(1 _ ^

e-^"1)" £ y€Zi

( t ; 0>

Vtr 0 ))

Pf + D («;0,j/)A^ z„o

x {pf+D (t - «; y, Vtft) - P? +D (<; 0, Vt/o)} x {pf+D (*-»»» + *. ^ 0 ) - P?+D («; 0, v^/o)} = Mi + M2 •

e Vtlo) I2

(6.23)

Phase Transitions of IPSs

134

To see this, Lemma 6.3.6 (2) implies that for any B C R d , - ZlD(ZdD)P(S°;*+D

E | Z?JJ(B)

f -2p?+D(t;0,B)E(ZlD(B)ZlD(ZdD))+p?+D(t:

=E | ZlD(B) =E < ZlD,lB

+2\J2 f z~0

€ B) | 2

0,£) 2 (£(Z t °, D (Z D ))) 2

< Z°t>D,lB > +p°+D(t;0,B)2E

> -2pf+D(t;0,B)E

<

Z%,1>

ds

Y,E < Z°t-s,D,6y >E< Z%, 1B>E< Z$, 1B >

Jo

y

-2p^D(t;0,B)\J2 ( dsJ2E< Z°_s,D,6y >E< ^ D , 1 B > E < Z%\\> z~0Jo

y

-2p^D(t;0,B)Xj2 J f*»Y,E< z~0 °

Zls,D,6y >E< Zfal >E< Z*+»,l >

y

+2pf+D(t-Q,B)2\J2 [ dsJ2E< 2?-.,D.*» > E < Z%,1 >E< Z»+D*,l > . z ~ 0 J<>

y

Lemma 6.3.3 gives E < Z;ID,1B

>=

ux m e (2dA-l)

J - p?+B(t;x>B),

so =e<2dx-1)tp?+D(P,0,B) - 2e^-^+D(t;0,Bf (*;0,B) 2 4+ e(2d\-l)t e«d^p?+D(t;0,Bf C»-l>* +2Xe + 2 A e(2*x-i)t

£ z~0

D f/ dse (s;(s; y+ ^ ((MX-», W A - i ) . £ Ppff++ DD (t (* -- s;,;0,0,„)pf y)pf+D (*5y,V,B)p? B)p+D y +z,B) z,B) +D(s; x+D Jo

y

-2p?+D(t;0,B)Xe^-»*

£

/ * d s e ^ ~ ^ £p?+D(t -

Z~0 J0

-2pD

dseVdX-V

( t;0,B)Xe^dX-^* J0

z~0

+2p?+D (|5 0,B) 2 A e * 2 ^ ^ £

s;0,y)Pf+D(s;y,B)

y

*;0,»/)pf »?«,(«: » +

yy

/ ' dse^-1)' £ J

z~0 °

Pf+D

*.«)

(| - 3; 0, y).

y

Note that the fifth term and seventh one in the last expression cancel, so we have x+D 0 +D E I e-^dx-^ZlD(B) - e-^x-^ZlD(ZdD)P{S0t;P{S B) ||22 6e B) t£ pf+D ft 0, B)} = e - ( 2 ^ - 1 ) l p l , + c {t.0,B){l-

t I dse-^-W-°^ ++ z,B) z,B) +2Xj:f + 2 A £ dse-^-^-'^ p?+D(t-s-,0,y) P?+D(s-,y,B) P?+D(s;y v> *)*&„(*:» + p^(t-s-,0,y)p^ D(s-,y,B)p^ D(s;y z~0

J0

y

-2Apf+D ft 0, B) £ > f + D (t; z,B)f z—t z~0

/:

efae-»«-W-4.

Diffusive Contact Processes I

135

On the other hand,

2 / ' du

»-<"»-i>« Y, P?+o («: 0- V» £
z~0

x {pf+D (* - «i w+»,*) - pf+D (*; 0,-B)} =2A ^

f «/„-("»-!)- ^ p f + D ( « ; 0 , y)Pf+D (*-«; J/, B)pf + D (t-«; V + , , * )

+2A ^

T &„-<*«-«» ^Pf+D („ ; o, j,)pf+D (t - . ; y, !)){-&,

+2A53 f due-^-V" z~0

53pf +D («;0,»){-pf +D (t;0,B)}pf +D (t

J0

(t; 0, B)} -u;y + z,B)

y

+APf+D(t;0,-B)253 /''due-l^-V" z~0Jo

=2A 5 3 / ' <{«-("*-«- 53Pf+D („; 0, »)pf+D (t - «;», 5)pf+D (t - „; y + z, B) z~aJo y -2Ap?+D (t; 0, B) 5 3 Pf+D («; z, B) / ' rf«-<«^>«. Jo

z~0

Therefore taking B = yLJa, the desired result can be obtained. Next, noting that p^(t;x,A) = P(S*'p 6 A), we will estimate M\ and M2. The first term Mi is bounded above by e -( 2 00. The second term M2 is divided by two cases, case 1, u > rot, and case 2, 0 < u < rot, that is, M2 = M21+M22For case 1,

M21 =2 / ' * , " ( « " * 53Pf+D(«;0,2/)A53 x(C(t-«;^/o)-pftDfto,^o)} x

{pf+D(*-«;»

du e - t 2 ^ - 1 ) " x 1 X 2dA X 1 X 1

<2 x / <2

X

+ *, V*/o) - pf+D(*; 0, v^/o)}

U

X

e-(2dX-l)rot_

2d\ — 1 So M21 goes to 0 as t —► 00.

Phase Transitions of IPSs

136

For case 2, M22 is devided into two terms as follows : rTot

M 2 2 =22 /

' E &° (a;0,2/)A^

due-' 2 "- 1 )"

z~0

y
x {?f+D (* - »5 9, >fa>) - pf+D (*; 0, Vtio)} x (p?+D (* - «; 2/ +*> V^o) - pf+D (*! 0, Vtio)} rrot

+2/

_

d«e-(MA-D-

£

P? + >;0,2/)A£

y>if(v^+l)

x {P?+D (* - »; 2/- V*/o) - p?+D (*; 0, Vtft)} x 0>? +D (t-u;y + z,Vila) - pf+D(*;0, Vtl0)} = M 2 21 + M 2 22-

For M221, using Eq.(6.23), we have Mm < 2 x / Jo

du e-fW*-1)" x 1 x 2dA x 2 0 ? x 2 0

^2dAx2rfT3Tx87'Similarly, for M222, using Eq.(6.23), M222 < 2 x / Jo

du e - ( 2 ^ - ! ) " x 7j x 2dA x 1 X 1

^ 2 d A X 2sfcl X 2 ? ? So

2dA M22 = M22i + M222 < 2dx_ 7 X IO77.

Since 77 > 0 is arbitrary, we have the desired conclusion. For any f\ > 0, F(| e-^^-D'Z^CV^/o) - W P ( B ( V D ) / D £ 4 ) l> 35?)

e - ( « W % ( / f t ) - e - ( 2 ^ - 1 ) ' Z ° D ( Z i ) ) P ( 5 ^ + D 6 0 J O ) l> 5)

+P( e -t«*»-i)*20 D (Z* ) I P ( S ° £ + D 6 VtJ 0 ) - P(P ( ° A + D ) / D £ lb) 1> 5) +P(I e -

0

" -

1

^ ^ ^ ) - W I P ( P ( V C ) / D e /o) > ij)

=J? 1 (t) + -ff2(t) + ff3(i). Combining Chebyshev's inequality and Lemma 6.5.3 gives Jfi(t) —> 0 as t —> 00. By Markov's inequality, Et{t) <| P(S t °,£ +D 6 v U ) - P(4x+D)/£) 6 Jo) I /if.

Diffusive Contact Processes I

137

So Lemma 6.4.2 implies H2(t) — ► 0 as t -> oo. Note that the definition of Brawnian motion yields, with A > l/2d fixed, e/

DW/(B(W

»)>0'

(6-24)

By Eq.(6.24) and Lemma 6.5.1 (1), we have F 3 ( t ) - f 0 a s ( - » o o . So we obtain the following result. Lemma 6.5.4. It D > 1, then, as i —> oo, e-^-^ZloiVtlo) (Vilo) -» WP(B°{X+D)/D

e Jo) in probability.

Moreover, by Eq.(6.24), Lemma 6.5.1 (2) and Lemma 6.5.4, we can prove Lemma 6.5.5. It D > 1, then 2

HP{ZlD{VLI0)>Klle)=

-%±.

2dX-l 2dX

'

On the other hand, to estimate the difference between the DCP and the DBP, we will need the following lemma. Lemma 6.5.6. With e > 0 fixed, for j = ±1, (1)

lim {P(ZlD(VZl0)

(2)

lim {P(Z02L,D{JLIj)

> K^e) - P(&:D(VLI > #i/£)} = 0. (VLIo) 0) > K^s) - P^oiVIlj)

> K^e)} = 0.

Proof. For part (1), P(Z°LiD(VIlo)

> KJe)

- .P(££,D(VM,) >

K,/e)

> £U(^/o)) - &,D(Zb)

> 1)

d

<E(ZlD(Z D)-tlD(Z D)) (Zf))) =-E(Ai ljD ). Lemma 6.3.9 (2) implies that, with e > 0 fixed, E(A°LD) -► 0 as D -> oo. This gives (1), and the proof of (2) is similar.

Phase

138

Transitions

of IPSs

Let 77 > 0. By Lemma 6.5.5, there is an e\ > 0 so that if 0 < e < £1, then

(1 - n)~^

< PtfUfV&b) > KI/B).

So, combining the last inequality and Lemma 6.5.6 (1) implies that, with £ G (0,£i) fixed, there is a D\ > 0 so that if D > D\, then (1 - IV)^^1

< Pi&A^)

^ ^x/£)-

(6-25)

Fix £ G (0,£i). Therefore, by Eq.(6.25), if D > Dlt then with probability greater than (1 - 2r))(2d\ - l)/2d\,ZlD n VZl0 contains a set A with | A \D= K^je. So, the last fact and a similar proof of Proposition 6.4.8 impliy that for any S, 77 > 0, there are £2 G (0,1), D2 and Ki >0,i= 1,2 so that if £ e (0,£ 2 ),£ = K2/e and D > D2, then < P{Z$LID(VLII)[SZlx) > Ki/e).

(1 - 2r,)(l - 6)^=^

(6.26)

On the other hand, by definitions, p,(D)

= Urn P ( £ ! L , D ( Z 2 > ) > 0),

(6.27)

and P$L,D(^LII)Wih) > KUe) < P(&L,D(ZdD)

> 0)-

(6-28)

By Eqs.(6.27) and (6.28), there is an £3 > 0 so that if 0 < £ < £3, then n l k D ( V £ I i ) . > Kile)

< (1 + r()px{D).

(6.29)

Using Eqs.(6.26) and (6.29),

«<^_n{D)
rAV J

H ii-n)H-s)\ux-i

~ \

I + 77

1

j

2<2A

{P(Z°2LiD(VLIj) > R\/S) - P^LA^)

> A'i/£)}

Since 77 and <5 > 0 are arbitrary, so by Lemma 6.5.6 (2), we have the desired result : . ._,

A c ( i » ) -►

2d\-l 2d\

—

a s D —► 0 0 .

Diffusive Contact Processes I

139

6.5.2. The Galton-Watson process In this subsection for convenience of readers we present some results for the GaltonWatson process which are used for the proof of Theorem 6.1.3. As for Galton-Watson processes, more details can be found in, for example, Athreya and Ney,14 Jagers.15 Let ZQ , Z\, Zi,... denote the nonnegative integer-valued Markov chain with Zo. Zn is interpreted as the number of objects in the n-th generation of a population. The probability distribution of Z\ is given by P(Z1 = k) = pk, for k = 0,1,2,..., where

CO

£

Pk

= 1.

fc=0 *:=0

Pk is interpreted as the probability that an object existing in the n-th generation has k children in the ( n + l)-th generation. We assume that pk does not depend on the generation n. For |s| < 1, generating function is defined by 00

k f(s) = J2p^ -

k=0

Let ,, = f'{l) = E{Z1), v = f"{l) + n-fi2

y=^™, =

E{(Z1-tf),

and

_ Ml-/*-")/(/*-1), ~\un,

Cn

if/*?*!. if/* = 1.

For any a > 0, let Ka = { Galton-Watson processes with w > a, and such that for any n > 0, there is afc(n)so that £*>*(„)fc2Pfc< "

}■

Then the following facts are obtained. Lemma 6.5.7. For any sequence of laws p<E> e ifa with ^ 71 —► C O ,

(1)

4«)p(4 s >>o)-i,

(2)

■IS^F'**°) '

-» 1 as z -> 0. Then as

140

Phase Transitions of IPSs

(3)

Zi ]

P (

. , > x I 4 e > > 0 ) -» e"*,

'

where / i ' e ' , i / e ) , and <& are t i e constants associated with p ' E ' . In our setting, set

p(:)=p(ziD(zdD)

So M <<)

=

oo

= k).

E(ZlD(ZdD))) = e2^-1

=

= e£,

jt=0

1 °°

„M

;(k - l)pW

^ f cI — = 0n

= i{£(Z10,D(Zi))2)-JB(210iD(Zi))} + l e *(e - 1), = ££±1^-1), =

E

and

,M

£

+ 2-e'(e° - 1).

So as e —» 0,

^ W -- 1, /« 1, * « - . 1, i/W v^ -» 2. u<£> - 2.

Moreover let

4 £ > = Z°, D (Z*,),

and <4«) = l / w

1

-^ ( ' ) )- n = l / w i-«-'" /i e - 1

e£ - 1

For simplicity, in the next subsection we omit superscript (e). 6.5.3. Proof of Theorem 6.1.3 This subsection is devoted to the proof of Theorem 1.3. Let e = 2dX - 1 = K4
Diffusive Contact Processes I

141

L e m m a 6.5.8. Let e = 2d\ - 1 and 0 < e < 1. There is a S(r) [ independent having S(T) -> 0 as r -«• oo], so that ift = T/E and D > 1, then for j = ± 1 ,

of D and

E | e-^-^ZlniVtli) zloiViij) - e-^-^ZlD(Zi)P(S°t;^D P ( S $ + D 66 Vtlj) Vfl/. | 2 < ?(r)/ £ . Proof. A similar argument in Lemma 6.5.3 implies that

(Vtij)

E | e-^^ZlviVtlj)

- e-W-V*ZlD(ZdD)P(S°;*+D

G Vilj) | 2

<e~ T + i ^ ( 2 e - r o T + 10fj), where 0 < ro < 1. Let 6 = S(T) = 8e- r » T . So if 0 < e < 1, then

e-r

+ i±£ ( 2 e -*r + 1(h?) < | + | A + 1 0 „\ <

*

1

«

1

2077 -■

- He 2s e If we pick 77 so that 2O77 < 36/8, then for large T, the upper bound is smaller than 6/e proving Lemma 6.5.8. By Lemma 6.4.2,

Jto^PiZg" Let 77 = P(B?X+D\/D

6 V^Ij) = P(tfx+D)(D 6 */)•

2

£ J j ) / - So there is an n 0 so that for any n > n0,

P(S°n£D e VKO) > v > o-

(6-30)

Observe that

e e£"-l 11 -- e~ e- ' cntiu = VX — £ — Xe£n = vY. e e - 1 e - 1' so Lemma 6.5.7 (3) implies that if £ is small then a s n - > o o ,

P(Z„

£

"

em-l

> x | Zn > 0 J ->e~x.

For 0 +r ' e V»i) > ^

I 2 . > 0)

>P (e-£"Zn/> > ? £ I Z„ > 0J =F

£ 7

> 2

^ i ^ l l Zn > OJ .

(6.31)

Phase Transitions of IPSs

142

On the other hand, if n > log2/£, then 2 p — — - < 4p. Hence combining Eq.(6.31) and the last inequality implies that if we pick p with e 4 " > 1 — 2i5, then a s n - t o o and £ —> 0 with n > log2/e,

P (e~'nZnP(S°n%+D e VST/) > 2 £ I Z„ > o) >p

N^~L> 4p\Z > OJ n

-4

-> e " > 1 - 2«. Therefore for any S > 0 and 0 0 so that if £ 6 (0,£i) and n> ni, then P (e-™ZnP(S°n'fiD

£ V»i) > ^

Let An = e~*»ZlD(V^h)

| Zn > o) > 1 - 26.

- e-™ZlD(ZdD)P(S°n#D

(6.32)

€ y/Sh).

By definition, A n = 0 on{Z° tD (Z£,) = 0 } .

(6.33)

On the other hand, Lemma 6.5.1 (2) gives « . D ( Z £ ) > 0 for all n) = ^

~

=

~ .

So if 0 < £ < 1, then P(Z°n,D(ZdD) > 0) > J^J

> \.

(6.34)

Hence combining Lemma 6.5.8 with Eqs.(6.33) and (6.34) we have

E(Al\ZlD(Zi) > 0) ~

£(AJ) P(ZlD(Z%) > 0) S(K2) 2 - 26 ?i X 22 = e£

e ~ £e '

so Markov's inequality gives

p (A L < - t | zlD(zdD) > o) = P ( - A L > £ | 2£j,(zi,) > o) < £ ( A i | 2 £ , D ( Z i ) > 0) x ( - I ) ' < ? * ^ > .

(6.35)

Diffusive Contact Processes I

143

Noting that m > log2/e and 6{K2) -> 0 as K2 -» oo, we pick K2 > 0 so that #2>log2, 2?(Jf2)

(6.36)

< '•

(6-37)

p e* > 3 ^ .

(6.38)

p2 2

2

Using Eqs.(6.35) and (6.37), we have Pie-^Zl^VLI,)(VT/i) - e-LZ°LtD(ZdD)P(g°L#D

£ VZh) < -Pj\Z°LtD{ZdD)

O

> 0) < S.

That is, P(e-eLZlD(VLh)VLh) - e-£LZlD(ZdD)P(S0^D+D D By Eqs.(6.32) and (6.39), P{ZliD(VZlx)

e v T / , ) > ^ | Z £ , D ( Z £ ) > 0) < 1 - 6. (6.39)

> e'<*£\Z°LtD(Z*D) > 0) = P C e - ' ^ t V I / ! ) > P-\Z\,D{ZdD)

> 0)

> 1 - 3(5. Prom the last inequality and Eq.(6.38),

p (zl^VIh) Hence

>^

| zlD(z%) > o).

p (zltD{JZh) > ^ | l ) > (i - 3«)P(z£,D(z£) > o) >(l-3<5)

So we obtain the following result.

e 2dA - 1 = (1 - 36 e-1' 2d\

Lemma 6.5.9. For any 8 and ifi > 0, tiere are £o £ (0,1) and Ki > 0 so that ife e (0,£o) and L = Kt/s, then for j = ±1, P(ZlD(y/LIj)

> 3KJe) > (1 - 3 « ) ~ .

By Lemma 6.4.7 for A = {0} and U = Z, P(D(L;0,VLIj;Z)

< A'i/e) > CK;1e1-d'2{'td(l/e)}-1m2-(L;0,B1n).

Bl).

(6.40)

0

Noting that D(L;0,VZlf, Z) > Z LD{VLIJ) - Z~l:D{y/Zlj), by Eqs.(6.40) and (6.7), we have P{Zl,D{4lli) - ZlD(VLJj) > iTi/e) < Cs/K3. Therefore, if Kz is large, then PiZloiVLIj)

- Z~lD(VLIj)

> ifi/ £ ) < fe/(e + 1).

Using Lemma 6.5.9 and Eq.(6.41), we have the following result.

(6.41)

Phase Transitions of IPSs

144

Lemma 6.5.10. For any S and Kx > 0, there are e0 £ (0,1) and Kt > 0,i = 2,3 so that ife £ (0,£o) and L = K2/e, then for j = ±1, P(ZlD(yfLIj)

> 2 ^ / e ) > (1 - 4 « ) J ^ Y .

