bootstrap percolation and some applications · 2019. 5. 6. · a mis padres eugenio y luz marina...

INSTITUTO NACIONAL DE MATEMÁTICAPURA E APLICADA

DOCTORAL THESIS

Bootstrap Percolation and someApplications

Author:Daniel R. Blanquicett T.

Advisor:Dr. Robert Morris

A thesis submitted in fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics to the

postgraduate program in Mathematics at the Instituto Nacional deMatemática Pura e Aplicada.

Rio de JaneiroFebruary 2019

http://www.impa.br

http://www.impa.br

http://quicetor.impa.br/Home

http://w3.impa.br/~rob/

iii

“Diviser chacune des difficultés que j’examinerais, en autant de parcelles qu’il sepourrait, et qu’il serait requis pour les mieux résoudre”.

René Descartes

v

Abstract

In this Ph.D. thesis we study bootstrap percolation, a monotone version ofthe Glauber dynamics of the Ising model of ferromagnetism, and some ap-plications to (non-monotone) dynamical models. The r-neighbour bootstrapprocess on a locally finite graph G is a monotone cellular automata on theconfiguration space 0, 1V(G), and the main question in the area is to deter-mine the so-called threshold for percolation.

In Chapter 1 we begin by discussing r-neighbour bootstrap percolation onZd, and then introduce the model of Bollobás, Smith and Uzzell for mono-tone cellular automata, we also state some recent two-dimensional results ob-tained in their extremely general setting. We end by discussing the Glauberdynamics of the Ising model on general graphs.

In Chapter 2 we explain in more detail the results about the general modelof monotone cellular automata: according to a geometric criterion, the so-called update families U were classified into three universality classes, namely,supercritical, critical and subcritical. We prove that for a wide class of sub-critical update families, the size of the cluster containing the origin decaysexponentially fast, in distribution.

One of the main contributions of this thesis is the determination of thethreshold for percolation for a family of models known as anisotropic bootstrappercolation, which are 3-dimensional analogues of a family of (2-dimensional)processes studied by Duminil-Copin, van Enter and Hulshof. In our modelsthe graph G has vertex set [L]3, and the neighbourhood of each vertex con-sists of the ai nearest neighbours in the ei-direction for each i ∈ 1, 2, 3,where a1 ≤ a2 ≤ a3. In Chapter 3, we resolve this problem in the caser = a3 + 1 for all triples (a1, a2, a3), and also in the case r = a3 + 2 unlessa3 = a1 + a2 − 1. To do so, we introduce a new technique called the beamsprocess, and use the exponential decay property about cluster size distribu-tions, proved in Chapter 2. Moreover, we determine upper bounds for allvalues r ≤ a2 + a3, which we believe are best possible up to a constant factor.

The fourth chapter deals with two applications of bootstrap percolation.The first part is devoted to the exposition of recent universality results ob-tained by Martinelli, Morris and Toninelli, for kinetically constrained spinmodels. In the second part, we generalise a result of Fontes, Schonmannand Sidoravicius, who showed the existence of a phase transition for the zero-temperature Glauber dynamics of the Ising model on Zd; we prove an anal-ogous theorem for the so-called U -voter dynamics, for various families U .

vii

Agradecimientos

Quiero darle gracias en primer lugar a Dios, quien me ha premiado con elentendimiento y la perseverancia necesaria para alcanzar mis objetivos.

A mi orientador Robert Morris por darme la oportunidad de compartirconocimientos, por su paciencia, compañerismo y creatividad inagotable.

A mis profesores del IMPA, Augusto Teixeira y Roberto Imbuzeiro, porsus grandes aportes en este proceso y excelentes consejos.

Al profesor Jorge Zubelli, ha sido un privilegio poder contar con su guía,amistad y ayuda incondicional.

A los miembros de la banca, Béla Bollobás, Hubert Lacoin, Renato Fontesy Simon Griffiths, por dar sus valiosos aportes a este trabajo.

A mis padres Eugenio y Luz Marina por su amor infinito.

A mi amada esposa Clarena, por ser la principal promotora y cómplice demis sueños.

A mis hermanos, sobrinos y tíos, por su alegría y por enseñarme a partirde sus experiencias.

A mis amigos, en especial, Argenis, Avila, Emily, Pedro, Reza, Walner yTie, por su fraternidad en el recorrido de este camino.

Al Instituto de Matemática Pura e Aplicada y su equipo, por propiciarmeun ambiente adequado y haberme permitido crecer profesionalmente.

A la CAPES, por su apoyo financiero.

ix

Contents

Abstract v

Agradecimientos vii

Contents ix

List of Figures xi

1 Introduction 11.1 Bootstrap percolation on Zd . . . . . . . . . . . . . . . . . . . . 21.2 Monotone cellular automata: The BSU model . . . . . . . . . . 51.3 The Glauber dynamics of the zero-temperature Ising model . 8

2 U -bootstrap percolation 112.1 The BSU model . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Universality in two dimensions . . . . . . . . . . . . . . . . . . 142.3 Exponential decay for subcritical families . . . . . . . . . . . . 16

2.3.1 Inwards stable droplets and the dilation radius . . . . . 162.3.2 Exponential decay . . . . . . . . . . . . . . . . . . . . . 19

3 Anisotropic 3-dimensional models 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Anisotropic bootstrap percolation on [L]3 . . . . . . . . 243.1.2 Critical families . . . . . . . . . . . . . . . . . . . . . . . 263.1.3 Induced processes . . . . . . . . . . . . . . . . . . . . . 303.1.4 Outline of the proof . . . . . . . . . . . . . . . . . . . . 35

3.2 Upper bounds for r ∈ c + 1, c + 2 . . . . . . . . . . . . . . . 363.2.1 Case III: c ∈ b + 1, . . . , a + b− s . . . . . . . . . . . . 363.2.2 Case IV: c = a + b− 1 and s = 2 . . . . . . . . . . . . . 393.2.3 Case I: c = b = a . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Case V: c = a + b . . . . . . . . . . . . . . . . . . . . . . 443.2.5 Case II: c = b > a . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Lower bounds via components process . . . . . . . . . . . . . 503.3.1 The components process . . . . . . . . . . . . . . . . . . 503.3.2 Case I: c = b = a . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Case II: c = b > a . . . . . . . . . . . . . . . . . . . . . . 533.3.4 Case III: c ∈ b + 1, . . . , a + b− s . . . . . . . . . . . . 54

3.4 Lower bounds via beams process . . . . . . . . . . . . . . . . . 563.4.1 The beams process . . . . . . . . . . . . . . . . . . . . . 563.4.2 Case V: c = a + b . . . . . . . . . . . . . . . . . . . . . . 59

x

3.4.3 The coarse beams process . . . . . . . . . . . . . . . . . 613.4.4 Case VI: c > a + b . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Upper bounds for r ∈ c + 3, . . . , c + a . . . . . . . . . . . . . 643.5.1 Cases III and IV . . . . . . . . . . . . . . . . . . . . . . . 683.5.2 Cases I and II . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6 Upper bounds for r ∈ a + c + 1, . . . , b + c . . . . . . . . . . . 76

4 Applications 814.1 Universality results for kinetically constrained spin models in

two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 824.1.2 Universality results . . . . . . . . . . . . . . . . . . . . . 84

4.2 Fixation for 2-dimensional U -voter dynamics . . . . . . . . . . 864.2.1 Motivation: The U -Ising dynamics . . . . . . . . . . . . 874.2.2 The U -voter dynamics: Main result . . . . . . . . . . . 884.2.3 The 1-dimensional approach . . . . . . . . . . . . . . . 93

4.2.3.1 A martingale argument . . . . . . . . . . . . . 964.2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . 97

4.2.4 The process inside rectangles . . . . . . . . . . . . . . . 1004.2.4.1 Bootstrapping the vertices in state − . . . . . 1014.2.4.2 Erosion step . . . . . . . . . . . . . . . . . . . 102

4.2.5 Wrapping up . . . . . . . . . . . . . . . . . . . . . . . . 103

A Proof of Lemma 3.6.3 107

B Probability on graphs 109B.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

B.1.1 Asymptotic notation . . . . . . . . . . . . . . . . . . . . 110B.2 Probability and expectations . . . . . . . . . . . . . . . . . . . . 110B.3 Product measure . . . . . . . . . . . . . . . . . . . . . . . . . . 112B.4 Classical Bernoulli percolation . . . . . . . . . . . . . . . . . . . 113

B.4.1 The critical probability . . . . . . . . . . . . . . . . . . . 114B.4.2 The subcritical regime . . . . . . . . . . . . . . . . . . . 115

B.5 Continuous time processes . . . . . . . . . . . . . . . . . . . . . 115B.5.1 Markov processes . . . . . . . . . . . . . . . . . . . . . . 116B.5.2 Interacting particle systems . . . . . . . . . . . . . . . . 118B.5.3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography 121

xi

List of Figures

1.1 6× 6 chessboard with 9 initial black squares. Squares labeledwith t become black at time t. The whole board is black aftert = 10 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1 A family U consisting of two rules, ±e1, (−1, 1)marked with∗, and ±2e1marked with×. ThenQ = ±e2, (1/

√2, 1/√

2)and D = ±e2, (1/

√2, 1/√

2),±e1. . . . . . . . . . . . . . . . 18

3.1 The set Na,b,c with a = 1, b = 2 and c = 4. The e1-axis istowards the reader, the e2-axis is vertical, and the e3-axis ishorizontal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 S11 is the big circle, S1

2 and S13 are drawn with dashed ellipses.

The vector u is outside S11 ∪ S1

2 ∪ S13 and H3

u contains all posi-tive multiples of e1,−e2 and e3. . . . . . . . . . . . . . . . . . . 27

3.3 [L]3 = R1 ∪ R2 ∪ · · · ∪ RL. We identify vertices with cubes ofsidelength 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 When s = 1, it is enough one single vertex. When s = 2, it isenough two vertices in diagonal. . . . . . . . . . . . . . . . . . 30

3.5 An s-pattern on the right-most side (s = 1), one copy of ∆a onthe front side, and one copy of ∆b on the top side. . . . . . . . 32

3.6 A 2-pattern on the right-most side (s = 2). The other sides re-quire α vertices in each of the lines along the e3-direction (α = 2). 33

3.7 a = 2, b = 3, c = 4, s = 2 and α = r− (a + b) = 1. . . . . . . . . 403.8 One copy of the right-angled triangle ∆a inside l + 1 × [w]2,

and one s-pattern on the top and right-most sides (s = 1). . . . 473.9 A beam w.r.t. the subcritical family N 1,2

4 . . . . . . . . . . . . . 583.10 To the left, a copy of the 3-board. To the right, a 3-pattern. . . . 65

4.1 The clock at vertex v rings at time τk, then the state of v is up-dated by using the two-dimensional family U = X1, X2, X3,where X1 = −e1, e2 is marked with ×, X2 = −2e1,−e2marked with

√, and X3 = e1, (2, 2), (−2,−2)marked with ?. 87

4.2 4 stable directions determining a (T , L)-droplet. . . . . . . . . 904.3 H2

y entirely − and Z2 \ (H2y ∪ $) entirely + . . . . . . . . . . . 93

4.4 3 or 4 stable directions . . . . . . . . . . . . . . . . . . . . . . . 954.5 A family without fair directions, its stable set and a failing con-

figuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xiii

List of Symbols

A initially infected set in bootstrap percolation〈A〉 closure of Ae1, . . . , ed canonical unit vectors in Zd

Hull(·) convex hull of ·I•(·) internally filled eventI×(·) internally spanned eventIV(·) covered beam eventK cluster of the origin in 〈A〉[L]d d-dimensional euclidean grid with sidelength LLc(U , p) critical length under U -bootstrap percolationN d

r d-dimensional nearest neighbours update familyN s,t

r anisotropic two-dimensional update familyN a,b,c

r anisotropic three-dimensional update familypc(G, r) critical probability under r-neighbour bootstrap percolation on GpIs

c (Zd,U ) critical probability for fixation of the U -Ising dynamics

pvotc (Zd,U ) critical probability for fixation of the U -voter dynamics

ru(·) number of rules disagreeing with vertex u in ·Sd unit d-sphereS stable setT(D) droplet erosion timeU update family

| · | cardinality of the set ·〈·, ·〉 usual inner productHd

u d-dimensional discrete half space orthogonal to uPp product of Bernoulli measures with density pQ1 set of rational directions on S1

V generator for the U -voter dynamicsZd d-dimensional euclidean latticeZd

L d-dimensional torus

α difficulty of Uβ dilation radius/bilateral difficulty of Uη, σ configurations in +,−L and +,−Zd

, respectivelyo(1) magnitude going to zero, as L goes to infinityO(·) magnitude upper bounded by · (up to a constant factor)Ω(·) magnitude lower bounded by · (up to a constant factor)Θ(·) magnitude upper and lower bounded by · (up to a constant factor)

Dedicado a Richard, LuzMa y Clare,mis fuentes de inspiración

1

Chapter 1

Introduction

The study of bootstrap processes on graphs was initiated in 1979 by Chalupa,Leath and Reich [19], and is motivated by problems arising from statisticalphysics, such as the Glauber dynamics of the zero-temperature Ising model,and kinetically constrained spin models of the liquid-glass transition (see,e.g., [30, 44, 51], and the recent survey [49]).

There are several interesting questions of different nature in bootstrappercolation, one typical example are the so-called extremal problems, like thefollowing combinatorial puzzle (see, e.g., [6]). Imagine that each square of anL × L chessboard can have one of two states: black or white. Initially, wehave some black squares, and each square of the board can change its statein discrete time according to the following 2-neighbour infection rule: blacksquares remain black forever, and a white square becomes black if it has atleast two black neighbours (two squares are neighbours if they have a com-mon edge).

FIGURE 1.1: 6 × 6 chessboard with 9 initial black squares.Squares labeled with t become black at time t. The whole board

is black after t = 10 steps.

What is the minimum number of initial black squares one needs to make thewhole chessboard black by the end of the process? We encourage the readerwho does not know the answer to solve this puzzle, and a nice solution willbe given in Chapter 2.

In this thesis, we will focus on random questions, which we describe inthe following (however, extremal results will be used in the sequel).

2 Chapter 1. Introduction

Given a locally finite graph G, we define the r-neighbour bootstrap pro-cess on the configuration space 0, 1V(G), (we call vertices in state 1 "in-fected"), evolving in discrete time in the following way: 0 changes to a 1when it has at least r neighbours in state 1, and infected vertices remain in-fected forever. We will always denote the initially infected set by A, and thefinal infected set by 〈A〉. The set A is chosen according to Pp, the productof Bernoulli measures with density p, and the main question is to determinethe so-called threshold for percolation: the value of p above which the entirevertex set is likely to be infected by the end of the process. More precisely,if the initially infected set A ⊂ V(G) is chosen to be p-random (meaning,Pp(x ∈ A) = p for all x ∈ V(G), independently), we consider the criticalprobability pc(G, r), defined as follows:

pc(G, r) := infp ∈ [0, 1] : Pp(〈A〉 = V(G)) ≥ 1/2. (1.1)

In this thesis we will study bootstrap percolation on Zd and its sublatticesZd

L and [L]d. We will draw the vertices in Z2 (respectively, in Z3) as squares(respectively, cubes) with sidelength one, as we did in the chessboard exam-ple. Nothing changes by adopting this convention, and there is a geometricadvantage: better visualization of the process.

1.1 Bootstrap percolation on Zd

The first rigorous mathematical results obtained on the graph with vertex setZd, and edge set given by the 2d nearest neighbours, were achieved by vanEnter [27] and Schonmann [55], who showed that if the initially infected setA ⊂ Zd is p-random, then percolation occurs almost surely for any p > 0 ifr ≤ d, and fails to occur almost surely for any p < 1 if r > d.

Theorem 1.1.1 (van Enter and Schonmann).

pc(Zd, r) =

0, if r ≤ d,1, if r > d.

(1.2)

The study of bootstrap processes on finite graphs was initiated by Aizen-man and Lebowitz [1] in 1988, who realized that interesting features arise inthe finite volume. They considered the cube [L]d with sidelength L as vertexset (instead of Zd), and determined the magnitude of the critical probabilityin the case r = 2, (we also say that they determined the metastability thresholdfor percolation).

Theorem 1.1.2 (Aizenman and Lebowitz). As L→ ∞,

pc([L]d, 2) = Θ(

1log L

)d−1

. (1.3)

In the case d = 2, Holroyd [37] determined (asymptotically, as L → ∞)the matching constant (this is usually called a sharp metastability threshold).

1.1. Bootstrap percolation on Zd 3

Theorem 1.1.3 (Holroyd). As L→ ∞,

pc([L]2, 2) =π2/18 + o(1)

log L. (1.4)

Building on work of [1] and [37], Gravner, Holroyd and Morris [33, 34, 50]determined even stronger bounds, giving a hint on the second order term inthe expansion of pc([L]2, 2). Later, Hartarsky and Morris [36] determinedsuch second term up to a constant factor.

Theorem 1.1.4 (Hartarsky and Morris). As L→ ∞,

pc([L]2, 2) =(

π2

18−Θ

(1

(log L)1/2

))1

log L. (1.5)

For the general case 2 ≤ r ≤ d, the threshold was determined by Cerf andCirillo [17] and Cerf and Manzo [18].