We turn to the proof of Theorem 6.1.3. By Lemmas 6.3.9 (1) and 6.3.1 (5), and the similar argument in Lemma 6.4.10, F(A° , D ) <

Cyt(D/e)/D

< ClKt(d). So we have P(A°LD > Kt/e) < Ce/K?d). Using A°LJ} > Zl^VIlj) - | £ , D (VZ/j), if K4 is large, then P(2lj>(VLIi) - ?L,D{VLIj) > Ki/e) < ie/(s + 1). (6.42) By Lemma 6.5.10 and Eq.(6.42), P(&,D{^LIi)

> Ki/e) > (1 -

&)-£i-

Therefore, for any S and K\ > 0, there are D0 and K{ > 0,i = 2,3,4 so that if e = Ki D0, then for j = ±1, P(&,D(JLIj) since

> Ki/e) > (1 - 5 * ) ^ = ^ , e 2rfA-l e + 1 ~ 2dA

So the proof is complete References 1. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985). 2. A.De Masi, P.A.Ferrari and J.L.Lebowitz, Reaction- diffusion equations for interacting particle systems. J.Stat.Phys. 44 (1986) 589-644. 3. R.Durrett and C.Neuhauser, Particle systems and reaction-diffusion equations. Ann. Fro6a6.22(1994)289-333. 4. R.H.Schonmann and M.E.Vares, The survival of the large dimensional basic contact process. Probab.Th.Rel.Fields. 72 (1986) 387-393. 5. M.Bramson, R.Durrett and G.Swindle, Statistical mechanics of crabgrass. Ann.Probab. 17 (1989) 444-481.

Diffusive Contact Processes I

145

6. M.Katori, Rigorous results for the diffusive contact processes in d > 3, submitted to J.Phys.A. 7. I.Jensen and R.Dickman, Time-dependent perturbation theory for diffusive nonequilibrium lattice models. J.Phys.A. 26 (1993) L151-L157. 8. R.Durrett, Lecture Notes on Particle Systems and Percolations (Wadsworth, Inc., California, 1988). 9. D.Amit, Field Thoery, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1984). 10. H.Matsuda, N.Ogita, A.Sasaki and K.Sato, Statistical mechanics of population - The lattice Lotka-Volterra model -. Prog.Theor.Phys. 88 (1992) 1035-1049. 11. N.Konno, Asymptotic behavior of basic contact process with rapid stirring. Preprint (1994). 12. G.F.Lawler, Intersections of Random Walks (Birkhauser, Boston, 1991). 13. R.Durrett, Probability : Theory and Examples (Wadsworth,Inc, California, 1991). 14. K.B.Athreya and P.E.Ney, Branching Processes (Springer-Verlag, New York, 1972). 15. P.Jagers, Branching Processes with Biological Applications (John Wiley & Sons,Inc, New York, 1975).

Phase Transitions of IPSs

146

CHAPTER 7 D I F F U S I V E C O N T A C T P R O C E S S E S II

7.1. Introduction In this chapter, we will prove the lower bounds on \0{D) given in Theorem 6.1.2. Note that the diffusive contact (branching) process is abbreviated to DCP (DBP) as in the previous chapter. By comparing with the DBP, in any dimension, mD(kt)

<

(mD(t))h

So if mD(t) < 1 for some t, then P(£? i£) ± <j>) < mD{t) -> 0 as t -> oo, i.e., the DCP dies out. Therefore, to prove the lower bounds, it is enough to show there is a C so that if D < Ctpd(l/e), then mD(t) < 1 for some t, where e = 2dX - 1. The key to doing this is the estimate on the interference term. Particularly, in d = 1,2, estimates on difference between the exclusion process and the independent random walk are essential to it. See Theorem 7.2.4 for d = 1 and Lemma 7.3.12 for d = 2. As for d > 3, some methods are applicable. 7.2. Lower B o u n d s for d = 1 In this section, we give the result in d = 1 : P r o p o s i t i o n 7.2.1. There is a C > 0 so that if D is large, then K(D)

*b c

DV3"

In the rest of this section, Ki(i = 5 , . . . , 9 ) are constants which will be chosen later. Note that Q%,{K6y/i) = [-K6\/i,K6Vi]. For * > 0, D > 0 and x e ZXD, define NtlD^)

=

{y e et,D

ifcc = {xe Ct,D =

tfj,

{yeet,D-0<\y-x\
\„> K7sVD}

(tlDnQ*(K6V~t))\r,t,D.

n

Q*(K6Vi)

147

Diffusive Contact Processes II

So, by definitions, we have &,D = Vt,D + C*,D + K?,D n Q*(K6Vin

(7.1)

First we estimate the second term on the right hand side of Eq.(7.1). Lemma 7.2.2. There are D0 and K5 > 0 so that if D > D0, e3 < Ks/D and l/10e < t, then E\Ct,D\D< 0.1. Proof. We begin by observing \G,D\D<

£

\(t,Dn[i,i

+ i)\D

i€Q«,(K-6x/i)

<

2K6K7eVDt.

If e is small and t > l/10e, then P{QD ^ 4>) < P{Z\D /)< C/t, where a constant C is independent of D. See Lemma 2.1 in BDS. Note that as in the previous chapter we call Bramson, Durrett and Swindle1 BDS for short. So from these facts it follows that if t > l/10e and e 3 < Ks/D, then E I Ct,D | D = E(\ C*,D \D\et,D t 4>)P(&,D * 4>)

<

CK6K7e^DJi

< CKeK7y/W5. Therefore, with Ke and K7 fixed, picking Ks to make the last quantity less than 0.1 proves Lemma 7.2.2. Next we estimate the third term on the right hand side of Eq.(7.1). Lemma 7.2.3. There are D0 and K& > 0 so that if D > D0, e3 < Ks/D and 0 < t < l/e,then E\tlDnQK(K6Viy\D<0.l. Proof. By the local central limit theorem, if t < l/e,D is large and e3 < K5/D, then E I &, n Q*(K6Viy

\„ <

=

E(ZlD(Q*(K6Viy))

e^+D(t;0,Q^(K6Viy)

<2ep^D(t;0,(KeVt,oo)) < -= I ~

- F =exp(-a 2 /2)do VDJK.12^

Phase Transitions of IPSs

148

So, with D fixed, E | QD n Q^,{K6y/t)c

| D -> 0 as K6 -» oo.

To estimate the interference term, we introduce the following processes. Let

ra

A?(t; (*,»)) = (KD be the exclusion process (see Liggett, ZdD and

2

Chapter VIII,) at rate Idv starting from (x,y)

B?(f,(x,y))

on

\Jt,DiJt,D>

-

be the independent random walk at rate Idv starting from (x,y) on Z^,. Moreover, define SlvyM = X*$ - X*£. Note that S^t2v = S$ - Sf£. In d = 1, the foUowing result is due to Andjel. 3 L e m m a 7.2.4. There is a Kg > 0 so that for any D > 0,

E i ns\'f'2D = x) - p t s j ^ = x) i< % xezD The local central limit theorem implies that there is a Kg > 0 so that if D is large, then for x e £ £ ( 1 ) , VD.2D P(S\^D = X)=P?D(1;1/VD,X) /D,X

= PI(2D;1,VDX) •JBX > ^ J .

Let /D.2D

«*(*) = {V € Q £ ( * ; 1) = P(Sl^2D

= y-x)>

>

^ L } .

By Lemma 7.2.4 and the last inequality, for any x 6 Z^,,

E

V5

I

P{S\'^D =

y-x)

- Pft'J/D,2L

=

V-x)\

*e<3£(x;l)\Q-(*)

>y=ra*;i)\
(7.2)

Next we present the following lemma which will be used for the proofs of Lemmas 7.2.6 and 7.3.15.

Diffusive Contact Processes II

149

Lemma 7.2.5. In any dimension, if x = (xi,x2),y = (3/1,3/2) £ Z j x Zj, with xx / x2 and 3/1 / 3/2, then P(A?(t;x) = y) = P(A?(t;y) = x), where D and v > 0. Proof. We prove only d = 1, since we can extend it to d > 2 in a similar way. For simplicity, we take D = v = 1. Let a: = (xi,x 2 ),3/ = (3/1,3/2) G Z 1 x Z 1 . Define g : Z2 x Z2 -► R 1 by ?(^>y)

( ■I: 1

if I xi -x2 |= 1= 1 and ||II xx-yy ||i= l|i= 1,3/1 l,«i ^3/2, ^ y2, or |I xjxi -- xx2 2 |=|= 11and and Xi i j==#3/2,x = 3/1, 2= or ,Z2 J/i, 2 or | xi - x2 |> 2 and \\ x - y ||i= 1,

or I xi - x2 |> 2 and || x - y ||i= 1, The above definition implies q(x,y) = q(y,x) for any x,y 6 Z . Moreover define 0 that otherwise. The above definition implies that q(x,y) = q(y, x) for any x,j/ 6 Z 2 . Moreover define i f Xl r(x v) = I 1 / ' 4 ' ' ~ X2 ' = 1&ndx = »> ' \q{x,y)/4, otherwise. Note that r(x,y) > 0 and ^2ur(x,u) = 1 for any x,y G Z 2 . So we can consider the discrete time Markov chain on Z2 defined by the transition probability r(x,y). Let Ak\x,y) denote the probability of a fc-step transition from x to y. A little thought implies that 7-'fc'(x,y) = r'fc'(3/,x) for any 1,5 £ Z2 and k = 1,2, Therefore, from the fact that P(Al(<;x) = 3/) = £ e - 4 ' ^ - r < * > ( x , 2 , ) , k 0

we obtain P ( 4 ( * ; x ) = 3,) = P(Al(t;2,) = x). The following estimate on the interference term is essential to the proof of Proposition 7.2.1. Lemma 7.2.6. There are Do,^5 and K7 > 0 so that if D > D0,e3 < K*,/D and l/10e < t, then ID{t) > 6eE I %D \D . Proof. By Lemma 7.2.5, for any a and x, y E Zjj with x / y, P(A°(1; (x, 3,)) = (a, a + ^ ) ) = P(Ag(l; (a, a + -^=)) = (x, y)).

(7.3)

For x 6 7?t_i,.D, we have | Q*(x;l) \D> K7eVD. SO using Eq.(7.2), \Q*[x)\D>KieSD-1£.

(7.4)

Phase Transitions of IPSs

150

By the definition of Q*(x), Eqs.(7.3) and (7.4), with K7, Ks and K9 fixed, if D is large then

lD(t) = ^

2*£P(a,a+^=etlD)

2 e _ 2

E

E

P(A%(l;(x,y))

=

(a,a+-^))P(xEVt-i,D)

xeZ\, !/SQ5,(a;;l)

= 2e_2 E

E

^(I;(M + 4 ) = ( ^ M ^ W ) V

sezj, !/egg,(x;i) 2

,

= 2e~ £ 2 J (^' *ezlD vegg,(x;i) > 2e-2 £

M>

= 3 ' - ^ ( ^ %-!,£))

^ l , ^ = 3/ - *)P(* 6 * - i ^ )

£

i e z ] D ye«*(i)

> 2 e - 2 ( W £ - ^)^E

| *_liD u

> C 8 r 7 e £ | W-I.D | 0 By comparing with the DBP, we have J5|IJM>L:S«,.E|I»-I^I0 So if e 3 < isTs/Ai then ID(t)

> Ce~cK7eE

\ r,t,D \D

>Cexp|-^y Therefore, picking K7 = 6exp{(Ks/Do)3}/C

1 K7eE I THJ, \D .

completes the proof.

The following lemma will finish the proof of Proposition 7.2.1. L e m m a 7.2.7. There are Da and K$ > 0 so that if D > D0, e 3 < Ks/D for l/10e < t < l / e , then

and mD{t) > 0.9

£»»(«) < -, Suppose Lemma 7.2.7 is valid. By Lemma 6.3.4 : 0 < mD{t) < mD{t) = ec\ and ex < 1 + 2x for 0 < x < 1, we see m D ( l / 1 0 e ) < mD(l/10e) < 1.2. So Lemma 7.2.7 implies mD{t) < 1.3 - Et for t > 1/lOe. If 0.4/e < * < l / e , then mD(t) < 0.9. Therefore mD{i) < 1 for some t, and the proof of Proposition 7.2.1 is complete.

Diffusive Contact Processes II

151

Proof of Lemma 7.2.7. Using A > 1/2, Lemma 6.3.8 (1) : ~mD(t)

= emD(t) -

\ID(t),

and Lemma 7.2.6, we have jtmD{t)

< emD(t) - 3eE | %tD \D .

(7.5)

By Eq.(7.1), Lemmas 7.2.2 and 7.2.3, mD(t) = E | r,tiD \D +E | Ct,D \D +E | £?,D n Q*(K6Vty <E\rit,D\D +0.1 + 0.1,

\D

that is, mD(t)<

0.2 +E\nttD\D

.

(7.6)

So Eqs.(7.5) and (7.6) yield jtmD{t)

< 0.2£ - 2eE \ m,D |D .

(7.7)

On the other hand, by Eq.(7.6), if mD(t) > 0.9, then E\m,n\D>

0.7.

(7.8)

Therefore combining Eqs.(7.7) and (7.8) gives 4 m D ( f ) < 0.2£ - 2e x 0.7 = -1.2e < - e .

7.3. Lower Bounds for d = 2 In this section, we give the lower bonnd in d = 2 : Proposition 7.3.1. Tiere is aC > 0 so that if D is large, then , ,„.

1

ClogD

Recall that 5^ i T l - [m,m + 1) x [n,n + 1) for m,n £ Z. For simplicity, from now on we omit superscript 0.

Phase Transitions of IPSs

152

Lemma 7.3.2. Tiere are D0 and C{ > 0,i = 1,2,3 so thai, if 0 < t < l/£ witi £ e (0,1) and D > D0, then for any a; £ Z2D, (1)

WD(«;«.J8m..)<j^,

(2)

Mfft*tfl»»)SC»^fep.

™Br,., n

w

{log(2

\ <- r

+ *)}2

Proof. (1) By Lemmas 6.3.1 (3), (4) and 6.3.3, mD(t;x,Bm
ectp?+D{t;x,Bm,n)

=

<ex?f +D (<;5 m ,„) CI

- 1+* (2) Using Lemma 6.3.6 (3) for k = 2, we have mf(i;x,B m ,„) =

ra°(t;i,Bm,n)

+ 2 A V y / dsmD(t - s;x,y)mD(s;y,Bmjn)mD(s;y

+

= h+h. (i) I\. From part (1),

"<-rU(ii) J2. Using part (1), we have / 2 = 2 A V / dsmD{t;x,Bm,n)mD{s;y

+ z,Bm}Tl)

t* C C < 2A x 4 / ds x Jo 1 + 11 + s <2(e+1)x ^ f< d ^ - v T ' 1 + tJo l + s ^ l o g C i + OHence "l 2 (.t,a;,i>m,nj - il + i2 < J V ^ + 0

——

< G2

——

.

z,Bm,n)

153

Diffusive Contact Processes II (3) Similarly using Lemma 6.3.6 (3) for k = 3, so m f (t;x,Bm
mD(t;x,Bm,n)

=

+ 3AEE/' 5Z5T /

+ 3A

D D lsm (t — Ds;x,y)m (* : y,B ■y,B {s;y + + mD(t-s;x,y)m (s m
z,B z,Bm,nn))

(t - s;x,y)m^< s :y,B :y,B n (s;y + + ™°{t-s\x,y)m${s , )m )i (s;y E Jo/ dsm

z,B z,Bmmin,n))

z~OyJo y z~0 A + 33A£ + X^5Z

z~0 z~0

ds

J0

ds

D

D

D m n mtn

Jo

y

= =h h+ +h h + + hh-

(i) Jj. From part (1),

C Ji< " 1+ + if (ii) Ji- Using parts (1) and (2), we have D J J 22 = 3. 3A / dsrn (t;y,Bmtn ismD(t;y,B m,)rilf(s;y nynl?{

z,Bmm,n) ,n) s;y + z,B

> gb( 2 + <3K ^)/ t *-gf + '>s) l +t C ' 1+1

< 3(e(e + 1)

~

'Jo

l+t

l+l

•

XogCg as iog(2 + + *J <) Ir' _ds_ l +t {log(2 + + t)} i)}22

-

1+s

(iii) J 3 . Let +1 B l,m++2)2)X [* [ n-- ll,n #m!n = [m -—1,W» , n + 2).

So a similar argument in Lemma 6.3.1 (4) gives

£,(*!*&)*^ifiif? ?f + D (*;
so

D m°(t;*,i?+i„) W ft* f i»&)< f . " lI +? t '

(7.9)

Using Eq.(7.9) and part (2), we have J 3 = 3A'V* /

dsmD(t;x,Sm,„-«)mf(s;j/,Bm,n)

J3 = 3 A V > / dsmD(t;x,Bm,n< 3(e + l)m"l(*;«.3&)

r _ds_ L i + .s < c log(2 + 0 / ' tfa log(2 + t) l +t

+iog(2 + t ) r i +
s {log(2 + *)}2 C 1 ^3 <■^ + mf^t; x, 5m,„) =- Ji T + J+2 + -

Therefore

Therefore -DU R

1

z)m£(8;y,Bm,n)

rise**2*'* Jo 1+*

7 +

f

+ 7 <

C

^

{log(2 + t)} 2 l+t "

, r , { l ° g ( 2 + *)} 2

"»f (t; i, Bm,n) = h + ^2 + Jz < j — ^ + 6

^+~t

C3

{log(2 + t)} 2 l+t '

{ t o g ( 2 + f)} a

1+1

Phase Transitions of IPSs

154

L e m m a 7.3.3. There are tQ,D0 and 6; > 0,i = 1,2 so that if t > t0,D > D0 and e <E (0,1), then (1)

e £i m mDD((tt; ;00, B , Bmm, n, )„ ) > 6 i j - j : ^ , 1 + t'

(2)

l0S(2 + < ) m?(<;0,5 , m f(t;0,£mm , „,„)>6 ) > 6 222 l5MH±0,

-

1+t

'

(m,ra) £ ££(•*/*). where (m,n) P r o o f . (1) Using the local central limit theorem, we have

j £ „ ft <»'*•».») =

E pf+D(*;o.»)

y£Bmm,„ »6B ,„

*E

y£Bm,„ neflm,„

(A + r»)t '

~2(A + D ) .

V

> | B-B„ m , n | D

g

3(1 + 0

>-£-.

-1+t

-

i+t

So Lemma 6.3.3 gives D c< = ee"pf p f + D(t(;°'t ; 00, ™.n) £ m , n ) > 6j — . m (t;0,B (t;0,minB)m ,„] = +D 1+t

(2) Using Lemma'6.3.6 (3) for x = 0 and fc = 2, we have mf(t;0,Bm,„) D

=m = m D{t-o, ( t ; 0 , BBmm, ,„n)) + 2 A V V /

> 22AAJ^ ^ / / *s 2 ~ 0 Jt» '''O z~0

DD

DD

£, D

s m (((t - s;0,i/)m s; 0, y)m (s; y, B ,„)m „ ) m (s;5 ( (5;!/,B (s;y + z2 , B 5 £ d
*~0 y J° ,f/2

rm ai

E

D D roD(t-s;0,y)m (t - s;0,2/)m (s; yD,(s;y Bm,n)mD(.s; y+ + z, Bz,B „ )).. E rn (s;y,B m ,mjn m,n)m ^llv-( .")IU<\/»

m w:||»-(m,i»)||i
From part (1), mD(s;y,Bm,n) and

m,„)

> &x—-— > ■1 + fl - 11 + + ^5 '

e£s" b61 mDD (s;2/ + z , B m , n ) > b e m {s\y + z,Bm,n)>h—— ■1 + s > - l + s'

Diffusive Contact Processes II

155

so ft/2

mP(i;0,B m , )>2(e + l) / mP(i;0,B m , nn )>2(e + l) /

ds ds ds

mD(t - s;0,y)

YY

^-^ATT)2 y:||y-(m,n)|| < '» 1

ft/2

Y, E

>C I2 ds >cf'Jtods

^-^Tihy '(1-M)2 ^-^(TTW

r-\\v-(m,n)\\1
(—)

v

79 ((7.9) ->

Note that if |\\y-(m,n) | 9 - ( m , n ) | |||i m^tt-s-.O.v)

"TDm (i D1(t-s;0,y) 5 ;0, j/)

Y

>

/

y.\\y-(m,n)\\i<^

!'*l|l<- v 8(<-»)

E

>C >c-

>c

m D (*-«;0,fi mD(t-s;0,B m ,min n ))

Y,

8

ss

ll(™,»)!l»SV' (*- ) ll(m,n)lliSv^(*{2y/E(t=7)y l1 + t t-s -s

>C-i-. ~- l ++ t Therefore Eq.(7.9) and the last inequality imply ft/2

m?(t;0,Bm,n)>C

{i'+t)(i + syda 2

f' r>

C c

s _s_

~ 1 + tJto

\2ds

r1' _ds_ - l+ + ttjt0 11+is ~

L

>^-t^(2

+

+ t).