Theorem 1.1.5 (Cerf, Cirillo and Manzo). Let 2 ≤ r ≤ d. As L→ ∞,

pc([L]d, r) = Θ(

1log(r−1) L

)d−r+1

. (1.6)

And the sharp threshold by Balogh, Bollobás and Morris [5] and Balogh,Bollobás, Duminil-Copin and Morris [7].

Theorem 1.1.6 (Balogh, Bollobás, Duminil-Copin and Morris). As L→ ∞,

pc([L]d, r) =(

λ(d, r)± o(1)log(r−1) L

)d−r+1

, (1.7)

and the constant λ(d, r) is also determined, for all d ≥ r ≥ 2.

Recently, more general rules for infection have been studied, for instance,consider the graph with vertex set given by [L]2, where the neighbourhoodof the vertex (x, y) ∈ [L]2 is the symmetric set

(x, y + 1), (x, y− 1), (x + 1, y), (x− 1, y), (x + 2, y), (x− 2, y) ∩ [L]2.

This model is called anisotropic, and was first studied by Gravner and Grif-feath [32]. Hulshof and van Enter [29] determined the threshold for r = 3.

Theorem 1.1.7 (van Enter and Hulshof). As L → ∞, the critical probability forthe anisotropic model is

Θ((log log L)2

log L

). (1.8)

The corresponding sharp threshold was determined by Duminil-Copinand van Enter [25]. Moreover, extremely precise results have been provedabout this model by Duminil-Copin, van Enter and Hulshof [24].


Theorem 1.1.8 (Duminil-Copin, van Enter and Hulshof). As L → ∞, the crit-ical probability for the anisotropic model equals(

log log L− 4 log log log L + 2 log9e2± o(1)

)log log L12 log L

. (1.9)

Another important studied model, usually called the Duarte model, is a(non-symmetric) model with drift, where the graph is taken to be the torusZ2

L instead of [L]2, and a vertex (x, y) ∈ Z2L becomes infected if there are at

least two infected vertices in the set

(x, y + 1), (x, y− 1), (x− 1, y). (1.10)

The threshold for the critical probability was determined by Mountford [52]by using a martingale approach, and only 22 years later the sharp thresholdwas determined by Bollobás, Duminil-Copin, Morris and Smith [12].

Theorem 1.1.9 (Bollobás, Duminil-Copin, Morris and Smith). As L → ∞, thecritical probability for the Duarte model equals(

18± o(1)

)(log log L)2

log L. (1.11)

Note that, despite of the substantial difference with the anisotropic model,the critical probability for the Duarte model has the same order of magni-tude. There is a reason for this to be the case: Universality. Indeed, Bollobás,Duminil-Copin, Morris and Smith [13] determined the threshold for all two-dimensional critical bootstrap families, a wide class of two-dimensional modelswhich includes all mentioned above (see next section for further details); intheir work, it becomes clear what is the relationship between the anisotropicand Duarte models. However, determining the sharp thresholds for suchgeneral models is still an open problem.

Anisotropic models in three dimensions

In this thesis, we determine the threshold for the three-dimensional versionof the anisotropic model (see Definition 3.1.2), and the problem can be statedprecisely as follows. Define a graph NL on vertex set [L]3 by connecting xand y if x− y = kei for some i ∈ 1, 2, 3 and some k ∈ [−ai, ai] \ 0, wherea1 ≤ a2 ≤ a3 are positive integers.

In the case r = a1 + a2 + a3, the critical probability for percolation on thisgraph was determined by van Enter and Fey [28].

Theorem 1.1.10 (van Enter and Fey). As L→ ∞,

pc(NL, a1 + a2 + a3) =

Θ(

1log log L

)1/a1

, if a1 = a2,

Θ((log log log L)2

log log L

)1/a1

, if a1 < a2.

(1.12)

1.2. Monotone cellular automata: The BSU model 5

For all other values of r it remained open. In recent work, we have suc-ceeded in determining the critical probability (see Remark 2.2.6) in the caser = a3 + 1.

Theorem 1.1.11. As L→ ∞,

pc(NL, a3 + 1) =

Θ(

1(log L)2

), if a3 = a2 = a1,

Θ(

log log L(log L)2

), if a3 = a2 > a1,

Θ((log log L)3

(log L)2

), if a3 ∈ a2 + 1, . . . , a1 + a2 − 1,

Θ(

1log L

), if a3 = a1 + a2,

Θ((log log L)2

log L

), if a3 > a1 + a2.

(1.13)

We are currently working on extending the proof to the general ‘2-critical’case: a3 + 1 ≤ r ≤ a2 + a3 (see Theorems 3.1.4 and 3.1.5).

Our proof is based on a new technique (which we call the beams process)of controlling the growth of anisotropic processes in dimensions larger thantwo, and a new exponential decay property for a family of subcritical (seeDefinition 2.1.2) two-dimensional bootstrap processes (see Theorem 1.2.7 or2.3.11).

In the ‘3-critical’ range a2 + a3 + 1 ≤ r ≤ a1 + a2 + a3 the critical lengthis likely to be doubly exponential in p (as opposed to singly exponential inthe 2-critical range), and the techniques required are likely to be rather dif-ferent (and more similar to those of [28], which were based on the methodintroduced by Cerf and Cirillo in [17]).

1.2 Monotone cellular automata: The BSU model

In this section we overview an extremely general class of d-dimensional mono-tone cellular automata, which were introduced by Bollobás, Smith and Uzzell[11], then we focus on dimension d = 2. In their groundbreaking work, theystudied in complete generality the family of two-state, deterministic, mono-tone, local, homogeneous cellular automata in Zd with random initial con-figurations.

Let U = X1, . . . , Xm be an arbitrary finite family of finite subsets ofZd \ 0. We call U the update family of the process, and the process itselfU -bootstrap percolation. Let the lattice Λ be either Zd, [L]d (the d-dimensionalgrid), or Zd

L (the d-dimensional discrete torus). Now given a set A ⊂ Λ ofinitially infected sites, set A0 = A, and define for each t ≥ 0,

At+1 = At ∪ x ∈ Λ : x + X ⊂ At for some X ∈ U. (1.14)


The set of eventually infected sites is the closure of A, denoted by

〈A〉U :=⋃t≥0

At.

We say that A percolates if 〈A〉U = Λ.The key question is that of how likely it is that a random set A percolates

on the lattice Λ; in particular, one would like to know how large p must bebefore percolation becomes likely. The point at which this phase transitionoccurs is measured by the critical probability.

Definition 1.2.1. Suppose that A ⊂ Λ is p-random. The critical probability is

pc(Λ,U ) := infp : Pp(〈A〉U = Λ) ≥ 1/2. (1.15)

Remark 1.2.2. The BSU model extends the notion of bootstrap percolationgiven in the previous section, when the vertex set is Λ. The r-neighbourbootstrap process on G = (Λ, E(G)) is the same as Ur-bootstrap percolation,where Ur is the family consisting of all subsets of size r of the set

N(0) := y ∈ Λ : 0, y ∈ E(G).

The Duarte model is the same as D-bootstrap percolation, where D is thecollection of all subsets of size 2 of −e1,±e2. Note that (1.1) and (2.2) arealso the same.

From now on, let us fix d = 2. Let S1 be the unit circle and denote thediscrete half space orthogonal to u ∈ S1 as Hu := x ∈ Z2 : 〈x, u〉 < 0.The stable set S = S(U ) is the set of all u ∈ S1 such that no rule X ∈ Uis contained in Hu. The following classification of families was proposed in[11]. A family U is

• supercritical if there exists an open semicircle C ⊂ S1 which does notintersect the stable set;

• critical if every open semicircle in S1 has non-empty intersection withS , and there exists an open semicircle C ⊂ S1 such that C ∩ S is finite;

• subcritical if every semicircle in S1 has infinite intersection with S .

The justification for this trichotomy should become inspired by the next re-sults. Bollobás, Smith and Uzzell [11] proved that the critical probabilitiesof supercritical families are polynomial, while those of critical families arepolylogarithmic.

Theorem 1.2.3 (Bollobás, Smith and Uzzell). Let U be a 2-dimensional updatefamily. As L→ ∞,

1. If U is critical thenpc(Z

2L,U ) = (log L)−Θ(1). (1.16)

1.2. Monotone cellular automata: The BSU model 7

2. If U is supercritical then

pc(Z2L,U ) = L−Θ(1). (1.17)

Later, Balister, Bollobás, Przykucki and Smith [4] proved that the criticalprobabilities of subcritical models are bounded away from zero.

Theorem 1.2.4 (Balister, Bollobás, Przykucki and Smith). If U is a subcritical2-dimensional update family, then

lim infL→∞

pc(Z2L,U ) > 0. (1.18)

Indeed, the critical probability has been determined by Bollobás, Duminil-Copin, Morris and Smith [13] (up to a constant factor), for all critical two-dimensional families.