So the desired result is obtained. Let J°(t) = {{m,n) e Qi(V~t) : £«?.D(*m.»)) > (1 - e)JS(Z?|D(JBm,n))} where 0 < 0 < 1 and Q^i/i) = {s e Z2 :|| x ||i< v^}- Then we have the following result.

Phase Transitions of IPSs

156

L e m m a 7.3.4. There are a,e0 £ (0,1) and D0 > 0,C; > 0,i = 1,3 and 6,- > 0,i = 1,2 so that ife £ (0,e0),8 £ (0,90),mD(t) > 1 - a for any t £ [a/2e,a/e] and i? > D0, then

£«?,D(Bm,„)2) > /?{log(l/£) + Ma/2)},

£ (m,n)6J 9 (()

wnere 60 = ( l / 6 i ) A ( 6 | / 4 C i C 3 ) and /3 = 62 - 2 V /0C 1 C 3 > 0. To prove this, we use the next Lemmas 7.3.5 and 7.3.6. L e m m a 7.3.5. If 0 < X < Y and EX > (1 - ff)£F, tien £ ( y 2 ) - E{X2) < 2^6EYE(Y3). Proof. Note that Y2 - X2 = (Y - X)(Y

+ X) = VY^X~{VY^X~(Y

+ X)}.

Using the Cauchy-Schwarz inequality and (Y - X)(Y + X)2 < 4Y3 for 0 < X < Y, we have E(Y2

- X2) < {E(Y - X)E((Y

- X)(Y

+

X)2)fn

<2y/E(Y-X)E(Y3). So the assumption, E(Y — X) < 0EY proves the lemma. L e m m a 7.3.6. There is an a £ (0,1) so that if a £ [a/2e, a/e], and mD(t)

> 1 — a, t i e n

I Jt l> \ I Q,(Vt) |. Let It = Q$\fi)

and Jt = jf. We assume | Jt \< \ \ It \, that is, \It\-\Jt\>\\Jt\-

Define Vm,n(t) =

E(ZlD(Bm,n)),

Um,n{t) = £«?iD(Bmi„)). So the definition of Jt yields

E

0W>« - u(m,n)(t)) >

(m,n)e/,

£

0W>(«) - uM(t))

(m,7i)e/iVi

>

E

eV

(m,n)(t)

(,m,n)£l,\J,

>(\It\-\Jt$9x

inf

V(m
Diffusive Contact Processes II

157

Hence from the assumption we have

E

(V(m,»)(0 - £W)W) > 5 I It I 9 X inf V(m,„,(t).

(7.10)

On the other hand, Lemma 7.3.3 (1) implies that if (m,n) G It, V(m,n)(t)

= mD(f;0,Bm,„) > h J - .

(7.11)

So combining Eqs.(7.10) and (7.11), we have E

(nm,n)W-P(m,B)«)>5l-rt|«X&1I^.

(7.12)

(m,n)6/i

Noting that if t is large then \It | = ( 2 V * + l ) 2 > 2 t + 2, from Eq.(7.12) we obtain

miD^D) =

E

mioiB^))

(m,n)€Z= (m,n)gZ=

• =(m,n)6Z E «w>(*) 2

(m,n)6Z 2

= E

*W)(*)+

(m,n)e/i

E

*W)(*)

(m,n)SZ 2 \/,

W1)V(2H2)x«x61I^+

< E (m,n)eii

' '

So

«(«?.D(Z2D))<

E

(m,n)€Z 2

*W)(0-**!«"

= £(2°^))-06^. If * < a/e then Piking a e (0,1) with

£(tf,D(z2D)) < (i - My* < a - M K . 0 < (1 - h8)ea

< 1 - a,

we have

E(tlD{Zl))
£(ff^(ZaD)) = m w ( 0 > l - a

E

*W)W-

(m,n)eZ 2 \/,

158

Phase Transitions of IPSs

gives

I Jt h l> \> \\ \h \h II •• We turn to the proof of Lemma 7.3.4 using Lemmas 7.3.5 and 7.3.6. P r o o f o f L e m m a 7.3.4. Let -^ ft,D(-^ni,ri)> Y ^ = — Zt,D\B™,n>Zt,D\Bm,n)X =—£t,D{B™-,n)>

Note that 0 < X < Y and if (m, n) e Jt then EX > (1 - 0 ) £ y . So Lemma 7.3.5 gives E(ZlD(Bm,n)2) n)2)-

- E(tlD(Bm,nf)

<

2y/0E(Z°iD(Bm,n))E(Z°iD(Bm,ny).

On the other hand, noting t < a/e < lit, the last inequality and Lemma 7.3.2 (1),(3) yield E(ZlD(Bm,n)2)

- E{HD{Bm,n)2)

<

-log(2 + t ) 2 ^ / 8 0 ^ ^ - ^

Moreover the last inequality and Lemma 7.3.3 (2) imply that if (m,n) E Jt, then t) 1 S ( 2 l0H 2+ > (h ~ 2 V / f l C ^ )2^C^' ° 1 3)+ ^; f ) 't ■.

Ei&ABm.n?) E(et,n(Bm,n)2) So

2 1 E(tin(Bm,r>)2) E(et]D(Brn,n) )>\Jt\(b2-2y/9C^-3) ^^-.

£ Lemma 7.3.6 gives

\Jt\>\\it\>t+x.

\>t+l.

Therefore Y,

E(tlD(Bm,n)2)

> (b2 - 2 V / 0C 1 C 3 )log(2 + t).

(mtn)€Jt

Observe that if t > a/2e, then log(2 + t) > logi = log (M

+ log(te) > log (j\

+ log ( | )

Let /3 = b2 - 2v/0CiC 3 . Then we have

£ (m,n)eJ (

so the proof is complete

Etiio(Bm,ny) > p {log ( I ) + log ( l ) } ,

159

Diffusive Contact Processes II

To obtain an estimate on the interference term (Lemma 7.3.15), we prepare several results. For a while, for simplicity, we consider A\(t;(x,0)) and B](t;(x,0)) on Z 2 . So we omit the subscript and/or superscript D = \. Following Durrett and Neuhauser, 4 we couple A(t;{x,0)) to B(t;(x,0)). We say Xf is crowded at time t, if || Xf - X°t ||j.= 1. When Xf is not crowded, we define displacements of Sf to be equal to Xf. When Xf is crowded, we use independent Possion processes to determine the jumps of Sf. For || x ||i= 1, let r% is the time of the ra-th return to crowded state for Xf and T£ = 0. Let Vn = | {T^_1 2} | for n = 1 , 2 , . . . , where | / | is the Lebesgue measure of J c R 1 and Sf '2 = Xf - X ( °. Note that both {V„ : n > 1} and {Wn : n > 1} are sequences of independent and identically distributed (i.i.d.) random variables. Moreover {Vn : n > 1} and {Wn : n > 1} are independent. L e m m a 7.3.7. There are A and to > 0 so that ift > to, then ^P([ er [ie(i (.ioo gg <)*] < ' ) <0)[M°S«*'1

<

*)

On the other hand, the following result is well known in a simple random walk : as t -> oo, P(Wi >t)r > t ) ~
^log4] <

P(r{k\ost] < * ) < < ! - 2 l o g r

since 1 - x < e~x. Set k = A l o g t . So if A > 2/C*, then e~c'k

£ -C*fc

= t~c'A

2 1}| |. . Let Xxf f =1 | , ==-. 1} = | {0 { 0<< u« << tf :|| :|| S^ ^ |lli

L e m m a 7.3.8. There are A,t 0 and C > 0 so that ift > t0, then 2 2 e 1 2 P(x t >CA(logt) )<2/t >CA(logt) ) . < 2/<2P(X?

< t~2.

Phase Transitions of IPSs

160

Proof. If T $ ( l o g t ) a ] > t, then

X?< Vi + --- + VWosm. Using a large deviations estimate for ii.d.random variables, (see, for example, Durrett, 5 ) if C > E{Vi), then P{X? < 5 A ( l o g i ) 2 , r ^ ( l o g t ) 2 ] > t) < P(Vi + ■ ■ • + V [ A ( I o g t ) a ] > C A ( l o g 0 2 ) <

e-K(C)^(logt)

2

<■ t -/c(C)Alog«

Lemma 7.3.7 and the last inequality impliy that, with C and A fixed, if t is large, then P(xV

> CA(logt) 2 ) < l/< 2 + i—* 5 )^ 1 "*' < 2/( 2

For x,3/ £ Z 2 , let P2(t;x,y) be the subtransition probability for the Markov process with Q matrix : » , , A _ /«(*>») A ?[(«,»), i f x / 2 / , where q{x,y) is the <5 matrix of 5 ' 1 ' and g ( i , y ) is the (J matrix of 5 ' 1 ' . Remark that if || a; | | i > 2, then q(x,y) = q(x,y) = q[x,y). Let pi(t;x,y) = P(Sj r ' 2 = y). So, for any 2 1,3/ G Z , P2(<;^.y) 2, we have » ( « ; » , » ) = P ( 5 J ' 2 = y, I| S*'2 | | i > 2 + P(S*' 2 = j , || Szu'2 HJ< 1

for any u e [0,*]) for some ti € [0,a])

< P2(«; *, 3/) + P{\\ SI'2 \\i < 1

for some u £ [0,«]).

The local central limit theorem implies that there is a Cj > 0 so that if || y — z | | i < 2^/7 and s is large, then P2(s;*ij0 > Cx/s. So, if || z | | i > 2, || y — z | | i < 2yfs and s is large, then Pi(s;z,y)>

Pi -j--

P{\\ Szu'2 | | i < 1

for some u e [0,s]).

We estimate the second term on the right hand side of Eq.(7.14).

(7.14)

161

Diffusive Contact Processes II

Lemma 7.3.9. There is a C2 > 0 so that if\\ z || x > s 3 / 4 /2, then P(\\ Szu'2 ||i< 1 for some u £ [0,«]) < C 2 /s 2 Proof. Let 0 ? =|| Sf'2 ||j . Since Uf is a submartingale, i " maximum inequality (see page 216 in Durrett, 5 ) implies that for p > 1 and I E Z 1 , £(( max tf„T) < -£-£!((l7f )P). 0
p— 1

(7.15)

On the other hand, it is well known in the case of a random walk that E{(U°SY) < Cps''2

(7.16)

So, by Markov's inequality, Eqs.(7.15) and (7.16), if || z \\^> s3'4/2, then P(\\ 5*'2 ||i< 1

for some u £ [0,a]) a 3 / 4 /2 for some it £ [0,a]) =P( max ^ > . 3 /"/2) 0
0 so that if II V — z ||i< 2>v/s,|| z ||i> s 3 / 4 /2 and s is large, then

( 7 - 17 )

S(i».»)sS/'. Moreover we have the following result.

Lemma 7.3.10. There are sQ and C3 > 0 so that Ifs>s0

and \\ y ||i> s3^, then

P2(t,e1,y) > - m i l S?J2S - y |t< V5) - F(ll s£2 - fe2 lb> V5)}. Proof. By the Markov property of S*'2, Eqs.(7.13) and (7.17), if || y ||i> s3/* and s is

Phase Transitions of IPSs

162

large, then fi(t;ei,y)=

^ zez2

P2{t-s;e1,z)f2(s;z,y)

> Yl P2(«-s;ei,z)P2(^;z,J/) z€Z 2

>

Yl

P2{t-s;e1,z)p2(s\z,y)

z:\\y-z\W<2^ Wli>»3/4/2

>—

Yl

V2^-s;e-i,z)

z:\\y-z\U<2^

=

Clp(\\S!Ll-y\\1<2VS)

> ~{P(\\ Setl'2s - y ||i< V~s) - P(\\ S& - S£'t ||i> v^)>First we estimate the term P(\\ S(ei'2 - y ||i< ^/s) in Lemma 7.3.10. Lemma 7.3.11. Let s = £ 2 (logt) 4 with B > 0. There are i 0 and C4 > 0 so that ift > t0 ands 3 / 4 <|| ?/ ||!< V<, then P ( | | ^ I ' 2 — y ||x< V~s) >Cs/t. Proof. To begin, P(\\ SHJ - y \\i< V^) =

£

P2(t-s;euz).

(7.18)

z-\\z-y\U<\/s

Note that if || y ||i< \/t, II 2 — y \\i< y/s and t is large, then || z ||i< 2\/t. So, the local central limit theorem implies that there are to and d > 0 so that if t > to, s = B 2 (logt) 4 and s 3 / 4 <|| j/ ||!< y/t, then M*-*;«!>•*) >CV*(7.19) Therefore, by Eqs.(7.18) and (7.19), we have the desired result. Next we estimate the term P(|| SfLJ - §(eI'2 ||!> s/s) in Lemma 7.3.10. Lemma 7.3.12. There are f0 and B > 0 so that ift >t0,s = B 2 (logt) 4 and s 3 / 4 <|| y ||i< \/i, then

p(\\s£2,-s?*

|ij> yi) < e/t2.

Diffusive Contact Processes II

163

Proof. By definitions,

P(\\s^-seti-2,\\i>V^) P{\\S£*-8tfb>Vi) V I ) = P

f e 'I {S" ~ X^ ~ <5° - X°J 11^ ^ ) - P f e II 5« _ X* ^ ^ + P ( 0 ?u^ I S° ~ X° ^ ^ /2) =Jl+J2.

Moreover, J\ is divided into two parts : h = P( omax( || S? - X « ||,> v^/2, x?1 < C4(logt) 2 )

+ p( wgt || a* - x<> |h> v»/2, x?1 > cyi(iogt)2) = J n + J12.

Note that when Xf1 is not crowded, || S?1 — X?1 || t is constant. Lemma 2.7 in Durrett and Swindle6 implies that when X f is crowded, ( ( Sf l t Xf l ) is an independent simple random walk on Z 2 . So we see that Ju < P(

|| S°u-2 \\x> P(logi) 2 /2 ).

max 0
By a large deviations estimate, we show that if B is large, then By a large deviations estimate, we show that if B is large, then Ju

< e -«(A,B,C)(log«) 2 < f -K(A,B,C)log<_

J

< e-K.(A,B,C)(\o6tf

u

< f-K(A,S,C)log(

(7.20)

2

On the other hand, Lemma 7.3.8 gives Ju < 2/t . Combining this fact and Eq.(7.20) yields Ji < 3ft2. Similarly, we have Ji < 3/t 2 , so the desired conclusion is obtained. Using Lemmas 7.3.10-12, there is a C6 > 0 so that if t is large, then ft(*;ei,y)

C4; - «l t

r
6. t>> 6

B2(logt)H2

}

t

We turn now to the setting of Z^,. By the above result, we have the following lemma.

164

Phase Transitions of IPSs

L e m m a 7.3.13. There are D0,B and Cb > 0 so that itD > D0,r(D) and r{D) <|| z |Ji< 1, then

=

B3'2{\ogD)3/s/D

p?D(*;-£=,*) >Cs/D.

On the other hand, the following result is also used for estimate on the interference term.

L e m m a 7.3.14. There are a,e0 £ ( 0 , 1 ) , B and C > 0 so that if e £ (0,e 0 ),< < a/e,D < K5 log(l/g)/e and r ( K ) = B3'■'n;(log £ ) 3 / ' / D , then

E J]

66

£

E

< C ( l o g £ ) llog(l/ o g ( l / e£))/.D. /.D. /■(i;0,z;0,3/) /(i;0,z;0,3,)
5&(wr(DV

Proof. Using Lemmas 6.3.1 (5) and 6.3.6 (6), for t < a/e,

E

£ E

£ x z

^l

/ ( *( ; 0 , Ix; ;00, ,!y/ ) < CC{r{D)VD} { r ( i } ) V f l2 } 2 j * Jo

yeQ£(z-,T(D)) ye§£,(a;;r(r>):

*p?+D(s)ds

L e m m a 7.3.15. There are a,e0 £ ( 0 , 1 ) , B and C > 0 so that its £ (0,£ 0 ),< £ [a/le + 1,0,/e] and D < Ks\og(l/e)/e, then

/D D

J (« t) * >

E E{Sl (B , )(Q (B "I (m.nJeZ E E(Ctjj(Bm,n)(eJ)(^,»)

|{

2

i

m,n)eZ

D

m n

D

m
-^W.)}.

-1))1)) _ M £ > ! l o g ( 1 / e ) \

a

Proof. Let Q " = Q £ ( s ; r ( D ) ) where r ( D ) = B3/2(\ogD)3/y/D.

Using Lemmas 7.2.5 and

Diffusive Contact Processes [I

165

7.3.13, ID(t) = 4

E p ( a > a + -7^tf,D) aezl

^

4e

~2

E

E

E

E

(m t n)eZ 2 x£Bm,n

= 4e 2

"

P(A^l;(x,y))

E

P(A%(l;(a,a+-^) — ) = (*>?))(x,y))P(X,yeet_liD)

(m,n)6Z 2 «6B m i I , j / e B m . „ \ { i }

= 4e'2 E 4e 2

^ "

E

E

pfcas 4f,»-*>*(*,» e &».D) Vi

(m,n)6Z"i6B„,„j6Bm,n\{i}

E

E

E

(m,n)eZ= i g B m . „ »6(B m ,„\{x})ne**(i)«

^§ E

E

E

(m,n)eZ 22 *eB m ,„\{x})nQ"( 3 ;)' m ,„\{x})nQ"(j;)= (m,n)€Z m ,„ 3/6(B m

= §{ E E

= C-+^)) (a,a+-^=))P(x,yeet-1,D)

E

veB m ,„\{x}

E

(m,n)eZ 2 i € B m , „ y £ B m , „ \ { i }

- E

^

?S3(i;^,2/-^Kj/ef?_1,D)

^,»6f?_ll0)

n^e^)

E

E

(m,n)eZ 2 *6B m ,„ y€(B m ,„\{x})nO"*(i)

>§{

E(et:D(Bm,n)(et,D(Bm,n)-l))- - i ) ) -

E (m,n)eZ=

n*,yt$-i,v)}

E

E

J D fcO,*;0,j,)}.

r e Z j , j, e $g,(x;r(D))

So Lemma 7.3.14 completes the proof. The following lemma will finish the proof of Proposition 7.3.1. Lemma 7.3.16. There are a,e0 € (0,1) and K5 > 0 so that if e e (0,eo),mD(t) for any t E [a/2e + l,a/e] and D < Ks log(l/£)/e, then ^-mD{t)<-Ze. at Suppose Lemma 7.3.16 is valid. Comparing with the DBP, mr>,± v

+ ]_) < 1RD( « + i) = e t+« < 1 + a + 2e. 2e 2e By the last inequality and Lemma 7.3.16, if e is small, then

^(j:e

/7

i.m»{a)d. + m B ( i + l )

7 f r + 1 ds

< - 3 e v( ^ - - l ) + l + a + 2£ ~~ 2e '

=-J+l+fe.

is

> l-a

166

Phase Transitions of IPSs

With 0 < a < 1/3 fixed, if £ is small enough, then mD(a/e) 7.3.1 is complete. P r o o f of L e m m a 7.3.16. From the fact that mD(t) and Lemma 6.3.8 (1), jtmD{t)

< 1, so the proof of Proposition

< mD(t)

et = e < 1 + 2a for i < a/e

\ID{t),

< 41 + 2a) -

(7.21)

where e = 4 A — 1. Let

U*(t)= £ (m,n)6Z 2

um= um=

£

(m,n)eZ 2

E($_ltD(Bm,n)($_1
£

(m,n)gZ 2

(tf-ijj(Bm,»)).

Note that f/*(i) = U?(t) - f2*(*)• By Lemma 7.3.4, we have

P?(t-l)>0{log(l/O + log(o/2)}.

(7.22)

On the other hand, comparing with the DBP implies for t < a/e, U£{t - 1) = mD(t - 1) < m D ( t - 1) < e a .

(7.23)

Using Lemma 7.3.15, Eqs.(7.21)-(7.23) and A > 1/4,

|m°(«) < £(1 + 2a) - §{ff*« - 1) - M ! l o g ( 1 / £ ) } = e(l + 2a) - £ { D * ( i - 1) - tf2*(i - 1) -

^

< e(l + 2a) - - [/3{log(l/e) + log(a/2)} - e a -

^-log(l/e)}, (1 S )6

° / log(l/ £) ]

< e ( l + 2 a ) - g [ ^ { i l o g ( l / £ ) + log(a/2)}-6a] ) where the last inequality follows from the fact that, if D is large, then

(f-(TV(l,.>,».

Let g(a) = -101og(a/2) + 10e"//3- So, if D < Id log(l/e)/e with iif5 = C/3/10, then d

D

9

4+, j°] A. (7.24) J m w(t)< £ {2a T dt - I log(l/e) J With a fixed, if e is small enough, then g(a)/ log(l/e) < 1/3. So, by Eq.(7.24) and picking a < 1/3, the proof is complete. 7.4. Lower B o u n d s for d > 3 In this section, we give the lower bounds in d > 3 :

Diffusive Contact Processes II

167

P r o p o s i t i o n 7.4.1. TAere is s C > 0 so that if D is large, then

*-c*>*s?-4 To do this, we will present three different proofs in this section as follows : (i) BDS method, (ii) Differential inequality method, (iii) Griffeath method. (i) BDS Method. The following estimate on the interference term is essential to our proof as in the proof of Proposition 3.1 in BDS. L e m m a 7.4.2. If D is large, then for t > 1/D,

ID(t)>^mD(t-±).

h>

Proof. It is sufficient to show that if D is large, then for any x G 7idD,

E

y€Q?(x;l/V5) y€Q?(x;l/y/D)

^■VEtfj,)^^6^)-

The inequality follows from combining these four facts, (i) The probability the particle at x survives from t — 1/D to t is e~llD > 1/2. (ii) If it survives, then the probability it will give birth to at least one particle is 1 - exp (-2dX/D) > 1/2D. (iii) If a particle is born the probability it will survive for at least 1/D units of time is e~^lD > 1/2. (iv) Moreover, the probability both particles will not jump for at least 1/D units of time is exp{2(2d — 1)}. P r o o f o f P r o p o s i t i o n 7.4.1. Let e = 2dX-l. imply that if t > 1/D, then

Lemma 6.3.8 (1), Lemma 7.4.2 and A > l/2d

lmD{t)<emD(t)-^mD(t-±).

»-*>

Comparing with the DBP gives mD(t) Therefore

< e'lDmD{t

-

~).

M^(.-M-%

168

Phase Transitions of IPS)

If D is large and e < C/3D, then e~BlD > 2 / 3 . So, using the last inequality, we show that if D is large, t > 1/D and g < C/3D, then D D {t) < e -&(*-•*! e-<'m -&m (l/D). (l/D). mD(t)

Therefore, mD(t) -* 0 as t —>■ oo. fi'ij Differential Inequality Method. By using the differential inequality of the DCP (Lemma 6.3.8 (2)), we can obtain the following more refined result. P r o p o s i t i o n 7.4.3. If D is large, then 1 — 1 1 Xc[D) C(D) >+ -2d {2d)t{2d-l)D2d + (2d)2(2d --1)DProof. Lemma 6.3.8 (2) yields \{lm^WsiCl + J ^ ^ + ici-J)^"-^ ™D{t)<\{l+°-)e^

+

= (t) ++ *?(*), = x? *f (*) »?«, where cr = u -=| {^{2(2d 2 ( 2 d - l)D \)D + 2{d 2(d + 1)A + 1}, 1}, - 1)D 1)K + (d + + 2y/d 2A/5 + + = [{(2d [{(2d+ (d

K =V

fi fi ===

i>r.

1)A + + i}{(2di}{(2d -- 1)£> 2s/d + 1)A + + -}] !)£> + (d ---2v/d+l)A 1)\

,

- { ( 2 d -- l ) J-{(2d-l)D-(d-l)\+l}. 3-(d-l)A+|}.

Let £ = 2dA — 1. If D is large, then /i < 0. So we see that x f ' W —* 0 as f —> oo. On the other hand, simple computation implies that, as D —t oo, M + ^ +

22 ,, 11 >. „ d„ A _-, 1 _22dA _^ I _ 11 ++0 °( ( _y )),h, i/, == 22dA - 1 - 2d-lD D

where / ( i ) is o(i) as t -> 0 means f(t)/t -> 0 as ( - t 0. Using this fact, we see that if £ < l/{2d(2d - l)D] and D is large, then s f (t) -> 0 as / —► oo, since A > l / 2 d . So the proof is complete. We have the following result by the Griffeath method, 7 which is found in Katori and Konno. 8 A similar argument is also applicable in the DCP on a tree, see Section 10.4.

169

Diffusive Contact Processes II

P r o p o s i t i o n 7.4.4. For any d > 1,

(2rf-l)J + l cV

;

- ( 2 d - 1)(2AD + 1)

£ 2d'

where 1? > 0. For simplicity, from now on we assume d = 1 in the rest of this section. (in) Griffeath Method. As in Section 5.2, we can obtain the following correlation equalities and inequalities. Note that, as we saw in Example 5.4.4, the DCP is coalesing self dual. L e m m a 7.4.5. (1)

- ( 2 A + 1)CT(0) + 2A
(2)

(7(0) - (A + l)cr(Ol) + ACT(012) + D[
L e m m a 7.4.6. (1)

(7(012) < 2
(2)

By using Lemmas 7.4.5 (2) and 7.4.6 (1),(2), we have (_A + 1 + 2D)(7(0) + [A - (1 + £>)]CT(01) > 0.

Lemma 7.4.5 (1) yields [(21? + 1)A - (D + 1)](7(0) > 0. So if (2D + 1)A - (D + 1) < 0, then (7(0) = 0. Note that a(0) = p(0) = px(D). Hence we have

K(D) > ^ t _ .

170

Phase Transitions of IPSs

References 1. M.Bramson, R.Durrett and G.Swindle, Statistical mechanics of crabgrass. Ann.Probab. 17 (1989) 444-481. 2. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York,1985). 3. E.D.Andjel, Preprint (1994). 4. R.Durrett and C.Neuhauser, Particle systems and reaction-diffusion equations. Ann. Pro6afc.22(1994)289-333. 5. R.Durrett, Probability : Theory and Examples (Wadsworth,Inc, California, 1991). 6. R.Durrett and G.Swindle, Coexistence results for catalysts. Preprint (1993). 7. D.Griffeath, Ergodic theorems for graph interactions. Adv.Appl.Probab. 7(1975)179194. 8. M.Katori and N.Konno, On the extinction of Dickman's reaction-diffusion processes. Physica A. 186 (1992) 578-590.

Long Range Contact Processes I

171

CHAPTER 8 L O N G R A N G E CONTACT PROCESSES

8.1. I n t r o d u c t i o n In this chapter we consider the following contact process with long range interactions, so we call it long range contact process (LRCP). For z e R d , define || z \\p= (|zi| p + 1|Z(i|p)i/j> with 1 < p < oo, and || z ^00= s u p 1 0 , M = 1 , 2 , . . . , vd{M) = | {x £ Z " : 1 < | | x || p < M} I, and Jjxiy) = Tj(y) ify^x, ijx(x) = l-n(x). When M = 1 and p = 1, the LRCP is equivalent to the basic contact process. For simplicity, from now on we asumme that p = 00, so set || • | | = | | • 11^, . In general case, p > 1, by using similar way, we can obtain the same results in this chapter. The LRCP is attractive, so the upper invariant measure is well defined : for any A > Q,M = 1 , 2 , . . . , V\,M = Urn S1S(t), where Si is the pointmass on f] = 1 and S(t) is the semigroup given by fi. As we saw in Example 5.4.6, the LRCP is coalescing self dual. So for x E f with £ C Zd,

£-f\{*}

at rate 1,

?-£U{j,}

at rate

vd(M)

for || y — x ||< M.

Phase Transitions of IPSs

172

Let £ ° M denote the LRCP starting from the origin. Define the survival event and survival probability by

ft» = { # , M #

for all * > 0},

Pxittco) = P($,M + <S>

for all t > 0).

Let £ j M be the LRCP starting from T] = 1. Then the order parameter and critical value are defined as follows : px{M) = vx,M(n ■■ nix) = 1)

= P(o e &M) AC(JW) = inf{A > 0 : p„{M) > 0} = inf{A > 0 : P(0 £ & , , M ) > 0} = inf{A > 0 : Px^oo)

> 0}.

Bramson Durrett and Swindle (BDS) 1 showed T h e o r e m 8.1.1. (1)

AC(M) -► 1

(2)

px{M) -► —^—

as

M -* oo.

as

M -► oo.

In this chapter we will give their proof of this theorem. Another proof can be found in Section 7 of Durrett. 2 For the convenience, as for the basic contact process, we change the scale of rate from 1 to l/2d. Let \c(d) be the critical value and p\(d) be the order parameter (survival probabolity) of the basic contact process. As we saw in Chapter 6, the following results are known (see, for example, Schonmann and Vares, 3 )

(1)

Ac(d) —> 1

(2)

Px(d) -> —r—

as

as

d ~» oo.

d —>■ oo.

A

Similarly the same rate change for the DCP implies that the above results are very similar to those of the DCP, that is,

Long Range Contact Processes I

173

Theorem 6.1.1'. (1)

AC(Z>) - 1

(2)

^(tfJ-Azl

as

D -» oo.

^

D^oo.

Therefore the asymptotic behavior of the LRCP with long range limit are similar to both that of the basic contact process with high dimension limit and that of the DCP with rapid stirring limit. As in the DCP, Lemma 8.1.1 can be obtained as follows. By the definition of ft, we have (A_1)p(0)

-OTTAx{

D

p(<>y)}=o,

(8.i)

where p(x1x2 . . . ! „ ) = VX,M{TI ■ V&i) = 1 for all i e { 1 , . . . , n}). As M —> oo, we assume that P(0y)

= P(0)\

(8.2)

that is, any two sites are independent. This assumption means that mean field theory is correct for the LRCP with long range limits. In other words, this limit has product measure as stationary measure as in case of the DCP with rapid stirring limits. Combining Eq.(8.1) with Eq.(8.2), we have

px{M) = p(0) = ^ 1 . As in the DCP, another interpretation of the mean field theory is given in the following. To do this, we introduce ZfM which is a continuous-time branching random walk on Zd with range M starting from A. The dynamics of this process is as follows : (i) each particle dies at rate 1, and gives birth at rate A, (ii) a particle born at x is sent to a site y chosen at random from {y £ Zd : 1 <|| y — x ||<

M}. We call this process the long range branching process (LRBP). As in the diffusive branching process (DBP), we define the survival event and survival probability of the LRBP by ftoo = {ZlM ± for all t > 0}, Px(Roo) = P(Z?,M +

for all t > 0),

where Z°M denotes the LRBP starting from the origin. It is well known that PA(fioc)=—^-, inf{A > 0 : iMfioo) > 0} = 1.

Phase Transitions of IPSs

174

So in this meaning, as M -* oo, the LRCP approaches to the LRBP. As we saw in Chapter 6, similar phenomena occur in the DCP as the stirring rate goes to infinity. The following results (Theorems 8.1.2 and 8.1.3) refine Theorem 8.1.1 as Theorems 6.1.2 and 6.1.3. Theorem 8.1.2.

fC/M2/3, I C(log M)/M2, { C/Md,

K(M) -In

d=l, d = 2, d > 3,

where m means that ifC is small (large) then right hand side is a iower (upper) bound for large M. We can rewrite this theorem in terms of volume v = vd(M) : Theorem 8.1.2*. [Ch*l\ d=l, A C ( M ) - 1 « I C(\ogv)/v, d = 2, [ C/v, d > 3, where « means that ifC is small (large) then right hand side is a lower (upper) bound for large M. Theorem 8.1.3. Let

(C/M2/3, A - l = < C(logM)/M 2 , { C/Md,

d=l, d = 2, d>3.

There is a constant 6d(C) with 64(C) —> 0 as C —> 00, so that if M is large, then PX(M)

> (1 -

Sd(C))^.

As in the Conjecture 6.1.4, we present the following conjecture. Conjecture 8.1.4. Ifl<M
then \C(M) > XC(M).

This chapter is organized as follows. Section 8.2 is devoted to some preliminary results. Section 8.3 gives the proofs of Theorem 8.1.1 (1) and upper bounds in Thorem 8.1.2. In Section 8.4, we prove lower bounds in Theorem 8.1.2. Note that the proofs of Theorems 8.1.1 (2) and 8.1.3 are similar to those of Theorems 6.1.1 (2) and 6.1.3, so we will omit them. (See details in BDS.)

Long Range Contact Processes I

175

8.2. Preliminary Results From now on we change the space from Zd to ZdM = Zd/M with M > 1. Note that this change does not have any effect on critical values and order parameters defined in the previous section. As in Chapter 6, first we introduce some notation. For z e R d define || z || p = (\Zl\" + ■•• + |z d |P) 1/p with 1 < p < oo, and || * || w = s u p ^ ^ |*;|. For A C R d , | A | is the cardinality of A in Z d and | A \u is the cardinality of A in ZdM. Moreover for x e ZdM, u e Kd, l 0, let Q P (z; r) = {3/ € Zd :\\y-x

\\p< r},

Q?{*\ «0 = {2/ € Z j , :|| v - x ||„< r}, QP(u;r) =

Qp(x; r) = Qp(x; r) \ {x}, Q f (x; r) = Q f (x;r) \ {x},

{yeKd:\\y-u\\p
When x is the origin, we omit x = 0. For example, Qp(r) = QP(0;T). subscript p, if p = oo. For n = (m,...,n<j) £ Z d , let

Moreover we omit

-8° = [n!,ni + l ) x ---X [nj.n.i + l), •Bi = [™i - !i rei + !) x • • ■ X lnd - 1) nd + 1)Note that

x

BlcBlcQR{n,l).

Let T)itM denote the LRCP with range M with state space {0,1}Z« : at each site £ ^il' Vt,M(x) takes t i e value 0 or 1. They evolve as follows: there are flip rates c 0 (x,7/) = 1, Cl (x, J7 )

= A|fnQf(x;l)|M,

so that if r]t,M(x) / i then P{qt+s,M{x) — * I '/(.M) ~ c,(x, 77^)3, as a -► 0 where /(*) ~ fl(«) means f(s)/g(s) -* 1. Let &,M be the set-valued process for 7 7 ^ on ZdM : 6 , M = {x £ZdM: T?(,M(X) = 1}.

Sometimes we identify ijt,M with ft,MLet f £ 0 be the LRCP starting from ££M = A and f^ M (B) =| f£ M n 5 | M where A C Zj{^ and B C R d . Moreover we introduce Z*M, a LRBP on Zdu with range M starting from A. The dynamics of the DBP is as follows: (i) each particle dies at rate 1, and gives birth at rate A. (ii) a particle born at x is sent to a site y chosen at random from QM(x; 1).

176

Phase Transitions of IPSs

Similarly ZfM{B) denotes the number of particles in B at time (. By definitions, the LRCP can be thought as a coalescing LRBP, that is, a system which follows rules (i)-(iii), and (iv) if a particle is sent to an occupied site, the two coalesce to one. We will review some definitions which were given in the previous section. Let floo = {f?,M 7^ <£ for all i > 0} where f ? M denotes the LRCP starting from £g M = {0} ( i.e., £fM when A = {0}). We define the critical value \C{M) = inf{A > 0 : i^fi,*,) > 0} and the survival probability ( order parameter ) p\(M) = P(f2oo). It is well known that the limits 1 and (A — 1)/A are the critical value and the survival probability for the LRBP respectively. That is to say, define n ^ = {Z^M ^ <j> for all r > 0}, so 1 = inf{A > 0 : P(fi«>) > 0} and (A - 1)/A = P(fJoo) where Z^ denotes the LRBP starting from Z £ M = {0} (i.e., ZfM when A = {0}). Here we would like to confirm our objective. In Sections 8.3 and 8.4, we will give the proofs of the following theorems following BDS. However the proof of Theorem 8.1.1 (2) is similar to that of Theorem 6.1.1 (2), so we will omit it. T h e o r e m 8.1.1. (1)

A C (M) -^ 1

as

(2)

Px(M) -> —^—

M -► oo.

as

M -^ oo.

T h e o r e m 8.1.2. For x > 0, let

r i/*v3 ;

d= i,

d = 2, d> 3 .

Then we have A C (M) - 1 w C % ( M ) , where ~ means that if C is small ( large ) then the right hand side is a lower ( upper ) bound for large M. From now on we present several preliminary results. Let ■Sf'" (resp. Sf'^f) be a continuous-time random walk starting from x and have jumps with the uniform distribution on Q(x;M) (resp. QM(x-1)) at rate v on Z d (resp. ZdM). Define Pt,(t;x,y)

= P(S?>" = y)

and

p f (t;x,y)

= P(S^

= y).

Note that Sf' (resp. St 'M) can be thought of as a continuous-time random walk starting from x and have jumps with the uniform distribution on Q(x; M) (resp. QM(x; 1)) at rate

Long Range

Contact

Processes

I

X77

f ( 1 + l/vd(M)) on Zd (resp. Zj^). \Ve can check this directly as follows. For simplicity, we assume that d = 1, so fli(Af) = 2M. Let p(n\x,y) be the probability of an n-step transition from i to y on Z with

_ 11/2M, if J. < \z - »| < M,

P 1 1 1l^,!/) P * , W - | \Q )0,

otherwise. otherwise.

So we have

E ^n! c , » ) .

P„(*; x >3) = «-*" *—* n=0 n=0

Using the inversion formular for characteristic functions,

I / " 2W-,

p ( n ) (*,
g

-i(2;-!/)(

where

'[«#)]-•»,

h{tcosm)-

m-- xez•^1i C f O j l e " *

\fc=i

I

So the last equality gives

_ i_ P„ (*»*»*)= ^ r / _ cos(x o s ( z—- 93/)0exp ) 0 e x Ip |~ut - i / t ( l - - ^ ]Tcos(fc0)J p„(*;s,sO ' 2TT

(i-iE^w; } d0.

I — IT

(8.3)

Similarly let p^ n ) (x, t/) be the probability of an n-step transition from x to y on Z with

t

if n < is - ?(l < M

r i/(2j *2ikf + 1),

IT- H .T. K 7

P

'^-|0,

Set 7 =v

otherwise. otherwise.

/

1

\

( 1 + »,fMW

^

2M + 1

:M

So we get

p„(*;s,z/) Hence

fn\x,y) where

=

=-7lE

t

T*)\ lUx.

IA

n=0

1

2*/-,

e-%*-v>)»$(B)]*d»,

l

M

i I MB) ^1 +(l 2+2^ cos(fc0) £>(**)) m == J2^(0,x)e^" 2=M— +1

Phase Transitions of IPSs

178

Then we have

i

x pPvit'^'V) =- ^— / u{^ ^y)

r ccos(z os x (

x j/)0e eexp - 3/) p -ut

2M + 1 1M

1 I 2M + 1 I

1 /" =: 2—^ // ccos(z o s ( s --yj /) )»0eexxpp | --i /*t I 1 1 -- 7— s(fc0)j I 7 2^ c.o«»(**)

M

+ 2V

cosffcfl)

<».

So Eq.(8.3) and the last equality give the desired result : p„ (*;z,sO = ?„(*;*, <>)■ For £ C R d put p„(f;z,.B) = £ p > , ( i ; z , 3 , )

and

P„ M (t;x,-B)p^{t;x,B)=^2i^(t;x, y). : yes

vefi

Define

q?(t;B) = sup i€Z{

P. •jf(Ux,B).

For i > 0, let 7<JM =

f V£, rf = i , n log 11, d = 2 , I 1. d>3 .

The following facts are very useful for our proofs. L e m m a 8.2.1. For v > 0 , M > l,t > 0,x,y

6 Zd,u,v

e %Am and A C Kd,

(1)

PKM(*;«,i;) = p 1 (W;M«,Af ? ;),

(2)

pf(t;s,A)< ? f(i;4).

There is a constant C so that if 1 < A < 2 and M > 1, then

(3) (4) where w £ R d and r > 0.

fteVi^c^/a+tf, / Jo

q™(s;QR(w;r))ds
I <*0

Long Range Contact Processes I

179

Proof. By the definitions, the proofs of (1) and (2) are obvious. Part (3) comes from pages 451-452 in BDS. The definition of fd{x) and part (3) give the proof of part (4). For A,B C Rd,x,y

e ZdM, and k = 1,2,..., define

m^(t; A,B) = E(tfM(B)k), mff(t) = E(t%M(ZdM)k),

E(Z*M(B)k),

m*?(t; A,B) =

m f (t) = E(Z?iM(ZdM)«).