Theorem 1.2.5 (Bollobás, Duminil-Copin, Morris and Smith). Let U be a crit-ical two-dimensional bootstrap percolation update family. There exists an explicitpositive integer α = α(U ) such that, as L→ ∞, either

pc(Z2L,U ) = Θ

(1

log L

)1/α

, (1.19)

or

pc(Z2L,U ) = Θ

((log log L)2

log L

)1/α

. (1.20)

Subcritical bootstrap percolation looks more like classical Bernoulli per-colation (see Section B.4), and the first paper studying these families in suchgenerality is [4].

For a large class of subcritical models, we have obtained an exponentialdecay property in the sense of Theorem B.4.6. More precisely, consider U -bootstrap percolation with initially infected set A ⊂ Z2, where S(U ) = S1;in particular, U is subcritical.

Definition 1.2.6. We define the component of 0 ∈ Z2 as the biggest connectedcomponent containing 0 in the graph Z2[〈A〉U ] (or subgraph induced by〈A〉U , see Section B.1), and we denote it by K = K(U , A).

Theorem 1.2.7. If S(U ) = S1 and p > 0 is small enough, then

Pp(|K| ≥ n) ≤ e−Ω(n), (1.21)

for every n ∈N.

We prove this theorem in Section 2.3, and remark that we do not knowwhether this property holds for subcritical families U satisfying S(U ) 6= S1.In order to determine the critical probabilities for general three-dimensionalcritical models (via the beams process), it could be useful to extend this resultto a wider class of subcritical families.


1.3 The Glauber dynamics of the zero-temperatureIsing model

Given a graph G, and any spin dynamics on G, we let σt ∈ +1,−1V(G)

denote the state of the system at (continuous) time t ≥ 0.In this thesis, we are interested in the zero-temperature limit of Glauber

dynamics on G, where the configuration σt evolves as t increases accordingto the the majority rule. More precisely, define the energy of a vertex v at timet as

Ev(t) := − ∑u:u∼v

σt(u)σt(v),

in words, the number of neighbours of v that disagree with v minus the num-ber of neighbours that agree with v. The Glauber dynamics of the Ising modelon G are defined as follows:

• Every v ∈ V(G) has an independent exponential random clock withrate 1, and clocks at different vertices are independent of each other.

• When the clock at vertex v rings, it makes an update according to thefollowing rules

σt(v) =

−σt−(v) if Ev(−t) > 0,±1 with probability 1/2, if Ev(−t) = 0,σt−(v) if Ev(−t) < 0.

Otherwise nothing happens.

In words, each spin flips with probability 1, (respectively 0, or 1/2) whenit disagrees, (respectively, agrees, or there is a tie) with most of its neigh-bours. In physical terms, these dynamics correspond to various stochasticIsing models at 0 temperature, for the Hamiltonian with uniform ferromag-netic interaction between nearest neighbors, with no external magnetic field.

Since this system evolves as a Markov process on the state space +,−V(G),we can alternatively define it in terms of its generator L (see Section B.5.2),which acts on local functions f as

(L f )(σ) = ∑v∈V(G)

cv(σ) [ f (σv)− f (σ)] , (1.22)

where σv is the configuration obtained from σ by flipping the spin at vertexv, and cv(σ) is the flip rate of the spin at vertex v, when the system is in thestate σ, i.e., if v has n(v) neighbours, then

cv(σ) =

1 if more than n(v)/2 neighbours of v have its opposite spin,1/2 if exactly n(v)/2 neighbours of v have its opposite spin,0 if less than n(v)/2 neighbours of v have its opposite spin.

The main question we are focused on is that of fixation.

1.3. The Glauber dynamics of the zero-temperature Ising model 9

Definition 1.3.1. Say that vertex v ∈ V(G) fixates if there is a time Tv ∈ [0, ∞)such that σt(v) = σTv(v) for all t ≥ Tv, in other words, if σt(v) only flipsfinitely many times. If moreover, σTv(v) = + for each vertex v ∈ V(G), wesay that the dynamics fixate at + (the state of every spin is eventually +).

Now, let the set v ∈ V(G) : σ0(v) = + be chosen p-randomly and writeP for the joint distribution of the initial spins and the dynamics realizations.We define the critical probability for fixation for the Glauber dynamics of theIsing model to be

pIsc (G) := inf p : P(dynamics fixate at +) = 1 . (1.23)

It was proved by Arratia [3] that pIsc (Z) = 1, basically, the reason is that

for every p ∈ (0, 1), in dimension 1 every site changes state an infinite num-ber of times. A well-known (and possibly folklore) conjecture states thatpIs

c (Zd) = 1/2 for every d ≥ 2. The first progress towards this conjecture was

the following upper bound, proved by Fontes, Schonmann and Sidoravicius[30].

Theorem 1.3.2 (Fontes, Schonmann and Sidoravicius). For every d ≥ 2,

0 < pIsc (Z

d) < 1. (1.24)

Moreover, the authors of [30] showed that this fixation occurs in time witha stretched exponential tail. Morris [51] combined this theorem with sometechniques from high dimensional bootstrap percolation to prove that criticalprobability is asymptotically 1/2.

Theorem 1.3.3 (Morris). pIsc (Z

d)→ 1/2, as d→ ∞.

Glauber dynamics have also been considered in other sublattices of Zd.For instance, Damron, Kogan, Newman and Sidoravicius [23] consideredslabs of the form Sk := Z2 × 0, 1, . . . , k − 1 with k ≥ 2. They proved aclassification theorem, which surprisingly holds for all p ∈ (0, 1); namely,they studied two scenarios for boundary conditions, free and periodic (inthe latter, vertices of the form (x, y, k − 1) and (x, y, 0) are neighbours, andwhen k = 2, this enforces two edges between (x, y, 1) and (x, y, 0), so thatin the computation of energy of a vertex, that neighbour counts twice), andshowed that Sk does not fixate at + (however, each single vertex in S2 fixatesat either + or −). Therefore, in this particular setting, which interpolates di-mensions 2 and 3, the critical probability is 1, and there is no phase transitionfor fixation at +.

Theorem 1.3.4 (Damron, Kogan, Newman and Sidoravicius). With free bound-ary conditions, each vertex in Sk fixates if and only if k = 2 (but the dynamics doesnot fixate at +). With periodic boundary conditions and k ≥ 2, Sk fixates if and onlyif k ∈ 2, 3. In particular, with both boundary conditions,

pIsc (Sk) = 1. (1.25)


Another related result for the symmetric case p = 1/2 (physically in theIsing model setting, this corresponds to an initial quench from infinite tem-perature) is due to Nanda, Newman and Stein [53]. They proved that indimension 2, every vertex almost surely changes state an infinite number oftimes, however, it is still unknown if the same holds for higher dimensions.

Theorem 1.3.5 (Nanda, Newman and Stein). In dimension d = 2, for everyv ∈ Z2, σt(v) flips infinitely many times, P-a.s.

On the hexagonal lattice the situation is different, where every vertex hasdegree 3; when the clock at vertex v rings it immediately changes state (ifnecessary) to agree with the majority of its neighbors. Howard and Newman[38] showed that some spins fixate at + and others at −.

Our last example of an important graph where Glauber dynamics has alsobeen studied in detail is the d-regular tree, Td (see for example [8, 16, 39, 47]),but even here very little has been proved about pIs

c (Td). Indeed, Howard [39]showed that pIs

c (T3) > 1/2, and it was proved by Caputo and Martinelli [16]that pIs

c (Td) → 1/2 as d → ∞ (in fact their result is more general), but forevery d ≥ 4 it is unknown whether or not pIs

c (Td) = 1/2.The second main problem addressed in this thesis is the extension of

the main result of [30] to other related models on Zd. In particular, wewould like to prove a conjecture of Morris [49], which states that for every‘critical’ d-dimensional update rule (see Chapter 4), the corresponding zero-temperature Glauber dynamics on Zd exhibit a phase transition. We have sofar proved the following partial result in this direction for the so-called votermodel (see, for instance, [43]).

Given an update family U , the U -voter dynamics on the space of configu-rations +,−Zd

is defined as follows:

• Every v ∈ Zd has an independent exponential random clock (rate 1).

• When the clock at v rings, the vertex v chooses X ∈ U uniformly atrandom. If the set v + X is entirely in state ∗ (where ∗ ∈ +,−), thenthe state of v becomes ∗, otherwise nothing happens.

The critical probability pvotc (Zd,U ) for this model is the infimum over p ∈ [0, 1]

such that this system almost surely fixates at + when the initial states for thevertices are chosen independently to be + with probability p and to be −with probability 1− p. We have proved that there exists a phase transition(see Theorem 4.2.8 for the full statement).

Theorem 1.3.6. For a wide class of critical two-dimensional families U ,

0 < pvotc (Z2,U ) < 1. (1.26)

11

Chapter 2

U -bootstrap percolation

2.1 The BSU model

The model of bootstrap percolation we study in this thesis is a special caseof the following extremely general class of d-dimensional monotone cellularautomata, which were introduced by Bollobás, Smith and Uzzell [11].

Let U = X1, . . . , Xm be an arbitrary finite family of finite subsets ofZd \ 0. We call U the update family, each X ∈ U an update rule, and theprocess itself U -bootstrap percolation. Let the lattice Λ be either Zd, [L]d (thed-dimensional grid), or Zd

L (the d-dimensional discrete torus). Now given aset A ⊂ Λ of initially infected sites, set A0 = A, and define for each t ≥ 0,

At+1 = At ∪ x ∈ Λ : x + X ⊂ At for some X ∈ U. (2.1)

Thus, a site x becomes infected at time t + 1 if the translate by x of one of thesets of the update family is already entirely infected at time t, and infectedsites remain infected forever. The set of eventually infected sites is the closureof A, denoted by 〈A〉U =

⋃t≥0 At. We say that A percolates if 〈A〉U = Λ.

The key question is that of how likely it is that a random set A percolateson the lattice Λ; in particular, one would like to know how large p must bebefore percolation becomes likely. The point at which this phase transitionoccurs is measured by the critical probability.

Definition 2.1.1. Suppose that A ⊂ Λ is p-random. The critical probability is

pc(Λ,U ) := infp : Pp(〈A〉U = Λ) ≥ 1/2. (2.2)

Let Sd−1 be the unit (d − 1)-sphere and denote the discrete half spaceorthogonal to u ∈ Sd−1 as Hd

u := x ∈ Zd : 〈x, u〉 < 0. The stable setS = S(U ) is the set of all u ∈ Sd−1 such that no rule X ∈ U is contained inHd

u. Let µ denote the Lebesgue measure on Sd−1. The following classificationof families was proposed in [11] for d = 2, and extended to all dimensions in[13].

Definition 2.1.2. A family U is

• subcritical if for every hemisphereH ⊂ Sd−1 we have µ(H∩ S) > 0.

• critical if there exists a hemisphere H ⊂ Sd−1 such that µ(H ∩ S) = 0,and every open hemisphere in Sd−1 has non-empty intersection with S ;

12 Chapter 2. U -bootstrap percolation

• supercritical otherwise.

The justification for this trichotomy should become inspired by the nextresults. In dimension d = 2, Bollobás, Smith and Uzzell [11] proved that thecritical probabilities of supercritical families are polynomial, while those ofcritical families are polylogarithmic.

Theorem 2.1.3 (Bollobás, Smith and Uzzell). Let U be a 2-dimensional updatefamily.

1. If U is critical thenpc(Z

2L,U ) = (log L)−Θ(1). (2.3)

2. If U is supercritical then

pc(Z2L,U ) = L−Θ(1). (2.4)

Later, Balister, Bollobás, Przykucki and Smith [4] proved that the criticalprobabilities of subcritical models are bounded away from zero.

Theorem 2.1.4 (Balister, Bollobás, Przykucki and Smith). If U is a subcritical2-dimensional update family, then

pc(Z2,U ) > 0. (2.5)

Hartarsky [35] studied subcritical families satisfying S(U ) 6= S1, withΛ = Z2, by considering another critical probability:

pc := inf

p : ∑n

nθn(p) < ∞

,

withθn(p) := Pp

(0 /∈ 〈A ∩ [−n, n]2〉U

),

and proved that above pc there is exponential decay of the probability of aone-arm event En, while below pc the event “En = ∂[−n, n]2 → 0” haspositive probability and the expected infection time is infinite. Let us set

θn(p) = Pp(En),

and θ(p) = limn θn(p).

Theorem 2.1.5 (Hartarsky). Consider subcritical U -bootstrap percolation on Z2

with S(U ) 6= S1.

• If p > pc, then there exists c(p) > 0 such that

max(θn(p), θn(p)

)≤ exp(−c(p)n).

• There exists c > 0 such that for p < pc

θ(p) ≥ c( pc − p) > 0.

2.1. The BSU model 13

• If p < pc, then there exists c(p) > 0 such that

Pp(τ0 > n) ≥ c(p)n

,

and in particular Ep[τ0] = ∞.

where τ0 is the infection time of the origin at density q of the update familyU :

τ0 := inf t ≥ 0 : 0 ∈ At .

Now, we introduce several important definitions and lemmas from [11],which will be used later in Chapter 4.

Definition 2.1.6. Let T ⊂ S . A T -droplet is a non-empty set of the form

D =⋂

u∈T(Hu + au), (2.6)

for some collection au ∈ Z2 : u ∈ T .

We refer to a T -droplet simply as droplet, if the set T is not relevant. Let uswrite diam(D) for the diameter of a droplet D, i.e., the maximum euclideandistance between two points in D, and given any set S we denote its cardi-nality by |S|.

Next, we introduce an algorithm whose importance is to provide 2 keylemmas concerning droplets; tecniques strongly used in computing lowerbounds for the critical probability.

Definition 2.1.7 (Covering algorithm). Suppose L is large and A ⊂ [L]2. Thefirst step is to choose a sufficiently large constant κ, fix a droplet D of diame-ter roughly κ, and place a copy of D (arbitrarily) on each element of A. Now,at each step, if two droplets in the current collection are within distance κ ofone another, then remove them from the collection, and replace them by thesmallest droplet containing both. This process stops in at most |A| steps withsome finite collection of droplets.

If a droplet occurs at some point in the covering algorithm, then let ussay that it is covered by A. If κ is chosen sufficiently large, then it followsthat the final collection of droplets covers 〈A〉U (see [11] for details). Now weare ready to state the 2 key lemmas: The first one is an "Aizenman-Lebowitzlemma", which says that an covered droplet contains covered droplets of allintermediate sizes.

Lemma 2.1.8 (Aizenman-Lebowitz lemma). Let D be a covered droplet. Then forevery 1 ≤ k ≤ diam(D), there is a covered droplet D′ ⊂ D such that,

k ≤ diam(D′) ≤ 3k.

Proof. See [11].

The second one is an extremal lemma, which says that an covered dropletcontains a linear proportion of initially infected sites.


Lemma 2.1.9 (Extremal lemma). There exists a constant ε > 0 such that for everycovered droplet D,

|D ∩ A| ≥ ε · diam(D).

Proof. See [11].

We conclude this section by solving the chessboard puzzle in the Intro-duction, which is a particular case of this second lemma and we can restateas follows. Consider N 2

2 -bootstrap percolation on [L]2, where N 22 stands for

the family of all subset of size 2 of the set ±e1,±e2. We want to determine

m2(L) := minA⊂[L]2

|A| : 〈A〉N 2

2= [L]2

.

Proposition 2.1.10 (Folklore). m2(L) = L

Proof. To deduce m2(L) ≤ L, note that we just need to start with the squaresof a diagonal in black, and after L− 1 steps all squares will be black. To provethat m2(L) ≥ L, we consider the perimeter of the black part of the board, thekey observation being that it can never increase underN 2

2 -bootstrap percola-tion. If we succeed in painting the whole chessboard black, then we will havea perimeter 4L, so we need at least L initiallly infected (black) squares.

Note that if we consider chessboard puzzle on the lattice Z2L (instead of

[L]2), then the minimum number of initial black squares would be L− 1 (in-stead of L), which is still a linear proportion of initially infected squares.

2.2 Universality in two dimensions

In this section, we state the results obtained by Bollobás, Duminil-Copin,Morris and Smith [13] in their groundbreaking work on universality of two-dimensional critical cellular automata. The first step is to give a quantitativemeasure of how hard it is to grow in each direction.

Let Q1 denote the set of rational directions on the circle, (that is, the setof all u ∈ S1 such that u has rational or infinite gradient with respect to thestandard basis vectors) and for each u ∈ Q1, let l+u be the subset of the linelu := x ∈ Z2 : 〈x, u〉 = 0 consisting of the origin and the sites to the right ofthe origin as one looks in the direction of u. Similarly, let l−u := (lu r l+u )∪0consist of the origin and the sites to the left of the origin. Note that the line luis infinite for every u ∈ Q1.

Given u ∈ Q1, the difficulty of u is

α(u) :=

minα+(u), α−(u) if α+(u) < ∞ and α−(u) < ∞,

∞ otherwise,(2.7)

where α+(u) (respectively α−(u)) is defined to be the minimum (possiblyinfinite) cardinality of a set Z ⊂ Z2 such that 〈H2

u ∪ Z〉U contains infinitelymany sites of l+u (respectively l−u ).

2.2. Universality in two dimensions 15

Proposition 2.2.1. α(u) > 0 if and only if u is a stable direction.