In particular, we omit the subscript when k = 1. Moreover, IM(t;x,a;y,b)

= E(^M(a)^M(b)),

rMw= E £/"(*;*, «;*>&), z

7M(t;x,a;y,b) =

/Mw = E

b

a

° e M ~"

z

e M

E(Z?M(a)ZlM(b)),

^M(t;x,a;x,b), b

~"

d

where 6 ~ a denotes a,6 e Z M with 0 <|| a- b \\< 1. Note that m^(t),m^(t),IM(t) and f (t) are independent of x. On the other hand, £fM and ZfM can be constructed on the same probability space with

tfM{B) < ZtAM(B). So the above definitions imply that for A,B C B,d,$,y,a,b £ Zff,k > 1 and t > 0, we have mf{t\ A, B) < mf{t\ A, B), IM(t;x,a;y,b)
mM(t) < mM{t), IM(t)<7M(t).

Moreover noting that d

-pM(t; x, y) = X E ? f ft »,» + *) - «d(M)Apf (t; ,, j/),

dt

z~0

we have Lemma 8.2.2. (1)

m Af (i;x, S ,) = e( A - 1 ' t pf(*;^3/)-

(2)

m M (*; :c ,B) = e ( A - 1 ) t pf(<; 2 : ,B).

(3)

mM« = e^1" Hence we have

Phase Transitions of IPSs

180

L e m m a 8.2.3.

o<

1 \x--!)(_ 0 < mM(t) < < mM(t) = e< eL*)*.

Sp this result allows us to conclude that 1 is the trivial lower bound on \C(M) : L e m m a 8.2.4. \C{M) A > 1. C(M) > lThe following results are also useful to analyze various behavior of the LRBP. Let Y denote the set of aU real-valued bounded measurable functions on ZM. In particular, for a e %\t^a, £ Y Js defined by Sa(a) = 1 and Sa(b) = 0 if 6 ^ a. For each / £ Y, let

E

Z < = J^ ZlZl¥(y)f(y)¥(y)f(y). t,MMJ>=

yezit L e m m a 8.2.5. For f,g £ Y, x,a,b £ ZdM, B C R d , M > 1, \,t > 0, p > 1, r > 0, and

J*

k E() £ ( < ^ MM,f> >) ' / )> ' : ) =

(i) (1)

+-- ^ E ( ' ;)EE ) E E (2)

)-^( > < M< zi%Z £ ? , / > « z^ ?-*,M>*y »E« ,/>IMJ /fdsEW >,M>

E(< Z?iM,g M<M M,g>)

J «y > ) £ ( < Z J,M Z

J

f >"-j3)),,

>) = =-E{< E(< Zfajg ^ M , / f f >>))

A

, * y »{£(< > ) £»E(< ( < Zl%^ +«d(M) VTm £E J2 E ffd»E{< ZU,M,Sy »{E(<7!/ ZlM/ ,f Z*%g dsE{) »

+

+E()E()}, , S > ) £ ( < z&J ■»}. ^s,M (3)

m^{t;x,B) m f(t;i,B) = = m*'((;i,B) m¥{t;x,B)

A

•tin

+ ^ M I E ( , - ) E

(4)

l^lt-r

E

/

n-r hi - t> (h)F,(Zx.

<^m M (*

^,y)mf(S;y,B)^{sly

(n\\4-

> z~0yeZ*M

z,B\

/ rf« ^ T M ^- s ; xc-^ , j / ).A

\~* V d{

+

■*«

Long Range Contact Processes I

181

X {mM(s; y, a)mM(s; y + z, b) + mM(s; y, b)mM(s; y + z,a)} ,

(5)

E E 7 ^ * ' * 1 ; * ' 6 ) ^ e(A_1)'pf(«;a;,4n B) aSAbeB

+\e^-V* j p f (i;*, A) jf' 9 f (S; B)ds + p?(f,x,B)

(6)

IM(f,W,x,b)<2\e^-Vi

E

E

»«!,

66Q"(«;r)

[

J* « f (s; A)ds} .

M

q™(s;Q

(r))ds.

°

Proof. As for parts (l)-(4), the proofs are the almost same as the those of Lemma 6.3.6 (l)-(4). Part (5), combining Lemmas 8.2.1 (2), 8.2.2, and the Markov property of Sfy implies A

E E E

/ dsmM(t-s;x,y)mM(s;y,a)mM(S-y

E

a£AbeBz~Oy£zdM

< e 2(A-i)< A ^ £

+ z,b)

°

fdspM(t-s;x,y)pM(s;y,A)pM(s;y

+ z,B)

f q™(s;B)ds. Jo

The rest of the second term is also obtained by a similar argument. So part (5) follows from part (2). As for part (6), using Lemmas 8.2.1 (2), 8.2.2, the Markov property of S*jg, and part (4), the desired conclusion can be obtained in the similar way. Lemma 8.2.6.

±mM(t) =

(1)

(2)

|

m

M

( t ) =

(\-l)mM(t).

(A_1)mM(i)__A_7M(i))

So we are interested in the term IM(t) which appears in Lemma 8.2.6 (2) but not in (1). We call this term the interference term as in the case of the DCP.

Phase Transitions of IPSs

182

Proof. For part (1), Lemma 8.2.2 gives the proof. For part (2), the definition of the LRCP implies

in* e &,M=) -P{*

6

A vd(M]

&,M)

P{y e Kt,M, &,M) y~X

-^E^^>"v

' y~x

Summing over x yields the desired conclusion. We need the following lemma to estimate the difference between the LRCP and the LRBP. Part (1) will be used in the proof of Theorem 8.1.3. Note that in part (2) a time t is fixed, so we apply this part to the proof of Theorem 8.1.1 (2). L e m m a 8.2.7. Let A * M = Z*M(ZdM)

2A2e2(A-l)t

(1)

(2)

£(Af, M ) -

vd(M)

with t > 0

^M(ZdM).

-

J (j\M(u;QM(l))du)ds,

fixed,

lim £ ( A ? M ) = 0.

M-+oo

'

Proof. The definition of Af M gives

T t

E

So we have £(A?,M)

Note that

x

-

^

iM(t)

E(&lu)

dlM{s)ds

A " vd(M)

fit)

Vd{M) J0

=E E

= 0.

IM(s)ds.

r^(i;i,o;s,6),

" 6 Z M 6€« M (a;l) >€« M (o;l)

so Lemma 8.2.5 (6) for r = 1 gives the proof. As for part (2), o2 A\ 2 e 22(A-1)«, ( A - l ) t < /•* £(A?,M) <

/

qf(u;QM(l))du

.In

Ctiijt) ~ Vvid((M)' M)' <

With < fixed, Cad{t)/vd(M)

- ^ O a s M - n s o , since u r f (M) = (2M + l ) d - 1.

183

Long Range Contact Processes I

8.3. Upper Bounds In this section we prove the upper bounds for any d > 1 in Theorem 8.1.2 by using comparison method which was studied in Section 4.6. That is, LRCP £t,M —* oriented site percolation £„ —* LRCP (,t,MTo do this, we need some notation. Let Ki,Kt,K3, will be chosen later. Define

and if4 be positive constants which

e = 2dX - 1, L = K2/e and /,■ = [2j - 1,2j + 1] x [-1, l]"" 1 , where j £ Z1. Let f t M ( r e s P- Z«,M) denote the modified LRCP (resp. LRBP) in which births outside {—2\/L,2ifL) x R d _ 1 are not allowed. Let S*'^ be the random walk S*'M killed when it leaves (-2-v/T, 2\/i) x R d _ 1 . Let Z^ M be the modified LRBP starting from A

x

ZQM = {x} (i.e., ZtM

when A = {x}.) Then we have

Lemma 8.3.1.

E(?tM(B)) = e ^ P O f t e B). Proof. Let pf(t ; a ; , 2 / ) = P ( 5 ^ = j,). From the definition we have

akfe-.*)-

\,:\y1+z1\<2VZ)PT(t>X'y+Z)-XP^(t'X^)-

So (A - l)E(Z*tM(y))

+ e ( A - 1 ) ( | p f (<;*,») = (A -

i)E(XM(y))

+ e (*-l)*[.

^ffi E W ^ K * ^ (*;*»»+*) -Apf(t;x ) 3 ,)j.

On the other hand we have ^(KM(J))

=^ ) E

W

+

* I < ^ ( ^

+ *» -

fi(^(f))

= (A - 1)£(KMM)

+ ^ ) E W * > H^E(XM(y

+ *)) " ^ , M ( r i ) .

Therefore the desired conclusion is obtained. We have the following lemma which is similar to Lemma 6.4.4. The proof can be found on page 467 in BDS.

Phase Transitions of IPSs

184

Lemma 8.3.2. Let L = K2/e. There is an e0 > 0 so that if e G (0,eo),K2 > 0 and j = ± 1 , then inf M(5"£M G VZlj) > p/2 > 0. M>1

Let vfM(B) yLIo, then

= e-cizfM(B).

By Lemmas 8.3.1 and 8.3.2, for any e 6 (0,£ 0 ), if A C

E(VtM(y/Lh)) = £ P0&

e VZh) > | | A L ■

(8.4)

On the other hand, using Lemma 8.2.5 (3) for k = 2 and 5 = ZJ^, we have E(ZlM(ZdMf)

= E{ZlM{ZdM)) + 2A / '

E(Z?_sM(ZdM))E(ZlM(ZdM)yds,

Jo

so

E(ZlM(ZdM)) = e*

gives „ef + i o2\e* \ „ e < // ee"ds £> E(ZlM(ZdMj&f) ^2^ _= e" Jo •2(e + l) £ + 2 _e _ c2Et f 2(e + 1) « + j * e - r i \

Hence

e-itLE{ZlM{VLhf)

<

e-^LE{ZlM{ZdMf) , 2(e + l)

e+ 2 ^

£

£

The rest is the same as the case of the DCP, so we have Proposition 8.3.3. For any 6 > 0, there are e0 E (0,1) and Kt > 0,z = 1,2 so that if e g (0,e o ),M > l , i = iT2/£ and A C T/IJO with | A \U> Kt/e, then PiztuWlh)

> 3Xi/»,

j = ±1) > i - &

The following lemma will be used for the proof of Proposition 8.3.6. Lemma 8.3.4. There are e0 £ (0,1), and K{ > 0,i = 1,2 so that if e e (0,EO),L Ktfe,M > 1, and A C vT/o with | A | „ < 2Kx/e, then

E(Zt, -^Cl/e). £ ( Z £MM{Blf) ( ^ ) 2 ) < C e ^CeW-^l/e).

=

Long Range Contact Processes I

Proof. To begin, note that E(Z£M(B\f) B\ and t = L implies

,-f(Z;z,Bi)= E E

185

= m?{L;A,B\).

Lemma 8.2.5 (5) for A = B =

7"(LiW,b)

nee; 6SBJ

<e^f(2r 2 /e;*X)

l + 2(l + c)e*» / " % ^ ( S ; B i ) d 5 ./o

(8.5)

Note that B\ C Q H (ra;l), so Lemma 8.2.1 (2)-(4) gives < q*(K2/e;QR(l))

pf{K2je;x,Bl)

< Ced'2,

2 [ \M(S;Bl)ds
mf(i;x,5i)
(8.6) (8.7)

(8.8)

By Eqs.(8.6), (8.8) and | A \u< 2Ki/s, we obtain m^(L;A,B1n)=J2^2I(L;x,Bl)+ xeA
E

rnM(L;x,Bl)mM(L;y,B1n)

£,y€A x?y

^ ( l / e ) + {\A |„ e * suppf (tf 2 / £ ;z,i?i)} 2

< CeW-^l/e)

+ Ced-\

so the proof is complete. Next as in the DCP, we prove the following lemma for a general process U*M on Z ^ starting from A. For t fixed, we can thin U£M by U£M{B\) = UfM{B\) A K3jd(l/e) for any n € Z d . Note that U*M is not uniquely determined in general, however it can be if we impose some rule. To estimate the difference between U^^iyLIj) and U^M(VLIj), we need the following result. Define D(t]A,B;U)

= E

U

hM(Bn)\u*„(Bl)>K3yd{l/e))-

n£B

Lemma 8.3.5. For j = ±1,

P(D{L;A,SLIf,U) > */.) < C ^ f e g ^ .

Phase Transitions of IPSs

186

This proof is the same as that of Lemma 6.4.7, so we omit it. Note that D(L; A,VLIf, Z) = ~ZAL,M(VZlj) -

T^jLIj).

So Lemmas 8.3.4 and 8.3.5 give PCzlM^LIj)

> * i / « ) < p(D(L; A, Jll,; Z) > JKj/e) E(ZJMM{Blf) E(Zl (Blf)

~ %1M(VLIS)

KiSse"*-1^/') C/K33.. << C/K So if K3 is large, the last inequality and Proposition 8.3.3 gives the following result. Proposition 8.3.6. For any S > 0, there are e0 £ (0,1), and K{ > 0, i = 1,2,3 so that if e 6 (0,e o ),I = KI/E,M > 1, and A C VZl0 with Kx/e <| A \M< 2Ki/e, then ■P(i£ M (vT/i) > 2 * i / « , i = ±1) > 1 - 25. The following lemma is essential to the estimate on the difference between the LRCP and the LRBP in order to obtain the upper bounds. Compared with Lemma 6.4.9 in the DCP, the different estimate is needed. Lemma 8.3.7. Tfcere are e0 G (0,1), and Ki > 0,i = 1,3 so that if e E (0,e0),< < K2/e,M > 1 and A C VLIo with \ A \„< 2K1/e and | A n Bn \M< 2K3td(l/£) for any n 6 Zd, then

J:E(ZUBI)2)
Proof. For simplicity, define J^ n = 5^ n e z j . Observe that ££(Z;V(Bi)2) = £ £ ^ M ( n

B

" )

2

) +E

n x£A

n

£

E(Z*M(Bl)Z»M{Bl))

xty£A

= K1 + K2. For Ki, as in Eq.(8.5), Lemma 8.2.5 (5) for A = B = B\ gives

** = £ m^(t; ,Bi) X

71

= £ £ £ £ 7^(1;s,a;x,6) i e X n aeBJ 66BJ

S £ £ (e*3*f &*.*!) f1 + 2(1 + £)e^ //C2/%f ( 5|j Bi)J 1 xgyl n l

L

■/o

JJ

Long Range Contact Processes I

187

So Lemma 8.2.1 (4) yields K1 JJi)X7-(l/e) M n

£(Zf,M(Bi))£(^M(5i))

^ E (E £(^V(^))' =^(£(^M(jBi)))2 n

x suVE(Z*M(Bl))

= K2l X K22. From Lemma 8.2.2 (3), we have K21 < 2dE(Z*M(ZdM)) < 2d | A |M xe'*" 1 '' 2 r f + 1 J g 1 e^ _ C ~

£

£ '

2K3~/d(l/e) for any n € Z d , we have

To estimate K22 by \ AnB\\M<

E^(*;^)= E <

E pf(«-.-.-»i)

V

|4n4l« x

sup i e A n B

m€(2Z)^

< 2ir 3 7 d (l/£) X V ) ag(2Z)"

sup p™(t;x,Bln) xeB

™

Note that for any n E Z d , 1 d SuppM(t;x,B n)<2

J2 m€(2Z)'"

x6B

pf(i;x,Bi) "

™

So Sup5>f(i;*X)
188

Phase TVansttions of IPSs

Hence from the last inequality and Lemma 8.2.2 (2), we get 5 ) p f;f*r x' X -ft' == sup ji'2222 = suP EEe ^^XW -) n " LUe/i

C sup 2^P7WX>*n) n

\

A

—A

notation. Let A ( M ( i ) = ZtM(x)

A

- f t i M ( z ) and A i M = E l € z J ,

A

\M(X)-

L e m m a 8.3.8. For any 6 > 0, there are £0 6 (0>1) a n d Ki > 0,i = 1,2,3,4 so that its 6 ( 0 , e o ) , i = K2/e,vd(M) > Ki*f(l/e)/e and A c - / t / o with | A | M < 2ATi/e and | A D Bl \M < 2K3id(l/e) for any n E Zd, then P(AL<M

> Ki/e)

< 6/2.

Proof. The definitions give jtE(A*M)

< (A-

1)E(A*M)

A

E EE^ha+^isivD^^^+z» + v (M) 2:€-A y z~0 d

< (A - 1 ) £ ( A " + ^^7(7M M )T £

E

y *~0

So

) 1

( 2 :|y 1 + ^|<2yi;)- E; (^,M(j/)^,M(3' + *))•

rf ^,-rA -E(A? M) < ££(< M ) + — - £ i W etfM ).

Noting that if 0 < e < 1, then A = £ + 1 < 2. So vd{M)

E

p

(°' 6

e

5*4

^ M ) < -v4d(M) TT E

E{Z£M{*)Z£MIV)

£ ^ " vAM) S Y,Evwin <«)'

(8.9)

Long Range Contact Processes I

189

Therefore, using Lemma 8.3.7, Eq.(8.9) and the last ineqaulity, we have the following differential inequality : ^tf(t)<ef(t) + g(t),

/(0) = 0,

(8.10)

where /(*) = E(A?M), 9(t)-

C

7d(l/e)

Vd(M)

£

So it follows from Eq.(8.10) that

E(E-l,M)
(8.11)

(8-12)

Jo On the other hand, the assumption Vd{M) > Ki-y(l/e)je gives

So Eqs.(8.11) and (8.12) yield E{KM)

< ^ -

Hence

P(&tM > Kile) < C/Kt Picking K\ to make the last quantity less than 6/2 proves Lemma 8.3.8. By definitions, we observe

(ZIM(BI)-IAL,M{B\)}

KM= = 2J £

n6(2Z)'

>

{KM(BI)-1AL,M{BD}

£ ne(2Z)

d

> ZLM{yz^iVIij) -ILM^1*)-"J& So combining Proposition 8.3.6, Lemma 8.3.8 and the last inequality gives the following result. Proposition 8.3.9. For any 6 > 0, there are £0 £ (0,1) and if; > 0,i = 1,2,3,4 so that ifs £ (0,e o ),i = K2/e,vd(M) > K4-y(l/£)/s and A C VLI0 with K^/e <| A | M < 2K1/e and | A n B\ \„ < 2K3ld{\/e) {or any n e z then rCVT/,) P&M^W

> A 'i/ £ ' i = ±i) > i - 3^.

190

Phase Transitions of IPSs

As in the case of the DCP, Proposition 8.3.9 shows that if A satisfies (1) Ki/e <\ A | M < 2A' 1 /e, and (2) | A n B\ | M < 2A 3 7d(l/e) for aU n e Zd, then with probability greater than 1 - 3(5, there are sets Aj C £L,M n V^-fj f° r J = ± 1 w ' t n 0-') I A? | M = A i / e , and (2') | A* n 5 i | M < KrtiiMe) for all n 6 Z d . So the rest of story is the same as the case of the DCP. That is, to use comparison method, we construct a general case. To do this, we let V = {(m,ra) 6 Z 2 : m + n is even }, / m = [2m - 1,2m + 1] x [ - 1 , l ] d - \ VZ]*-1,

v T / m = [(2m - l ) V T , ( 2 w + 1)VT] x [-VI, -1

Hm = ((2m - 2)-v/I,(2m + 2)VZ) x H"* , where m e Z. For construction of the ra-th stage, for ( m , n ) 6 V, sets ^4(m,n) C \fLIm given by

(2)

2iJTi

— <1 A(m,n) L <

(1)

are

for all k e Zd ,

I A ( m , n ) n B\ \u < 2K3ld(l/e)

where Bn = [n1-l,n1+l)x---x and

[nd - l,nd + 1),

I

v/x, d= 1, | log i |, d = 2 . 1, <2 > 3 . | l o g s |, d = 2 , A site ( m , n ) £ V is called open if a modified LRCP starts with A(m n^ occupied 1, <2 > £J3 ^. at time nL and is not allowed to give birth and is killed outside Rm survives up to time (n + l)L and there are sets A^mn+1) C ft^"+ n VT/m+i for j = ± 1 , which have fd(x)!

I A{m,n+1) \M~ ~ >

(!') (2')

I 4m,»+i)

n B

l \u < * V w ( V « ) for all k g Z<* .