Proof. Follows from properties of stable sets (see [11, 13]).

The authors of [13] determined the critical probability up to a constantfactor, for all two-dimensional critical bootstrap families. The form of thethreshold function depends on two properties of U : the difficulty of U , andwhether or not U is balanced.

Definition 2.2.2. Let C denote the collection of open semicircles of S1. Wedefine the difficulty of U to be

α = α(U ) := minC∈C

maxu∈C

α(u). (2.8)

It was proved in [11] (see also [13]) that if u ∈ S(U ) then α(u) < ∞ if andonly if u is an isolated point of S(U ). It follows that α = 0 for every super-critical update family, and that α is finite for every critical update family.

Proposition 2.2.3. An update family U is critical if and only if 1 ≤ α < ∞.

Proof. Follows from properties of stable sets (see [11, 13]).

Definition 2.2.4. We say that a critical update family U is balanced if thereexists a closed semicircle C such that α(u) ≤ α for all u ∈ C. It is said to beunbalanced otherwise.

Finally, we are ready to state the universality result.

Theorem 2.2.5 (Bollobás, Duminil-Copin, Morris and Smith). Let U be a criti-cal two-dimensional bootstrap percolation update family.

1. If U is balanced, then

pc(Z2L,U ) = Θ

(1

log L

)1/α

. (2.9)

2. If U is unbalanced, then

pc(Z2L,U ) = Θ

((log log L)2

log L

)1/α

. (2.10)

Various special cases of Theorem 2.2.5 had already been proved in theliterature; for example, the 2-neighbour model (Theorem 1.1.2 with d = 2),the anisotropic model (Theorem 1.1.7), and Duarte model (determined byMountford [52]). All of these models have difficulty α = 1, the first one isbalanced, while the anisotropic and Duarte models are unbalanced.

Remark 2.2.6. When Λ is finite (either [L]d or ZdL), some authors consider

instead the critical length for percolation for p small (compare with (2.2))

Lc(U , p) := minL ∈N : Pp(〈A〉U = Λ) ≥ 1/2, (2.11)


which is basically the inverse function of the critical probability, in the sensethat determining the former is equivalent to determining the latter. Fromthis point of view, Theorem 2.2.5 can be rewritten as follows: If U is a criticaltwo-dimensional bootstrap percolation update family, then, as p→ 0

log Lc(U , p) =

Θ(p−α) if U is balanced,Θ(

p−α(log p)2) if U is unbalanced.(2.12)

2.3 Exponential decay for subcritical families

In this section, we develop new machinery from bootstrap percolation; con-sider U -bootstrap percolation in Z2 with U subcritical. The first paper study-ing these families in such generality is [4], it turns out that these families havebehavior quite similar to models in classical site percolation, for instance, in[4] it is proved that pc(Z2,U ) > 0, for every subcritical family U (see Theo-rem 2.1.4 above).

We will only deal with subcritical families U satisfying pc(Z2,U ) = 1; theauthors of [4] proved that this condition is equivalent to S(U ) = S1. Our aimis to show that for such families, if we choose the initial infected set A to beε-random with ε small enough, then the size of the cluster in 〈A〉U containingthe origin decays exponentially fast; here we summarize the core idea. First,we need to guarantee the existence of inwards stable droplets, which are, basi-cally, discrete polygons that can not be infected from outside, it is possible toshow the existence of such droplets by using the condition S(U ) = S1. Afterthat, we introduce the dilation radius, which is a constant depending on U ,used to prove an extremal lemma that gives us a quantitative measure of theratio |〈A〉U |/|A|. Finally, we combine ideas used by Bollobás and Riordan inclassical percolation models to conclude.

2.3.1 Inwards stable droplets and the dilation radius

Imagine for a moment that we have a convex set D in the plane and supposeit is inscribed in the ball Bρ of radius ρ, then we know that any other ball withradius ρ and center outside B3ρ is disjoint from D. This simple remark is atool to prove exponential decay, if D is inwards stable (see Lemma 2.3.8).

Given x, y ∈ R2 we denote the usual euclidean distance between x and yby ‖x− y‖, and Bρ(y) is the ball of radius ρ > 0 centered at y:

Bρ(y) := x ∈ R2 : ‖x− y‖ ≤ ρ. (2.13)

For simplicity, we denote Bρ := Bρ(0).

Definition 2.3.1. Let us define a rounded droplet D as the intersection of Z2

with a bounded convex set in the plane. We say that D ⊂ Z2 is inwards stableif

〈Z2 \ D〉U = Z2 \ D. (2.14)

2.3. Exponential decay for subcritical families 17

We need to guarantee the existence of inwards stable (rounded) droplets,note that they are finite; this is the only point where we use S(U ) = S1 toprove the exponential decay property.

Lemma 2.3.2 (Existence). If S(U ) = S1 then, there exist an inwards stable dropletD such that 0 ∈ D.

The origin 0 ∈ Z2 has no special role here, it is just a reference point tolocate the droplet D. Any translate of D is inwards stable too.There are several choices for the shape of inwards stable droplets, the follow-ing proof is the same in spirit as that in [11] which shows that, inside dropletswith sides perpendicular to quasi-stable directions, the infection grows all wayto the corners (see Lemma 5.3 in [11]).

Proof. Given a point t 6= 0 in the plane, we denote t the (π/2)-rotation of tclockwise. Consider the following collection of unit vectors:

Q :=⋃

X∈Ut/‖t‖ : t ∈ X ⊂ S1.

Since S(U ) = S1, every rule X ∈ U contains the origin in its convex hull,which we denote as Hull(X), in particular, 0 ∈ Hull(Q). Moreover, a con-vex region enclosed by sides perpendicular to the vectors in Q, so that eachvector inQ is orthogonal to exactly one (bounded or unbounded) side of theregion, is a (bounded) polygon if and only if 0 is in the interior of Hull(Q),in this case we define

D = Q.

In the other case, 0 is in the boundary of Hull(Q), so the region is unboundedbecause there exists a direction u ∈ S1 such thatQ∩H2

u = ∅, in this case weset

D = Q∪ u,−u.Let us denote by Dρ the polygon with sides perpendicular to the vectors inDwhich circumscribes the ball Bρ (Bρ ⊂ Dρ). We claim that if ρ is large enough,in particular every side of Dρ is bigger than∇(U ), then Dρ is inwards stable,where

∇(U ) := maxX∈U

maxx,y∈X

‖x− y‖.

In fact, let us fix a counterclockwise enumeration D = u1, . . . , u|D| of D ⊂S1 and set u|D|+1 := u1. There exist positive scalars γ1, . . . , γ|D| such that, forevery k = 1, . . . , |D|, the k-th side of Dρ is tangent to the boundary of Bρ atthe point tk := γkuk.Call zk the corner of Dρ which is between tk and tk+1, and set

Qk = Hull(tk, zk, tk+1, 0).

Each Qk is a quadrilateral and Dρ = Q1 ∪ · · · ∪Q|D|.To finish the proof, it is enough to show that for every v ∈ Dρ ∩Z2 and

for each rule X ∈ U we have (v + X) ∩ Dρ 6= ∅. Assume by contradiction


FIGURE 2.1: A family U consisting of two rules, ±e1, (−1, 1)marked with ∗, and ±2e1 marked with ×. Then Q =±e2, (1/

√2, 1/√

2) and D = ±e2, (1/√

2, 1/√

2),±e1.

that some v ∈ Dρ ∩Z2 and X′ ∈ U satisfy (v + X′) ∩ Dρ = ∅. Then v ∈ Qkfor some k, and it is enough to consider the worst case; v = zk.

By construction of D, we have that

zk + X′ ⊂ zk + (H2−uk∪ Sk),

or equivalently X′ ⊂H2−uk∪ Sk, where Sk denotes the discrete segment from

0 to∇(U )sk, and sk ∈ S1 is defined by sk = uk. But this implies 0 /∈ Hull(X′).This contradiction shows that if the droplet D := Dρ ∩Z2 is initially healthyand ρ is large enough, then D will be healthy forever.

Remark 2.3.3. As we said, this is not the only way to construct the droplet,for instance, there is an alternative proof which is included in [4]: Accordingto that proof, D could not be a polygon anymore, and this fact justifies therounded term in the definition.

Alternative proof of Lemma 2.3.2. Suppose that Bρ is initially healthy. If ρ islarge enough then every rule X can only infect sites in disjoint circular seg-ments ’cut off’ from Bρ using chords of length at most ∇(U ), and parallel tothe sides of Hull(X), and these segments are all either disjoint or containedin each other for different rules, since ρ is large. No additional infection takesplace in Bρ, therefore D = Bρ \ 〈Z2 \ Bρ〉U is inwards stable.

Now, given ρ > 0 we denote the discrete ball as B′ρ := Z2 ∩ Bρ. An imme-diate consequence of the above lemma is the fact that every vertex which iseventually infected should be within some constant distance from an initiallyinfected vertex.

Corollary 2.3.4. If S(U ) = S1, there exists ρ > 0 such that, for every y ∈ 〈A〉U ,

A ∩ B′ρ(y) 6= ∅. (2.15)

Proof. Let D be an inwards stable droplet with 0 ∈ D and, ρ > 0 such thatD ⊂ Bρ. Given y ∈ 〈A〉U , it follows that the translation D(y) := y + D is also


inwards stable and y ∈ 〈A〉U ∩ D(y). Therefore

A ∩ Bρ(y) ⊃ A ∩ D(y) 6= ∅.

Definition 2.3.5 (Dilation Radius). We define the dilation radius β := β(U ) tobe the smallest radius ρ ≥ 1 satisfying the conclusion in Corollary 2.3.4.

Note that|B′3β| ≤ 30β2. (2.16)

2.3.2 Exponential decay

Consider U -bootstrap percolation with initially infected set A ⊂ Z2, whereS(U ) = S1. Here is the main object in this section:

Definition 2.3.6. We define the component (or cluster) of 0 ∈ Z2 as the biggestconnected component containing 0 in the graph induced by 〈A〉U , and wedenote it by K = K(U , A). If 0 /∈ 〈A〉U , then we set K = ∅.

Before stating the main lemma, we give a name to an specific collection offinite subtrees of Z2.

Definition 2.3.7. For n ≥ 0 we let T0,n to be the collection of all trees T ⊂ Z2

containing the origin 0 ∈ Z2 and other n vertices (|T| = n + 1). We alsodefine the collection of all trees containing 0 and having at most n vertices(|T| ≤ n) as

T≤n :=n⋃

k=1

T0,k−1. (2.17)

Let β be the dilation radius.

Lemma 2.3.8 (Extremal lemma for K). If |K| ≥ n then, there exists a tree T ∈T≤n such that

|A ∩ T| ≥ (30β2)−1n. (2.18)

Proof. In fact, let us suppose that |K| ≥ 30β2n, and recursively find n distinctvertices x′1, . . . , x′n ∈ A ∩ T, for some tree T ∈ T≤30β2n.

By definition of β, for x1 = 0 ∈ 〈A〉U there exists x′1 ∈ A ∩ B′β(x1), thenset K1 = B′3β(x1), and since |K1| ≤ 30β2 we can find a vertex x2 ∈ K \ K1,which is at distance 1 from K1; now we apply the lemma to x2 ∈ 〈A〉U andfind a new vertex x′2 ∈ A ∩ B′β(x2). Proceed in this way, assume we havefound vertex x′i ∈ A ∩ B′β(xi−1) then set

Ki = B′3β(xi) ∪ Ki−1.

Since |Ki| ≤ 30β2i, for i = 1, . . . , n− 1 we have

|K \ Ki| ≥ 30β2n− 30β2i ≥ 1,


so we can find a vertex xi+1 ∈ K \ Ki, which is at distance 1 from Ki. Observethat at step n− 1 we still have |K \ Kn−1| ≥ 30β2 ≥ 1, so for xn ∈ K \ Kn−1we can apply the lemma one more time to get our last vertex x′n ∈ A.

For i = 1, . . . , n, the vertices x′i are all distinct because all balls B′β(xi) arepairwise disjoint by construction.

Finally, consider a spanning tree T ⊂ Kn of Kn, and note that xi, x′i ∈ Tfor all i = 1, . . . , n. In particular, |A ∩ T| ≥ n, and the fact that T ∈ T≤30β2nfollows from 0 = x1 ∈ T and |T| ≤ |Kn| ≤ 30β2n.

The same proof allows us to prove another similar extremal lemma:

Lemma 2.3.9. There exists a constant λ ∈ (0, 30β2] such that, if 〈A〉U is connectedthen,

|〈A〉U | ≤ λ|A|. (2.19)

Proof. If A is infinite we have nothing to show. Assume A is finite, then itis contained in a big rectangle R ⊂ Z2, since ±e1,±e2 ∈ S , so 〈A〉U ⊂ Ris also finite. Since 〈A〉U is connected, the above proof shows that |〈A〉U | >30β2n implies |A| > n. In other words, |A| = n implies |〈A〉U | ≤ 30β2n =30β2|A|.

Our last ingredient to prove the exponential decay theorem is an upperbound for the number of subtrees of Z2 containing the origin and having nvertices. The following proposition is a particular case of a beautiful problemin the book The art of mathematics: Coffee time in Memphis (see Problem 45 in[9]). For the sake of completeness we write down their proof.

Proposition 2.3.10. For every n ≥ 1 we have |T0,n| ≤ (3e)n. As a consequence,|T≤n| ≤ (3e)n.

Proof. Consider the rooted Cayley tree T ∗4 of order 4, this is, the infinite reg-ular tree where every vertex has exactly 4 neighbours and we fix an specialvertex 0∗ ∈ T ∗4 which we call the root. By an usual argument we can see thatevery finite tree T ⊂ Z2 containing 0 can be embedded as a subtree T∗ ⊂ T ∗4containing 0∗, this embbeding sending 0 to 0∗, |T∗| = |T|, moreover, the mapT 7→ T∗ is injective. Therefore it is enough to show that the number Xn oftrees T∗ ⊂ T ∗4 such that 0∗ ∈ T∗ and |T∗| = n + 1 is at most (3e)n.

In fact, let Yn be the number of subtrees of T ∗4 with n + 1 vertices, oneof which is the root, labelled 0, 1, . . . , n, with the root labelled with 0, thusYn = n!Xn. Let Yn be the set of trees with vertex set 0, 1, . . . , n, by Cay-ley’s formula Yn has (n + 1)n−1 elements. Given T ∈ Yn, let emb(T) be thenumber of ways of embedding T into T∗, with 0 put into the root. Then

Yn = ∑T∈Yn

emb(T).

To bound emb(T) from above, let us embed T vertex by vertex. We start withthe vertex labelled 0: that goes into the root. Next, we embed a neighbour of0: we have 4 choices. From then on, we always embed a vertex with at least


one neighbour already embedded: at each stage we have at most 3 choices.Therefore emb(T) ≤ 4 · 3n−1, and

Xn =Yn

n!≤ 1

n!(n + 1)n−14 · 3n−1 ≤ (3e)n.

The second part is immediate.

Finally, we consider subcritical U -bootstrap percolation where A ⊂ Z2 isε-random, S(U ) = S1, and let β ≥ 1 be the dilation radius. This is the maintheorem of this section, whose proof is inspired by lines through the bookPercolation of Bollobás and Riordan (see pp. 70 in [10]).

Theorem 2.3.11 (Exponential decay for the cluster size). If 0 < ε < e−150β2

and C = C(ε) := − 160β2 log(ε), then

Pε(|K| ≥ n) ≤ ε1

60β2 n= e−Cn, (2.20)

for every n ∈N.

Proof. By Corollary 2.3.8 and Proposition 2.3.10, with δ = (30β2)−1, we ob-tain

Pε(|K| ≥ n) ≤ Pε

⋃T∈T≤n

|A ∩ T| ≥ δn

≤ ∑

T∈T≤n

Pε(|A ∩ T| ≥ δn)

≤ ∑T∈T≤n

(nδn

)εδn

≤ ∑T∈T≤n

(eδ−1ε)δn

≤ ([3e][eδ−1ε]δ)n

≤ e−Cn,

and we are done.

This theorem can be generalized to all dimensions d ≥ 3 and all familiesU such that S(U ) = Sd−1. However, we do not know if this property holdsfor subcritical families U satisfying S(U ) 6= Sd−1. In order to determine thecritical lengths for general critical models, it could be useful to extend thisresult to a wider class of subcritical families.

Problem 2.3.12. Characterize the subcritical d-dimensional update families U suchthat K has the exponential decay property.

23

Chapter 3

Anisotropic 3-dimensional models

3.1 Introduction

In this chapter we consider a three-dimensional analogue of the anisotropicbootstrap process studied by Duminil-Copin, van Enter and Hulshof [24, 25,29]. We will determine the critical length (equivalently, the critical probabil-ity, see Remark 2.2.6) for a subfamily of anisotropic models.

Fix dimension d = 2 for a moment. Given s ≤ t and r positive integers,let us consider the family N s,t

r consisting of all subsets of size r of the set

Ns,t := s′e1 : ±s′ ∈ [s] ∪ t′e2 : ±t′ ∈ [t]. (3.1)

The model of N s,tr -bootstrap percolation on [L]2 is called isotropic when s =

t and anisotropic when s < t, for example, the 2-neighbour model N 1,12 is

an isotropic model. Hulshof and van Enter [29] determined the thresholdfor the first interesting anisotropic model given by the family N 1,2

3 , and thecorresponding sharp threshold was determined by Duminil-Copin and vanEnter [25]. The threshold was also determined in the general case r = s+ t byvan Enter and Fey [28] and the proof can be extended to all t + 1 ≤ r ≤ s + t,we state the general result here, since we will use it later.