If (m, n) is not open, we set ^(m,n+l)

=

#•

To continue the construction on level n + 1, define Am-l,n+l) = 4(m-2,„+l) U ^ , n + 1 ) , A ( m + l , n + l ) - 4 ( m , „ + 1 ) U A -1 m+2,n+l)-

Long Range Contact Processes I

191

To use comparison thorem, for x, y e V, we write x -> y if there is an open path from x to y. That is, there is a sequence a;0 = x,... ,xn = y o{points in V so that all sites x{ is open and for 1 < i < n, x{ is x;_i + (1,1) or xi+1 + ( - 1 , 1 ) . Let C° = { m e Z : ( 0 , 0 ) ^ ( m , n ) e V } . Hence the comparison method guarantees that if the LRCP starts with A^to) satisfying (1) and (2) occupied and there is an infinite open path on V starting as (0,0) : P(C f 4> for all n ) > 0, then the LRCP survives : P(8,M

+ forall n ) > 0.

That is, the comparison method gives the following flow chart : LRCP £fM

-> modified LRCP £*M -»■ oriented site percolation C° -> LRCP f? i M .

We should remark that Hi and Hj are disjoint for \i — j \ > 2, so given what happened at level k < n — 1, the fates of sites (m,n) on a fixed level n are independent. That is, { (m, n) is open }, (m, n) G V, are independent. However they are not necessarily identically distributed, since initial configuration A( mi „) is not identical in general. Let p (m,n)

_ p ( ( m ) r e ) i s o p e n ),

so Proposition 8.3.9 gives p<°>°) > 1 - M, moreover the above observation implies that for any (m, n) e V, p(m,n)

>

x

_

3S

Therefore comparing with the oriented site percolation, the result that upper bound on the critical probability for this model is 80/81 (see Theorem 4.2.4 (2)) yields that if

p(-.")>l-3*>g, that is, 35 < 1/81, then the LRCP survives. So the argument above has shown that if vd(M) > C 7 d ( l / e ) / e and C is large, then the LRCP survives. Note that e > C C ' 7 ( l / e ) / g when M is large. Therefore we can prove the upper bounds on XC(M) given in Theorem 8.1.2. Moreover, combining this result and the trivial lower bound, 1 (see Lemma 8.2.4) gives the proof of Theorem 8.1.1 (1) : lim \e(M) = 1.

192

Phase Transitions of IPSs

8.4. Lower B o u n d s for d = 1 In Sections 8.4 - 8.6, we will prove the lower bounds on \C{M) By comparing with the LRBP, in any dimension, mM(kt)

<

given in Theorem 8.1.2.

(mM(t))k

So if mM(t) < 1 for some t, then P($M / < / > ) < mM(t) -> 0 as t -* oo, i.e., the LRCP dies out. Therefore, to prove the lower bounds, it is enough to show there is a C so that if Vd(M) < C 7 d ( l / e ) / £ , then mM(t) < 1 for some t, where £ = A - 1. The key to doing this is the estimate on the interference term, so basic strategy is the same as the DCP. But, in the cases of d = 1,2, the estimates are easier than those of the DCP. In this section, we give the result in d = 1 : P r o p o s i t i o n 8.4.1. There is a C > 0 so that if M is large, then

A

<w * \+-£?•

In the rest of this section, K$ and Ke are constants which will be chosen later. Note

that Q*(K6Vi) = [-K6V~t,K6Vi]. For t > 0,M > 1 and x £ ZlM, define NtMx)

= {y£ 8,M : 0 < | y - x ]< 1},

%,M = {xe

$ju

:| Nt,D(x)

\u > 6eM} n

Q*(KeVi),

Ct,M = (f° M n QZ{KBVi)) \ r,tMSo, by definitions, we have f?,M = m,M + Ct,M + (&°iM n Q^(K6Vi)c).

(8.13)

First we estimate the second term on the right hand side of Eq.(8.13). L e m m a 8.4.2. There are e 0 £ (0,1) and Kb > 0 so that if £ £ (0,e 0 ), £ 3 / 2 < Kb/M l/10e < t, then E\Ct,M\M<0.l. Proof. Observe that IC«.ML<

J2

ICt,Mn[i,« + i)| M

< 6EM x 2KeVi

=

12K6MeVi.

and

Long Range Contact Processes I

193

If £ is small and i > l/10e, then P(QM ^ <j>) < P(Z°M ^ <j>) < C/t, where a constant C is independent of M. (See Lemma 2.1 in BDS.) So from'these facts it follows that if t > 1/lOe and e 3 / 2 < Ks/M, then E I C*,M U

= E{\

Ct.M U

&M

< 12K6MeVtx <

* *)P(8M

±

*)

-

CK5Ke.

Therefore, with K6 fixed, picking ii"5 to make the last quantity less than 0.1 proves Lemma 8.4.2. Next we estimate the third term on the right hand side of Eq.(8.13). L e m m a 8.4.3. There are £„ g (0,1) and Ke > 0 so that ife G (0,e 0 ), e 3 / 2 < Ks/M 0 < t < l/e,then

and

E\8j*r\Q*{KuViy\M
\M < E{Z°iM(Q* =

then

(Key/if))

e**pM(t;0,QK(K6V~ty)

<2epM(t;0,{KeVi,oo))

i

r°° < 2e /

I -=exp(-a2/2)
JIUIUM V27T

So, with Af fixed, £ | $M

n <52,(/i-6V<)c L -> 0 as j r 6 -+ oo.

The rest is easier than the case of the DCP. That is, the following lemma will finish the proof of Proposition 8.4.1. L e m m a 8.4.4. There are e0 and K& > 0 so that ife e (0,£o), £ 3 / 2 < Ks/M 0.9 for l/10e < t < l/e, then ! » « ( « )

<

and mM{t)

>

-£■

Suppose Lemma 8.4.4 is valid. By Lemma 8.2.3: 0 < mM{t)

< mM{t)

= eet,

and e- < 1 + 2x for 0 < x < 1, we see m M ( l / 1 0 £ ) < m M ( l / 1 0 £ ) < 1.2. So Lemma 8.4.4 implies mM(t) < 1.3 - et for t > 1/lOe. If 0.4/e < t < l/e, then m M ( i ) < 0.9. Therefore mM(t) < 1 for some t, and the proof of Proposition 8.4.1 is complete.

Phase Transitions of IPSs

194

P r o o f of L e m m a 8.4.4. By the definition, observe that 1 ■

MW/MW x

■>

1M ■

'

El xx

'■{««»««?,»} -

3e

I * ' * r 'M

(8.14)

y^x y~x

So using A > 1 and Lemma 8.2.6 (1) : 5 m-(l)

= «m"(l)-^Sji-W,

and Eq.(8.14), we have ^ m M ( t ) < £ m M ( t ) - 3 e £ | r,t,M U ■ at By Eq.(8.13), Lemmas 8.4.2 and 8.4.3, mM(t)

= E | VtM

\M +E | Ct,M L +

£

(8-1S)

I #,M n Q £ ( ^ 6 V / t ) c | M

< ^ U i , M | M +0.1 + 0.1, that is, mM{t)

< 0.2 + E | 7/t,M L ■

(8-16)

< 0.2£ - 2eE \ r,tM \„

(8.17)

So Eqs.(8.15) and (8.16) yield jtmM{t)

On the other hand, by Eq.(8.16), if mM(t)

> 0.9, then

■Eh«,M|M>0.7.

(8.18)

Therefore combining Eqs.(8.17) and (8.18) gives — m M ( t ) < 0.2E - 2e x 0.7 = - 1 . 2 e < at

-e.

8.5. Lower B o u n d s for d = 2 In this section, we give the lower bound in d = 2 : P r o p o s i t i o n 8.5.1. There is a C > 0 so that if M is iarge, then

A0(M) >

1 ClogM 4+ M2 '

Recall that B ^ „ = [m,m + 1) x [n,n + 1) for m , n € Z. For simplicity, from now on we omit superscript 0. A similar argument in Lemma 7.3.2 (details can be found in Lemma 4.2 of BDS) gives

Long Range Contact Processes I

195

Lemma 8.5.2. Tiere are d > 0,i = 1,2,3 so that if 0 < t < 1/e with e G (0,1), then for any x e Z2M,

(!)

™M{t;x,Bm,n) < ^ ,

(2)

™f(t;s,B m , n )
(3)

^(«;x,Bm,n)
Lemma 8.5.3. Tiere are to and 6,- > 0,i = 1,2 so that ift > to, and e G (0,1), tien WM(t;0,Bm,n)>hjj-r

(1)

(2)

m?tea,B^)>bJ?&±&,

where (m,n) g Q*,{Vi). Let /•(«) = {(m,n) G Qi(Vi) : E(^D(Bm,n)) > (1 - 0)£(Z° D (£ m , n ))} where 0 < 0 < 1 and Qi(Vi) = {x G Z2 :|| a; ||i< \ / t } . Then we have the following result as in Lemma 7.3.4 (see details in pp.459-460 in BDS). Lemma 8.5.4. Tiere are a,£ 0 G (0,1) and C; > 0,i = 1,3 and bt > 0,i = 1,2 so that if e G (O,£O),0 G (0,90),mM{t) > 1 - a for any t G [a/2e,a/e], tien £

E(tlM(Bm
> /»{log(l/e) +log(a/2)},

(m,n)6J»W

wiere 0O = (l/&i) A (bl/AC^)

and P = b2 - 2y/0O[Cl > 0.

Compared with the DCP, we can easily estimate the interference term.

Phase Transitions of IPSs

196

L e m m a 8.5.5. There are a,e 0 e (0,1), so that if e e {0,eo),t M

I (t)

e [a/2e,a/e],

> (3 {log(lA) + log(«/2)} -

then

e\

Proof. By definition,

7M(<)= X)

P

(*,V£&,M)

x,y~x

>^2E{tlM(Bm,n)(ZlM{Bm,n)-l)) m,n

= $>tf?,M(*m,»)J) - E- £ «tV( B -.")) m,n m.n

m.n m,n

— K\ — K2. As for K\, Lemma 8.5.4 gives # i > / 3 { l o g ( l / £ ) + log(a/2)}. As for K2, by Lemma 8.2.2 (3) and t < a/e, we have

K2 < J2E(ZlM(Bm,n)) = E(ZlM(ZdM)) = mM{t) = e^*

< e\

Therefore the proof is complete. The following lemma will finish the proof of Proposition 8.5.1. L e m m a 8.5.6. There are a,e 0 6 (0>1) and Ks > 0 so that if e e (0,e 0 ), for any t £ [a/2e,a/e] and v2(M) < Kslog(l/e)/e, then ^-mM(t) at

< -3E.

Suppose Lemma 8.5.6 is valid. Comparing with the LRBP, mM{")<JnM{») xx

2e'

-

yy y

2e' 2e' 2e'

=

e i < l

~

+ a.

By the last inequality and Lemma 8.5.6, if e is small, then M 1 m (-)= ( - ) = I1~m ~mMM(s)ds (s)ds €

J_2_ (IS

<-3e(^) + l + a

= *-i-

+ +

mmMM(-) 2,£

mM(t)) >

1-0

Long Range Contact Processes I

197

The last inequality gives mM(a/e) < 1, so the proof of Proposition 8.5.1 is complete. Proof of Lemma 8.5.6. By Lemmas 8.2.6 and 8.5.5, we have jtmM{t)

< emM(t) - - A _ / J { i o g ( l / £ ) + log(a/2)} - e\

(8.19)

From the fact that mM(t) < mM{t) = eet < 1 + 2a for t < a/e and Eq.(8.19), we get ^

W

<

£

( l

+

2a)-^A_/-(t).

(8.20)

Let g(a) = -51og(a/2) + 5e a //3. So Eq.(8.20) implies that if v2(M) < Kh log(l/ £ )/e, with Ks = /3/5, then

kw^-'+dsy-

^

With a fixed, if £ is small enough, then j(a)/log(l/e) < 1/3. So, by Eq.(8.21) and picking a < 1/3, the proof is complete. 8.6. Lower Bounds for d > 3 In this section we give the lower bounds in d > 3 : Proposition 8.6.1. There is a C > 0 so that if M is Jarge, then

M M ) i 5 + 3p. The following proof was called BDS method in Section 7.4. So the estimate on the interference term is essential to this proof. Lemma 8.6.2. If M is large, then for t > log 2, J M ( i ) > g m M ( t - l o g 2). Proof. It is sufficient to show that for any x G ZJ^, if A > 1, then

INP(*,ve&j*)

>^e^0-log2,M)-

y~x y~x

The inequality follows from combining these three facts, (i) The probability the particle at x survives from t - log2 to t is e - l o g 2 = 1/2. (ii) Kit survives, then the probability it will

Phase Transitions of IPSs

198

give birth to at least one particle is 1 - exp ( - A l o g 2 ) > 1/2. (iii) If a particle is born the probability it will survive for at least log 2 units of time is e _ ' ° s 2 = 1/2. P r o o f of P r o p o s i t i o n 8.6.1. Let e = A - 1. Lemma 8.2.6 (1), Lemma 8.6.2 and A > 1 imply that if t > log 2, then |

m

M

( f )

^

m

M

( t )

__l__

M

m

( t

_

l o g 2 }

.

8vd(M) Comparing with the LRBP gives mM(t) Therefore

If e < l/9vd(M),

< eel°z2mM(t

- log2).

M ^-m ^ATT^0^)™™^)v (t) <(ev dt ' - \ $vd(M) J '

then for large M,

Moreover comparing with Moreover comparing with

Therefore if e < l/9vd(M), Therefore if e < l/9vd(M),

<* M / , W °-01 _ M A A —m (I) < ; , ,. m it). K v ; dt > vd(M) the LRDP gives the LRDP gives mM(log2)<e£l°82. mM(log2)<eEl°82. then then

mM(t)

( '°g 2 m W<<eex *p P ^(M)

_ 0.01(*-log2)\ JVi{M)

So, m M ( i ) -> 0 as t -» oo.

References 1. M.Bramson, R.Durrett and G.Swindle, Statistical mechanics of crabgrass. Ann.Probab. 17 (1989) 444-481. 2. R.Durrett, Ten Lectures on Particle Systems (St. Flour Lecture Notes, 1993) Springer Lecture Notes in Mathematics, to appear. 3. R.H.Schonmann and M.E.Vares, The survival of the large dimensional basic contact process. Probab.Th.Rel.Fields. 72 (1986) 387-393.

Uniform Nearest-Particle

Systems

199

CHAPTER 9 UNIFORM NEAREST-PARTICLE SYSTEMS

9.1. Introduction This chapter is devoted to some nearest-particle systems with long range interactions in one dimension. In Section 9.2, we consider uniform nearset-particle systems with finite ranges. Section 9.3 is devoted to the uniform nearest-particle system with infinite range. In general, the methods in both sections are combination of correlation identities and the Harris-FKG inequality. In Section 9.4, we will mention a result about finite system of uniform nearest-particle system with finite ranges. 9.2. U n i f o r m N e a r e s t - P a r t i c l e S y s t e m s : Finite R a n g e s In this section we study the uniform nearest-particle system n\ with infection rate A and the distance of range M on state space { 0 , 1 } Z . (Results here except Section 9.2.2 appeared in Konno and Katori. 1 ) Then for a site i £ Z and a, time t > 0, n\ (x) takes values in {0,1}. If i is vacant, n(x) = 0, and if a; is occupied by a particle, n(x) = 1. Let lx(n) and rx(n) denote the distances from x to the nearest particle to the left and right respectively. The dynamics of the evolution is described by the following transition rate : at site x g Z, 0^1

at rate

1 —► 0

at rate

/3{M\lx(v),rx{v)), 1,

where /3W(/,r) = 2Axa(f;i)1, with a(M,k)

effi a

_ i n

for

i < „ < 2M, 1 < fc < ra,

= a^lM+m+1-k)

2M+m = °

for

= —, m>l,M

for

+ l
m > 1, 1 < k < M, + m,

200

Phase Transitions of IPSs /3<M>(/,oo) = /3<M>(°°>J') = T7 M

()( \l,oo)

M

= P<- \oo,r)

= 0

for for

lM

+ l,

^ W (00,00) = 0. We call this process UNPS(M)

for short. For example, when M = 1, /3( 1 '(1,1) = 2A, /3( 1 '(1,2) = /?( 1 )(2,1) = A, /3<1)(1,3) = /3( 1 )(3,1) = A, / ^ ' ( l , 00) = /jW(oo,l) = A,

and ^x\l,r) = 0 otherwise. This process is equivalent to the basic contact process in one dimension. Similarly when M = 2, /3<2>(1,1) = 2A, /?<2>(1,2) = /?<2>(2,1) = A, /?<2>(1,3) = /J<2>(2,2) = /3<2>(3,1) =

^ ,

/?<2>(1,4) = /3<2>(2,3) = /3<2>(3,2) = /jW(4,l) = \ , /? ( 2 ) (1,5) = /?<2>(2,4) = /3<2>(4,2) = /?<2>(5,1) = £ ,

/3<2>(l,oo) = /3<2>(2,oo) = /3<2>(oo,2) = /3<2>(oo,l) = £ , and f}(2\l,r) = 0 otherwise. In the next subsection we present a general class including { UNPS(M) : 1 < M < 00}. From now on we consider of infinite systems with the exception of Section 9.4. 9 . 2 . 1 . Attractive nearest-particle systems with finite ranges Assume that {/?(/, r) : 1 < l,r < 00} is a, collection of nonnegative numbers which satisfies, sup,,.f3(l,r) < oo,/3(/,r) = p(r,l),p(l,oo) = /3(oo,l) > 0, and /?(oo,oo) = 0. Then we consider the following one-parameter family of the nearst-particle systems with birth rates 0(1, r) = px(l, r) : B = { /3{l, r) = 2Xa\'lr_1

(0

: a™, A > 0,

am

=

0(n-*+l)>

for n > 1, 1 < k < n,

Uniform Nearest-Particle Systems .(*)

(it)

(in) (Hi)

(fc+1)

201

(fc)

£<#> = !, a' 1 ' - a (1)

(to) \™J

°n

(v) (o)

> a (1)

"n+l ^

a

n+l

- a (1)

°n+2>

n a2M_ x #/ 0 aand £ £ ^ j == 0-} 0.1 there is an M > 1 such that
For example, in the case of UNPS(l),

„(i) _ am _ I „(1) _ .(3) _ I

°3

and a„

— "3

— 9'

= 0 otherwise. In particular,

<41> = l > a 2 1 > = a31> = . . - - i

2

Similarly, in the case of UNPS(2),

«?> = 1, a2

= o2

= -,

„(i) _ u_(*) _ a.(3) —_ I "3

— 3

— 3

o>

„(i) _ „m _ „(3) _ „(4) _ I a4 - a4 - a4 - o4 — , „(D _ _(2) _ .(4) _ .(5) _ I

and ak

= 0 otherwise. So we have „(i) _ il>>, „(i) _- a (D at — a2 _- I- v> „(D a 3 _- i- >> ao(i) a 5 - - . . . - — I- . 4

Phase Transitions of IPSs

202

We can easily confirm that the processes introdued in the previous section satisfy the above conditions. The above condition (i) is the symmetricity of /3(l,r). (ii) means that this process is attractive. The condition (iv) is necessary to use the Harris-FKG inequality. By (ii), (v) implies that a 2M+m

0

for m > 1 and k E {M + 1,M + 2 , . . . , M + rra}.

That is, this process has a finite range of the distance M. We define the order parameter PA by Px = E¥x(ri(x)) = Mv ■ V{x) = 1), where u\ is the upper invariant measure. Note that p\ is independent of % E Z. The conditon (ii) of attractiveness gives p\ is a [0,l]-valued nondecreasing function of A. So as usual the critical value Ac is defined by Ac = s u p { A > 0 : p * = 0}. Moreover the following results are easily obtained by conditions (i) and (ii). L e m m a 9 . 2 . 1 . For fi(l,r) e B, a(l)-1 >a(1)-i>a(1)>--->a(1)

m

(2)

4l+m

= a[%

- a (1)

a^r+1-fc)=42M+1-fc)

form>l,l
In the rest of this section we sometimes omit the superscript (1) of an ' : an = an ■ For n > 0, we let fc(n) = K({1, ...,n})

= Eux f 1,(0) f [ ( l - V(i))V(n

+ 1) J .

For example, fc(0) =

^(r) 1J(0) = 7?(1) = 1).

Hi) = Mi

T ? (0) = 1 , ) ? ( 1 ) = 0,17(1) = 1).

fc(2) = Mv 7?(0)

= 1,7,(1) = 7,(2) = 0,77(3) = ! ) .

Next we present correlation identities for k(n). For n > 1, we let

b(n)= £

0(l,T).

l+r=n+l From Eqs.(5.1) and (5.2) on page 347 of Liggett, 2 we have

Uniform

Nearest-Particle

Systems

203

Lemma 9.2.2. oo

(1)

,» = £)6(n)*(n), n=l OO

(2)

*(0) = £)/J(i»,l)*(n). oo

(2)

*(0) = £/J(n,l)*(n).

71 = 1 71 = 1

Note that as in the Lemma 4.6.1 (1) we have the following sum rule : oo

/* = 5>(»)-

(9.1)

n=0

Moreover observe that n

b(n) = 2A Y^ «-(n] = 2 ^

a

nd ft(n, 1) = an.

fc=i

So we obtain the following correlation identities as a consequence of Lemma 9.2.2. Corollary 9.2.3. 9A — 1

(i)

*(o) = - ^ - ^ > OO

(2)

fc(0)

*(0) = 2A^a n fc(ra). n=l

By Lemma 9.2.1 (1), Corollary 9.2.3 (2) can be rewritten as

{

*(0)i = 2 A -

So the sum rule (9.1) gives

2M-1

oo

~j "j

5Z anKn) + » £ fe(") ■

(9.2)

Y, ank(n) + a2M £ fc(n) fc(n) || . J2 "nk(n)

(9.2)

nn = = ll

nn=2M =2M

IJ

2M-1

2A ^

K - o2M )fc(n) - (2o2M A + l)fc(0) + 2a2M A W = 0.

ra=l

Let

*(») = p,({l,...,«}) = En (f[(l - iK0)j

(9.3)

204

Phase Transitions of IPSs

Remark that p\ = 1 - i ( l ) and Jfc(n) = x(n) - 2x(n + 1) + x(n + 2)

for n > 0.

(9.4)

Combining Eq.(9.3), the definition of x(n) and the last remark gives 2M-2 2 a

(

2M-i

- ° 2 M )Ax(2itf + 1) + 2 ] T K + 2 - 2
- (3A + l ) i ( 2 ) + 2(A + l)a;(l) - 1 = 0. On the other hand, from the definition of x(n), Corollary 9.2.3 (1) can be written as ,(2)=(2A

+

1M1)-Ij

(95)

By the last two equations we have the following correlation identity. L e m m a 9.2.4.

1 l(1) =

2M_1

f

2A2 + A + 1 1

4A

' S

1 (""+2 -

2a

" + ! + a « ) I ( n + 2) + A + 1 L

The Harris-FKG inequality (Corollary 3.3.3 (2)) gives L e m m a 9.2.5. For n > 1,

x(n) >

x(l)n.

Combining Lemmas 9.2.4 and 9.2.5 with the condition (iv) (i.e., an+i — 2 a „ + 1 + o n > 0 for n > 1,) implies

1 l(1)

2M-1

f

- 2A* + A + 1 I

4A

' £

1 (a"+2 -

2a

" + i + <*")*(l) n + 2 + A + 1 \

Moreover the last inequality can be rewritten as

| £ (Qn-i - «»>(!)" - ~^r J (*(1) - 1) < 0. where aQ = 3/2. So if A > Ac (i.e., 0 < x ( l ) < 1) then

J2 (a„_i - a„)i(l) n - - i - > 0.

(9.6)

Uniform Neareat-Particle Systems

205

In general for A > 0 and x £ [0,1] we let

F,(x) = G(x)-±±± where 2M

G(x) = £ > » - i - an)xn. n=l

Let

3 q = G(l) = a0 - a2M = - - a2M. Note that ? > 1. Attractiveness (i.e., a„_i > a n for n > 2) and a0 = 3/2 imply that Fx(x) is a nondecreasing function of x for fixed A. Observe that ^(0) =

i^W =^

A+ l ^ -1AT<0'

FA(1) = 9

> 0. lim^D^oo,

A+ l ~1A^' &ft(l)

= ? >0.

Therefore there is the unique A(c2) > 0 so that f \ ( l ) > 0 if A > A(c2), J \ ( l ) = 0 if A = A(c2), and FA(1) = 0 if A < A(c2). So A(c2) is a positive root of A+l 9 = 4A2 Another root is denoted by Ac . So / Al 2) = ;i { l + v i6^TI},

3J») = l { i _ yie^TT}. Note that

A<2> + 1 4(A(02))2

gives

A?>:

3 *

2

11 2*

Let x(A2) be the unique positive root of F^x) = 0 for A > Ac2). That is, G(xl2>)=^i. Define p(A2) = 1 - s(A2) for A > A(c2), = 0 otherwise. The above observation gives A<2> < Ac>

(9.7)

206

Phase Transitions of IPSs

p,f.

i Moreover let

X x(D ,

- 2J1A_ _

A —1 1 «i i _ Wn i _ 22A ~ . X p„ A« -- 1 xx 2A

and and

Simple calculation yields G{x<») < G(xl 2) ), so

Pl2) < Pl1}

A>A<2>.

for

The definition of px ' and Eq.(9.7) give P!2) = I - 4 2 ) A+ l - 11-_ J 2 ) _ J. = G(4 2 ) ) 4A2 1

1 where Q(x) = G(x)/x.

A+ l

2) 4A2

o(4 )

'

Note that 0 ( 1 ) = G ( l ) = ?. For any x 6 [0,1], we let 2M-1

ij(x) =

^ ^ - a ^ x " "

1

,

l

71=1

so

R(x) = Observe that (2) Px

= 1-

1

A+ 1

Q(l)

4A2

Q(x)-Q(i) x-1 1

+

Q(l)

A+ 1 4A*

_L

= ^ = ^ + (42,-i)^f. Therefore we have 2

m (2) _ 4gA - A - 1 J Px 4< 2

#' =

^^

£(42>)A + l '

I 1 + g
Note that 4g

- * - < » .-42>)(A-A(2>):

1(2) _ 1(2) _ V 1 6 ( ? + 1 c c ~ 4q '

A+l

Q ( X C « ) ) 4A2

Uniform Nearest-Particle

A<2> + 1

207

= 1,

4g(A c 2) )2

xm

Systems

- 1

So lim

-?±-

J^n-Jil'1

V16q + 1

9+r

8g + 1 + y/16q + 1'

where

r = R(l)= J2 K-«=«)• Therefore we obtain Theorem 9.2.6. A c > A ( 2 ) = l±vgI±T > A W = l.

(1)

P\ < Px] < Px]

(2)

for A > 0.

2

p ,( ) lim * = 8g2yi6?TT 2) ( 2) M>< A - A c (9 + 0(8? + 1 + VWgTT)'

(3) where 1 — p\

is the unique positive root of \

,

i

x;(o„-i-«»)* B -^ i =o

(A>AWJ,

71 = 1

2M-1

9 = 9 - fl2M .

r

= 5Z (fln ~ a=M )■

Z

n=l

9.2.2. Better lower bounds As for lower bounds on the critical value, we can get better estimates Ac without condition (iv) which was used for the Harris-FKG inequality. Let y{n) = x(n) - x(n + 1), so k{n) = y(n) - y(n + 1) for n > 0. From Eq.(9.2) we have "2M-1

*(0) = 2A

^

oo

an{y(n) - y(n + 1)} + o2M £

{y(n) - y(n + 1)}

= 2A "lK 1 ) - ^2 (an - a.n+\)y{n + 1) 71=1

2M-1

= 2A x(l) - x(2) - ^2(an-

an+1){x(n + 1) - x(n + 2)}

Phase Transitions of IPSs

208

On the other hand, fc(ra) > 0 gives x(n)-x(n and an > a „ + J , we get

+ l) > i ( n + l ) - a ; ( n + 2 ) . Using this inequality 2M-1

fc(0)>2A \ x(l)-x(2)-

Y,

(an - an+i){x(l)

-

x(2)} ■

71 = 1

= 2a 2M A{. 2a,MX{x(l)-x(2)}. = Let B' denote the collection of the birth rates P(l,r) satisfying conditions (i),(ii),(iii), and (v). So B C B'. Then we have L e m m a 9.2.7. For /3(l,r) e B', k(0)>2a2M\{x(l)-x(2)}. ■ *(2)}. By Corollary 9.2.3 (1), we get

fc(0)

9A — 1

2A - ( ! - * ( ! ) ) •

Combining Lemma 9.2.7 and the last equality gives 9\ _ i

" -(l-*(l))>2a2MA(z(l)-z(2)). 2A By Eq.(9.5), x{l)-x(2)

=

l-z(l) 2A

(9.8)

(9.9)

Prom Eqs.(9.8) and (9.9), we have 2A-1 l-x(l) ( l - x ( vl ) ) > 2 a 22M MA 2A v " 2A so

{2(l-a3M)A-l}PA>0. Therefore we obtain T h e o r e m 9.2.8. For P(1,T) e B', Ac > A?> =

2

^ V (1 - < W

Next we consider the relation between A^ ' and A^ . Note that A^ ' is the positive root of /(A) = 0 where /(A) = 4gA2 - A - 1 with q = 3/2 - a 2 M . Then

^3,) = =

i(3)

So noting that Ai

/(

2TT^)) ^:

'■tril -")>•■

(1-,

> 0 and / ( 0 ) = - 1 , we have

Uniform Nearest-Particle

Systems

209

Corollary 9.2.9. For j3(l,r) G B, Ac > A<3> :=

1

> A (2)

2(1 - f l 2 « )

1 + ^^6^TT 8g

=

1 2

where 9 = 3/2 — a2

9.2.3. UNPS(M) In this subsection we apply the results in Sections 9.2.1 and 9.2.2 to the UNPS(M). Noting a„ = 1/ra for 1 < n < 2M, the following result comes from Theorem 9.2.6 and Corollary 9.2.9 immediately. Corollary 9.2.10. In UNPS(M), (1)

XC(M) > X™(M) = ^

(2)

(3) 1 ;

-

> A «

W

=

M +

< P(?{M)

P^M)
A2)W

Um 2

>JM(25M 2 8- )8)

f f

^ -

> mM)

forA>0,

8g2viog+T

2)

AIA< >(M) A - A< (M)

(« + r)(8j + 1 + 7169TT)'

wiere 1 — p\ (M) is the unique positive root of 2M

Xnn

E' (n x- l)n n=2 *

'

_3

J_

2

2Af'

9 _

X x

A A ++ 1l

2

4A* r

„

0

_2^rl l

^

Tl =

n

/-.

(2\\

(A>A),

J_.

2M;'

l

For example, in UNPS(1)= basic contact process, Ae(l) > A<3>(1) = 1 > A<2>(1) = i ± ^ I M 0.640 > A»>(1) = \ , 1 — p* (1)

is the positive root of

1^ P(A2)(1) = M42)(i) A - A^2)(l)

x2 x A +1 ~r- + -r = .! 2

I g V g « 1.676. 3(9 + ^ )

= |,

Phase Transitions of IPSs

210

In UNPS(2),

Ac(2) > A<3'(2) = I > A<2>(2) = i ± ^ « 0.558 > A « ( 3 ) = I, 1 — /rA '(2)

x4

is the positive root of

x3

12

+

6

x2 +

2

x_ A+l +

2

4A2 '

^2)(2) 75^21 lim ,„/— = T=T- Ki 1.575. HA»C») A - A(c2)(2) 14(11 + V H ) The asymptotic behaviors with large M are interesting as follows. From Corollary 9.2.10 (1), we have 1 Xc{M)-\>\f\M)-\ \M)-\ 2 i M _ 1 ~~ 2M - 1 2 rA

- i [ j L (—Y ~ 2 2M

+

\2M)

+ ' " •

Therefore we have the lower bounds on XC(M) — 1/2 for large M M. Therefore we have the lower bounds on A C (M) — 1/2 for large M.

Corollary 9.2.11. In UNPS(M), Corollary 9.2.11. In

UNPS(M),

Urn A<3>(M) = 1

(1)

(2)

M-+00

z

1 [ 1 A,(M)-I>i 2M 2 2M

+

\2MJ

+

""

In particular, for targe M , A

(3)

1

^ ^ 2 ++ i x ¥ -

Note that the following weaker estimate is given by Ac ( M ) in the similar way : K(M)

>- +— x —

for large M.

9.3. Uniform Nearest-Particle S y s t e m s : Infinite R a n g e s In this section we consider the uniform nearest-particle system with infinite range by using the results in the previous section. Results here appeared in Konno and Katori. 3 The

Uniform Nearest-Particle

Systems

211

uniform nearest-particle system with infinite range is defined by the following transition rate : at site x G Z, 0 - 1 1 —* 0

W*(v)Mv)), W*(n

at rate at rate

1,

where

f}(l,r) =

2A l+ r-l'

This process is also called the uniform birth process process, Mountford 4 showed Theorem 9.3.1.

,.=i

in Chpater VII of Liggett. 2 In this

1 2"

On the other hand, the same result as Corollary 9.2.3 (1) is obtained in the uniform nearest-particle system with infinite range, so the Harris-FKG inequality gives 2A — 1 Px < - j j p

1 for A > Ac = - .

(9.10)

So combining Theorem 9.3.1 with Eq.(9.10) implies Corollary 9.3.2. Pxc = 0. T h a t is, the second-order phase transition appears in the uniform nearest-particle system. Furthermore the next result comes from Eq.(9.10). Corollary 9.3.3. logPx Um > i. xixc log(A - Ac) _

So this result means if p\ ~ (A - A c )" for A > Ac and A RJ AC then /3 > 1. Next more detailed information on px near critical value will be given. Noting an = 1/n for n > 1, Eq.(9.6) implies if A > Ac (i.e., 0 < x ( l ) < 1) then °° ^ ( a „ - i - an)x{l)n n=l

x f, s(l)» ~ 2 n=2 4 i » ( « - i)

-

A+ 1 - ^ -

A+l 4 2 ^ ~

0.

Phase Transitions of IPSs

212

So p\ = 1 — x(l) gives 3 A+ 1 pxlogpx + - ( 1 - px) - - j j 5 - > 0. Therefore

px

<

*±±1

(9.11)

- 2A2(3-21ogpA)'

X-Xe

Prom the last inequality and Corollary 9.3.2, we have Urn - ^ - = 0. XIK A - Ac

(9.12)

On the other hand, Eq.(9.11) can be rewritten as n^

Px t,

- A-Ac V PA

&r

f,

3 \3A+1

27 ' 2A* Px

,,

n

v

,,

31

=X3A;{1OS(A3X)+1OS(A-AC)-2/

+

3A + 1

^ A ^

6V g (A-Acc;) T+ A^ g7 A "* - A -lo -V A c l 5°(A -r \c^ ) -! 2^ A -VA+ c c

3A±A

2A* "

So from Eq.(9.12) we get more detailed information on the critical behavior of px as follows. T h e o r e m 9.3.4. Let Ac = 1/2. m

|log(A-Ac)|

(1)

& ' *

A-Ac

^

There is a C so that for A > Ac, (2)

px < C

A-Ac log(A-Ac)|"

Results here about critical exponent /?i of this process, i.e., /3i>l, contrasts with that 02 of the basic contact process in one dimension, since as we saw in Section 5.1, the estimated value is as follows : fc « 0.277 < 1.

Uniform Nearest-Particle

Systems

213

9.4. U N P S ( A f ) (Finite M and Finite S y s t e m ) In this section we consider the finite system of the UNPS(M). Let (fM denote the UNPS(Af) starting from A C Z. That is, the process is Markov process birth rates /3<M>(/, r) and death rate 1 . A-+A\J{x}

at rate

P(M)(lA(x),rA(x))

A -» A \ {x}

at rate

1

for each x i A,

for each x & A,

where 0{M)(i K

T)=l ' '

2
for / A r < M, for I A r > M,

where 1 < / , r < oo, and IA(X) — x — max{y : y < x and y € A}, rA(x) = min{j/ : )/ > a; and y e A} -x, IA(X) or rx(a;) is oo if the maximum or minimum is not defined. Note that this ^M~>{i,r) is equivalent to the definition of that which appeared in the Section 9.2. Let "oo = Ut,M ? forall t > 0}, where £ ° M is the TJNPS(M) starting from the origin. So the survival probability is defined by Px(fl00). We take it as the order parameter of the UNPS(Af). Then we define the critical value A*(M) by K(M) = inf{A > 0 : Px^) > 0}. We used superscript * in order to distinguish the finite system and the infinite system. In this setting, Bramson 5 proved the following result by using block construction and comparison theorem. T h e o r e m 9 . 4 . 1 . For any A > 1/2, there is an M = M(A) so that P*(fioo) > 0. Some stronger statements can be found also in Bramson. 5 References 1. N.Konno and M.Katori, Upper bounds for order parameters of a class of attractive nearest-particle systems with finite ranges. J.Phys.Soc.Jpn. 61 (1992) 806-811. 2. T.M.Liggett, Interacting Particle Systems (Springer-Verlag, New York,1985). 3. N.Konno and M.Katori, Analysis of the order parameter for uniform nearest particle system. J.5iat.Pftys.65(1991)247-254. 4. T.S.Mountford, The critical value for the uniform nearest particle process. Ann. Probab. 20 (1992) 2031-2042. 5. M.Bramson, Survival of nearest-particle systems with low birth rate. Ann.Probab. 17 (1989) 433-443.

Phase Transitions of IPSs

214

C H A P T E R 10 PARTICLE SYSTEMS ON TREES

10.1. Introduction In this chapter we consider some particle systems on the homogeneous tree, T , which is also called Bethe lattice. This is an infinite graph without cycles, in which each vertex has the same number of nearest neighbors, which we will denote by K > 3. Let o be a distinguished vertex of T , which we call the origin. For x,y G T , the natural distance between x and y, | % — y |, is defined by the number of edges in the unique path of T joining x to y. We introduce the diffusive contact process (DCP), 7]t,D, on T , which is a continuoustime Markov process on {0, l } r The formal generator is given by

n/00 = £

v(x) + (i-r,(x))\

xeT |_

Yl rtv) [/(*)-/(?)]

yeT:\y-x\=l

+ ^ E v(x)(i-n(y))p{*,v)lf(n*,v)-f{i)l where A > 0, D > 0, p(x,y)>0,

^p(x,y)

= l,

p(x,y)

=

p(y,x),

Vx(y) = V{V) tfy^v, »?«(*) = 1 - V(.v), and nxy(x) = ri(y),nxy{y) = n{x),T)xy{u) = n{u) otherwise. The first term of right hand side is equal to the basic contact process and the second one is the exclusion process (or stirring system) on T . As in the Chapters 6 and 7, we consider only the nearest neighbor stirring mechanism : p(x,y)-.

for

2/1=1.

So, for the stirring mechanism, the following (i) and (ii) are equivalent :

Particle Systems on Trees

215

(i) for each x,y e T with | x — y |= 1, we exchange the values at x and y at rate D. (ii) a particle at x waits an exponential time with stirring rate KD and then chooses y, i.e., one of K nearest neighbors randomly with probability 1/K. If y is vacant it jumps to y, while if y is occupied, it remains at x. This process is attractive, so the upper invariant measure is well denned : for A, D > 0, V\,D = lim SiS(t), t—*oo

where Si is the pointmass on r) = 1 and S(t) is the semigroup given by Q. As in Example 5.4.4, the DCP on T is also coalescing self dual. So for x e £ with £ C T, £— ► f \ {x} at rate 1, £— ► £ U {t/} at rate A for | y - x |= 1, £ -► (C U M ) \ {x} at rate £ for \y-x\=l. Let f °£, denote the DCP starting from the origin. Define the survival event and survival probability by &°o = {t°,D ? forall*>0}, PA(fioo) = P(tt,D t for all t > 0). Let (} D be the DCP starting from 77 = 1. Moreover we let £fD be the DCP starting from £frD = A and £fD(B) =\ £fD n B \ where A,B C T. Then the order parameter and critical value are defined as follows : px(D)

= I>\,D{V ■ V{x) = 1)

= P(o £

&,D)

= ft(0»). Af (D) = inf{A > 0 : px(-D) > 0} = inf{A > 0 : P(o e &,,£>) > 0} = inf{A > 0 : P ^ o o ) > 0}. For particle systems on T, we introduce the new type of critical value : \[(D) = inf{A > 0 : P(limsupffD]r)(o) > 1) > 0} t—+ 00

= inf{A > 0 : P(o £ # >D infmitly often ) > 0}. The definitions give {lim sup £?,D(o) > 1} C {a,D ^4> for all t > 0}, t—+oo SO

A»(X?) < Ai(2?). (10.1) From now on, when D = 0, we will omit D, for example, Af = \ac(D = 0). Note that as for the basic contact process,( i.e.,D = 0,) on Z d , Bezuidenhout and Grimmett1 showed A? = X'c. (10.2) On the other hand, on T, Pemamtle2 proved

Phase Transitions of IPSs

216

T h e o r e m 10.1.1. For « > 4, K < A' So the following is expected. Conjecture 10.1.2. For K = 3,

A?
As for bounds on \ C and Aj., Theorem 2.2 in Pemantle 2 gives As for bounds on \9C and Aj., Theorem 2.2 in Pemantle 2 gives T h e o r e m 10.1.3. For K > 3. T h e o r e m 10.1.3. For K > 3, 1

(1)

K-1

1

(2)

4+

2^/^T - "

(3)

~

c

?

^\ / 99 ++ ^^ - -1 ! ~ 2/c

1 K - 2 '

- ■Y 1 /88++ N /.. i_i I _ I V2J / S = T-, - 1-, ++ (v^=T-l)

2(V«^T-1)

lim KA» = 1,

4

(4)

^ 8 ss 0.586 < Urn inf ( V ^ T Xlc) < lim sup ( V ^ 7 ! A^) < e fa 2.718.

In fact Theorem 10.1.1 and Theorem 10.1.3 (3) come from Theorem 10.1.3 (1) and (2). Next we introduce a diffusive branching process (DBP), ZfD, on T at stirring rate D starting from A. The dynamics of the DBP on T is as follows: (i) each particle dies at rate 1, and gives birth at rate KA. (ii) a particle born at x is sent to a site y chosen at random from « nearest neighbors, (iii) each particle jumps to a randomly chosen neighbor from K nearest neighbors at rate K.D.

Similarly ZfD(B) denotes the number of particles in B at time t. By definitions, the DCP can be thought as a coalescing DBP, that is, a system which follows rules (i)-(iii), and (iv) if a particle is sent to an occupied site, the two coalesce to one. Let Z° D denote the DBP starting from the origin. As in the DCP on T , define the survival event and survival probability by Hoo = {Z°tiD £$ for all r > 0}, P\(Q-co) = P(Z°D + 4> for all i > 0).

Particle Systems on Trees

217

So the critical values are defined as follows : XlD) = inf{A > 0 : P*(H») > 0}, \'C{D) = inf{A > 0 : P(limsup Z°tD{p) > 1) > 0}. t—foo

As in the DCP, note that

As in the DCP, note that

'

X(D) < x'jiD).

X(D) < \'C(D). On the other hand, it is well known that

(10.3)

ft(5oo) = ^ i ,

(10.4)

X = inf{A > 0 : P^fioo) > 0} = - .

(10.5)

3

Moreover in our setting Madras and Schinazi proved Theorem 10.1.4. For K > 3,

\[{D)=—/L=

2V/K - 1

+D

K

2VK-1

-0-

Then Eq.(10.5) and Theorem 10.1.4 imply that X(D) < t(D) for any A,I> > 0, and K > 3. The proof of Theorem 10.1.4 will be found in Section 10.3. Prom the above mentioned facts, the following results are obtained. Theorem 10.1.5. For A, D > 0 and K, > 3,

(i) (2) (3)

X(D) - i < tW = ^=l

yL= -1) < *m,

X(D) = i < A»(X>) < A<(C), lim A'(£») = lim \'C(D) = oo,

D—*oo

(4)

+n

D—*oo

p»(U) = P^fico) < P A ^ ) = ^ - ^ V 0,

where a V b is the maximum of a and b. Moreover Madras and Schinazi3 studied also the biased voter model on T. (As for the biased voter model, for example, see Chapter 3 in Durrett.4) Now we present some conjectures as follows.

218

Phase Transitions of IPSs

Conjecture 10.1.6. For A > 0 and K > 3, (1)

(3)

\°(D)

-> -

px{D) -* ?~-

as D -+ oo.

-1

as D -

oo.

As for part (1), Lemma 10.2.4 gives the lower bound on A£(Z>) : Af(D) > 1/K. Moreover, in Section 10.4, we will give the following more refined result by the Griffeath method : T h e o r e m 10.1.7. Let K > 3. There is a C so that if D is large, then

1. ^)-\>-CB-

K

>-iN

Of course, the BDS method and the differential inequality method will be also applicable to this proof. So we expect the following result : Conjecture 10.1.8. Let K > 3. There is a C so that if D is large, then

i.c^ ~ - ^ D As we saw in Theorem 6.1.2, this conjecture is related to the result of the DCP on Zd with d > 3 :

*">-s"£ where sa means that if C is small (large) then right hand side is a lower (upper) bound for large D. As in the DCP on Zd, we give Conjecture 10.1.9. Let K > 3. If 0 < D < ~D, then (1)

\«(D) >

Xl(D),

(2)

\'C(D) < \'C(D).

Moreover, when D = 0, results on the continuity of px at A£ are given by Pemantle 2 for K, > 4 and by Morrow, Schinazi and Zhang 5 for K = 3. That is,

Particle Systems on Trees

219

T h e o r e m 10.1.10. For re > 3, P\i = 0. So we expect that this holds even for D > 0 : C o n j e c t u r e 1 0 . 1 . 1 1 . In the DCP on trees with re > 3, P\UD)(D) = 0.

As for the critical exponent, we present the following conjecture. C o n j e c t u r e 10.1.12. Let re > 3 and D > 0. If we assume that there is a critical P = P(K,D) SO that as A J. A3(X>),

Pi{D)~{\-\l{D)Y,

exponent

imf,

then /? = !■ Note that /3 = 1 is the same as that of the DBP on T and Z d This chapter is organized as follows. Section 10.2 is devoted to the preliminary results for the proofs of Theorems 10.1.4 and 10.1.7. In Section 10.3, we give the proof of Theorem 10.1.4. In Section 10.4, the proof of Theorem 10.1.7 is given by the Griffeath method mentioned in Section 7.4. 10.2. Preliminary R e s u l t s As in Chapter 6, from now on we present several preliminary results. Let S*'" be a continuous-time simple random walk starting from x in which a particle jumps to a randomly chosen neighbor at rate KV on T . For x,y e T , v > 0,B C T, define p„(t;x,y)

= P(S^"

= y)

and

p„(t;x,3)

The following facts are very useful for the proofs. L e m m a 1 0 . 2 . 1 . For v > 0,< > 0,x,y (1)

6 T,

p„(*;z,y)
= £?„(<;*,#).

Phase Transitions of IPSs

220

(2)

lim —

t-nx

KVT

logy„(i; o, o) = 2 ^ — ^ - 1. K

There is a constant C so that

(3)

pi (*;»,«>} < ^TJ.

P r o o f . The proof of part (1) is the same as that of Lemma 6.3.1 (2). As for parts (2) and (3), see Madras and Schinazi. 3 For A, B C T , x € T , and A; = 1 , 2 , . . . , define m £ (t; A,B) = £ ( ^ D ( S ) f c ) ,

E(Z*D(B)k),

m f (f; A, B) =

«?(<) = £«?,Dm*)-

™£(0 = £(^ D (T)*).

In particular, we omit the subscript when k = X. Note that m ^ ( t ) and m£(t) are independent of x. On the other hand, (fB and Z/^, can be constructed on the same probability space with

tfD(B) < ZtAD(B).

So the above definitions imply that for A, B C T, k > 1 and t > 0, we have mj?(t;A,B)<

mf(t;

mD{t)

A, B),

<

mD(t).

By the definition, jtP^D(t;x,y)t;x,y) = (A + D) Y,Px+D(*;*,y *,y + *) - *(A + D ) P A + D ( * ; X , K ) .

(10.6)

z~0

From Eq.(10.6) and Lemma 10.2.1 (1), we have L e m m a 10.2.2. (1)

mD(t;x,y).»)<=

(2)

mD(t;x,y) y) = ^ w

:«A-II

e^-Vtp>+D(t;x,y).

D

( t -

3]X,Z)

s;x,z)mD(s;z,y).

z€T

(3) (4)

fnD(t; x, y) > mD(t - s; x, z)mD(s;

mD(t

z, y).

mD(t;x,y)<mD(t;o,o).

Note that our notation pt(x,y),y) is equal to pKt(x,y)y) in Madras and Schinazi.3 Hence we have

Particle Systems on Trees

221

Lemma 10.2.3. 0 < mD(t) < mD{t) =

e^'^K

So this result allows us to conclude that 1/K is the trivial lower bound on Af (D) : Lemma 10.2.4.

\i(D) > i . From Lemma 10.2.2 (3) for x = y = z = o and superadditivity, the following limit exists : 1\{D) = Urn - log m D (t;o,o) = sup -\agmD{t;o,o). l-»oo t

(10.7)

t>tj t

On the other hand, Lemma 10.2.2 (1) gives -logmD(t;o,o)

= KX-1 + - log p A+D (i;o,o).

(10.8)

By Lemma 10.2.1 (2), we have Hm -tPx+D («j o, o) = «(A + 0 ) Urn <

= K(k +

x

\

D)t

l°gP 1+D fo o, o)

D)(2?£~-l\

= 2(A + D)y/1T^1 - K(X + Z>). From Eqs.(10.7), (10.8), the last equality and Theorem 10.1.4, we have Lemma 10.2.5. (1)

y x (U) = 2(A + D)y/^l \JK — 1 — KD -KD-1.

(2)

7A(£) = 2 V ^ T [ A - X ( £ ) ] .

So Lemma 10.2.5 (2) gives the following characterization of AC(D) : Al(P) = i n f { A > 0 : 7 A ( £ ) > 0 } = sup{A > 0 : jx(D) < 0}. By using similar proofs of Lemma 6.3.6 (1) and (3), we have the following results. Let Y denote the set of all real-valued bounded measurable functions on T. In particular, for a £ T,Sa E Y is defined by Sa(a) = 1 and Sa{b) = 0 if b £ a. For each f € Y, let

=ZyeTZZD(y)f(y).

Phase Transitions of IPSs

222

L e m m a 10.2.6. For / £Y, x e T, B C T , D, A, t > 0, and k = 1 , 2 , . . . , E(k)

(1)

E()

=

+ A E (*) E E fdsE{< Z?_s,D,6y »E(< ZloJ >j)E« Z*%,f >k~*) z~0y£TJ°

j=l ^^

(2)

= mD(t; x, B)

mf(t; x,B) +

A

E ( (

f c ■

)

) E E >

ftdsmD(t-a;x,y)rnf(s;y,B)m^J(S;yJ , ^

1

+

z,B).

:

)y£T-

Moreover to prove Theorem 10.1.4, we use Moreover to prove Theorem 10.1.4, we use L e m m a 10.2.7. m f ( i ; o , o ) < m D ( t ; o , o ) j l + 2A« /

(1)

mD{a;o,o)ds\

m D ( i ; o , o ) 2 < Tnf (i;o,o)P(Z° I ,(o) > 1).

(2)

Proof. As for part (1), Lemma 10.2.6 (2) for k = 2,x = o, and B = {o} gives mj (*i°!°)

=

"*(*> °>°) + 2A Y j / J

/ dsIff (t — s;o,y)m

(s; y,o)m

(s; y +

z,o).

So from Lemma 10.2.2 (2) and (4), we have m®(t;o,o)

< mP{t;o,o)

+ 2AK / Jo

D mD(t;o,o)m >)mD( (s;o,o)ds,

since mD(s;y + z,o) < mD(s;o,o). The desired result is obtained. For part (2), using the Cauchy-Schwarz inequality, we get mD{t;o,of

~

E{ZiD(o)f

= W » l E(zi = (!.D | , ) iKoia}m > l ) )y' | t{o)i < E{Z$iD{of)P{ZlD{o) > 1) = mf(i;o,o)P(^D(o)>l). So the proof is complete.

Particle Systems on Trees

223

10.3. \'C(D) . In this section we give the proof of Theorem 10.1.4 following the arguments in Madras in Schinazi. 3 Let

^=vCT+pbfcr-4

so we will show XC(D) = x{D). P r o o f o f \[{D) > \{D).

We assume A < X(D)- Set 7 = fx(D).

Lemma 10.2.5 (1) :

7 = 2V^T[A-X(£>)] gives 7 < 0. Note that we don't use Theorem 10.1.4 to prove Lemma 10.2.5 (1). On the other hand, Lemma 10.2.1 (3) and the definition of 7 imply that there is a, constant C so that mD{t;o,o)
,

e'-

So P(KD(O) P(KD(O)

> 1) < C

> 1) < C

±g.

^

.

By 7 < 0, the last inequality and the Borel-Cantelli lemma, we have P{Z°,D{°) so A < \[{D).

> 1 a* arbitrarily large t) = 0,

Hence we have x(-D) < A~l(£>).

P r o o f of ~)?C(D) < x{D)- We assume A > x(D)- Lemma 10.2.5 (1) gives 7 > 0. So there is a C > 0 so that mD(t; o, o) > ect for all t large enough. (10.9) On the other hand, combining Lemma 10.2.7 (1) and (2) implies P(ZID{°)

mD(t;o,o) > !) > - , , . ,t_g, 1 + 2AK J 0 mr{s;

TT-

o, o)ds

Hence by Eq.(10.9) and the last inequality, we get , „ , , , 1 f,. ftmD(s;o,o)J liminf P(Z? (p) > 1 > — - H m s u p / ) ' v ZDUK t—00 ' 2AK (_ i_c» J0 mu{t;o,o)

"I" 1 'ds \ . J

Prom Lemma 10.2.2 (3), we have liminfP(Zt°D(o)> 1 ) > - p - ^ / <-oo *• t , i , v 2A« [J0

zrrn r\ mu(s;o,o))

.

224

Phase Transitions of IPSs

Eq.(10.9) gives

r J0

ds

< 00,

m"(s: mD(s;o,o)

so liminf t _ > o o P(2 t 0 D (o) > 1) > 0. Noting that P(lim sup Z?D(o) > 1) > lim sup P{Z°D(o)

> 1) > liminf P(Z?D(o)

we have P(lim s u p ^ ^ Z°tD{o) > 1) > 0. Hence A > j'c(D), 10.4. Lower B o u n d s o n

> 1) > 0,

so AC(D) < xC°)-

X°(D).

In this section we will give the proof of the following theorem : T h e o r e m 1 0 . 4 . 1 . Assume K > 3. For D> 0, (K - 1)JP + 1 (1)

A?(jD)

1

- ( « - i ) ( « J 3 + i) > «'

If .D is large, then

(2)

Af(Z>)>i + . K

!

!

K 2 ( K - 1)L>'

Part (2) follows from part (1) and refines Theorem 10.1.7. Note that parts (1) and (2) can be obtained by translation 2d —» K in Propositions 7.4.4 and 7.4.3 respectively. So the proof of part (1) are the almost same as that of Proposition 7.4.4. However for the convenience of readers, we will give it in the following way. Note that when K = 3 and D = 0, Griffeath 6 obtained an Improved lower bound on Af as in the proof of Theorem 5.2.1 : 109-1 Ag> ^ f - 1 M 0.524 > 1 . c 18 2 Proof. (Griffeath Method.) For A e Y, we let o-(A) = P(£fiD / <j> for all t > 0). Note that the coalescing self-duality of the DCP on T gives a(A) = v\tD{v '■ Jy(^) = 1 for some x e A ) .

Let o ~ *i ~ S2 with S2 ^ o. Set
o-({o,Si}),...

Particle Systems on Trees

225

L e m m a 10.4.2. (!)

(2)

- ( « + l ) a ( o ) + K
a(o) - [(K - 1)A + l]<7(o<5i) + (K - l)Ao-(oM 2 ) + {K- \)D[a(o62)

-
On the other hand, as in the case of Zd, the following submodulality for o(A) holds in this case. L e m m a 10.4.3. For any A,B C Y, a(A UB) + c(A n B) < c(A) + cr{B). From Lemma 10.4.3, we have Corollary 10.4.4. (1)

a{oi-ih)

(2)

< 2(7(0(5!)

-cr(o).

a(oS2) < 2a(o). Using Lemma 10.4.2 (2) and Corollary 10.4.4 (1),(2), we have [ - ( « - 1)A + 1 + 2(K - l)D}a(o) + [(K - 1)A - 1 - (K - X)D]cr{oSi) > 0.

By Lemma 10.4.2 (1) and the last inequality, [(it - 1 ) ( « J D + 1)A - {(K - 1)D + 1}] 0.

So if (K - 1)(KD + 1)A - {(« - 1)D + 1} < 0, then a(o) = 0. By a(o) = p\{D), conclusion is obtained.

the desired

References 1. C.Bezuidenhout and G.Grimmett, The critical contact process dies out. Ann.Probab. 18 (1990) 1462-1482. 2. R.Pemantle, The contact process on trees. Ann.Probab. 20(1992)2089-2116. 3. N.Madras and R.Schinazi, Branching random walks on trees. Stock. Proc. Appl. 42 (1992) 255-267. 4. R.Durrett, Lecture Notes on Particle Systems and Percolations (Wadsworth, Inc., California, 1988). 5. G.Morrow, R.Schinazi and Y.Zhang, The critical contact process on a homogeneous tree. J.Appl.Probab. 31(1994)250-255. 6. D.Griffeath, Ergodic theorems for graph interactions. Ann.Probab. 7(1975)179-194.

226

Phase Transitions of IPSs

SUBJECT INDEX

additive system, 73 attractive, 17 basic contact process, 7, 15, 89, 171, 200, 209 BDS method, 167, 197, 218 Bethe lattice, 214 biased voter model, 217 binary contact path process, 103 bond, 53 branching annihilating random walk, 69 Brownian scaling relation, 121 coalescing dual process, 71, 94 coalescing duality, 84 comparison method, 57, 130, 131, 190, 191 contour, 55 contour method, 54 correlation identity, 18, 59, 62 critical exponent, 74-75, 103, 212, 219 critical value, 17, 215 crossover, 103 differential inequality method, 168, 218 diffusive branching process (DBP), 102, 110, 216 diffusive contact process (DCP), 92, 100, 109, 214 diffusive ^-contact process, 92 duality equation, 71, 76 duality theorem, 88 edge process, 72

Subject Index

edge speed, 72 ergodic, 16 exclusion process, 91, 100 Galton-Watson process, 138 Gauss's hypergeometric series, 81, 84 generalized contact process, 70 generator, 6 graphical representation, 15, 57 Gray duality, 93 Griffeath method, 75, 169, 218, 224 Harris-FKG inequality, 21-22, 204, 211 Harris-FKG inequality method, 21 Harris lemma, 28, 80 Holley-Liggett method, 59, 67, 80 identity matrix, 10 independent and identically distributed (i.i.d.), 4 interference term, 118, 181 Liggett and Ziezold-Grillenberger methods, 33, 42, 78 local central limit theorem, 121 long range branching process (LRBP), 173 long range contact process (LRCP), 92, 171 lower invariant measure, 17 Markov chain, 1 Markov extension, 33 Markov extension method, 27 Markov process, 7 Markov's inequality, 125 mean field theory, 102, 173 modified threshold contact process, 13 order parameter, 17 oriented bond percolation, 53 oriented site percolation, 56 origin, 214 Paierls argument, 54 pair annihilation model, 69

227

Phase

228

Transitions

periodic threshold contact process, 80 random renewal renewal renewal

walk, 2 chain, 4 decoupling, 66 measure, 62, 66, 81

site, 56 second-order phase transition, 74, 211 self dual, 74, 91, 92 state space, 1 stationary distribution, 1, 6 stationary measure, 1, 5 steady state concentration of particles, 74 submodular, 77 temporally homogeneous, 1, 5 0-contact process, 70, 91 threshold contact process, 11, 41 transition matrix, 1, 5 transition probability, 1, 5 tree, 214 uniform birth process, 211 uniform nearest-particle system, 199 universality class, 75, 103 UNPS(M), 200, 209 upper invariant measure, 17 zero matrix, 10

of

IPSs

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