Theorem 3.1.1 (Anisotropic 2-d). If t + 1 ≤ r ≤ s + t and 1 ≤ s ≤ t, there existconstants Γ > γ > 0 depending on t such that, as p→ 0,

Pp(〈A〉N s,tr

= [L]2)→

1 if L > exp(

Γ p−(r−t)(log p)2·1t>s)

,

0 if L < exp(

γ p−(r−t)(log p)2·1t>s)

,(3.2)

and hence,

log Lc(N s,t

r , p)=

Θ(

p−(r−t))

if t = s,

Θ(

p−(r−t)(log p)2)

if t > s.

This result is also a particular case of Theorem 2.2.5, since N s,tr has diffi-

culty α = r− t, and it is balanced if and only if t = s (see (2.12)).Very little is known about bootstrap percolation in higher dimensions,

and our aim is to contribute in this direction, by determining the threshold

24 Chapter 3. Anisotropic 3-dimensional models

for a class of anisotropic models, which are three-dimensional versions ofN s,t

r , in the sense that all update rules are contained in the coordinate axis.

3.1.1 Anisotropic bootstrap percolation on [L]3

Here, we consider bootstrap percolation on Λ = [L]3. Let us set the specificupdate families we are interested in.

Definition 3.1.2. Given a ≤ b ≤ c and r positive integers, define the familyN a,b,c

r as the collection of all subsets of size r of the set (see Figure 3.1)

Na,b,c := a′e1 : ±a′ ∈ [a] ∪ b′e2 : ±b′ ∈ [b] ∪ c′e3 : ±c′ ∈ [c]. (3.3)

FIGURE 3.1: The set Na,b,c with a = 1, b = 2 and c = 4. The e1-axis is towards the reader, the e2-axis is vertical, and the e3-axis

is horizontal.

There is an advantage in considering this model of percolation, whichwe will be able to exploit, namely, the presence of symmetry, this propertyimplies that growth is equally difficult in every pair of opposite directions.These families were studied by van Enter and Fey [28] for r = a + b + c; theydetermined the following bounds on the critical length for N a,b,c

a+b+c.

Theorem 3.1.3 (van Enter and Fey). As p→ 0,

log log Lc

(N a,b,c

a+b+c, p)=

Θ (p−a) if b = a,Θ(

p−a(log p)2) if b > a.(3.4)

We would like to generalize the result of van Enter and Fey to all a, b, cand r; note that, by (3.4) the critical length is doubly exponential in p whenr = a + b + c. It is possible to show that the critical length is polynomial in pif r ≤ c (see Proposition 3.1.18 below). On the other hand, the critical lengthis singly exponential in the case r ∈ c+ 1, c+ 2; indeed, we determined thecritical length up to a constant factor in the exponent, for all triples (a, b, c),except for c = a + b− 1 when r = c + 2.

The following is our main result.

3.1. Introduction 25

Theorem 3.1.4. Fix s := r− c ∈ 1, 2. As p→ 0,

log Lc

(N a,b,c

r , p)=

Θ(

p−s/2) if c = b = a,

Θ(

p−s/2(log 1p )

1/2)

if c = b > a,

Θ(

p−s/2(log 1p )

3/2)

if c ∈ b + 1, . . . , a + b− s,

O(

p−1(log 1p )

2)

if c = a + b− 1 and r = c + 2,

Θ (p−s) if c = a + b,

Θ(

p−s(log 1p )

2)

if c > a + b.(3.5)

We are able to deduce the lower bound log Lc

(N a,b,a+b−1

a+b+1 , p)≥ Ω(p−1),

corresponding to case IV. However, we think that the threshold should be

log Lc

(N a,b,a+b−1

a+b+1 , p)= Θ

(p−1(log 1

p )2)

.

A problem which remains open is the determination of the threshold forother values of r. We believe that the techniques used to prove Theorem 3.1.4can be adapted to cover all c + 2 < r ≤ b + c (though significant technicalobstacles remain), and the critical length should be singly exponential. How-ever, to deal with the cases b + c < r < a + b + c, the techniques requiredare likely to be more similar to those of [17] and [28], and the critical lengthshould be doubly exponential.

We have made some progress in this direction, by determining an upperbound in the general case r ≤ a + c, which, we believe gives the right orderof the threshold length.

Theorem 3.1.5. Fix s := r− c ∈ 3, . . . , a. As p→ 0,

log Lc

(N a,b,c

r , p)=

O (p−αs) if c = b = a,

O(

p−αs(log 1p )

(t+1)/(t+2))

if c = b > a,

O(

p−αs(log 1p )

(t+3)/(t+2))

if c ∈ b + 1, . . . , a + b− s,

O(

p−ms(log 1p )

2)

if c = a + b− s + m

O(

p−s(log 1p )

2)

if c ≥ a + b.(3.6)

where, the sequence αss≥3 is given by

αs =t + 1t + 2

(s− t/2) , (3.7)

t = ts :=

⌈√9 + 8s− 5

2

⌉,

and for m ∈ [s− 1],ms := maxαs, m. (3.8)


Here there are some numerical values of ts and αs, for s = 3, 4, . . . , 14.

s 3 4 5 6 7 8 9 10 11 12 13 14ts 1 1 1 2 2 2 2 3 3 3 3 3αs 5/3 7/3 3 15/4 18/4 21/4 6 34/5 38/5 42/5 46/5 10

TABLE 3.1: Some values of ts and αs.

Indeed, we have upper bounds for all values of r in the general 2-criticalcase.

Theorem 3.1.6. Set r ∈ a + c + 1, b + c. As p→ 0,

log Lc

(N a,b,c

r , p)=

O(

p−(s−a+αa))

if c = b,

O(

p−s log 1p

)if c > b,

(3.9)

where α1 = 1/2, α2 = 1, and αa is the same as that in Theorem 3.1.5 for a ≥ 3.

3.1.2 Critical families

Now, we discuss 3-dimensional critical families. Following Definition 2.1.2,we claim that the family N a,b,c

r is critical if and only if

r ∈ c + 1, . . . , a + b + c.

In fact, if r > a + b + c then every u ∈ S2 is in the stable set, since there isno rule of N a,b,c

r contained in H3u. Thus S(N a,b,c

r ) = S2, and the model issubcritical. For each i = 1, 2, 3, let us denote by

S1i := (u1, u2, u3) ∈ S2 : ui = 0

the unit circle contained in S2 that is orthogonal to the vector ei.When r ≤ c, for every u /∈ S1

3 either r′e3 : r′ ∈ [r] or r′e3 : −r′ ∈ [r] iscontained in H3

u, so u is not in the stable set. Therefore S(N a,b,cr ) ⊂ S1

3, so thehemisphereH3 pointing in the e3-direction satisfiesH3 ∩ S = ∅ andN a,b,c

r issupercritical.

Finally, when r ∈ c + 1, . . . , a + b + c, every canonical unit vector is inthe stable set since r > c ≥ b ≥ a, so every open hemisphere in S2 inter-sects S(N a,b,c

r ). Moreover, for each u /∈ S11 ∪ S1

2 ∪ S13, H3

u intersects all threecoordinate axis (see Figure 3.2), hence there is a rule contained in H3

u sincer ≤ a + b + c. It follows that S(N a,b,c

r ) ⊂ S11 ∪ S1

2 ∪ S13 and every hemisphere

H ⊂ S2 satisfies µ(H∩ S) = 0, so N a,b,cr is critical, as claimed.

Now, by the universality result in [13], Theorem 2.2.5, we know that givenany critical two-dimensional family U , there exists a positive integer α de-pending on U such that, as p→ 0, either

log Lc(U , p) = Θ(

p−α)

, (3.10)


FIGURE 3.2: S11 is the big circle, S1

2 and S13 are drawn with

dashed ellipses. The vector u is outside S11 ∪ S1

2 ∪ S13 and H3

ucontains all positive multiples of e1,−e2 and e3.

orlog Lc(U , p) = Θ

(p−α(log 1

p )2)

. (3.11)

Trying to prove a universality result of this kind for three (or higher) di-mensions is a challenging open problem. However, there is a weaker con-jecture concerning all critical families and all d ≥ 3, stated by the authors in[13]; here we state the only case we are considering, d = 3:

Conjecture 3.1.7. Let U be a critical three-dimensional family. As p→ 0, either

log Lc(U , p) = p−Θ(1), (3.12)

orlog log Lc(U , p) = p−Θ(1). (3.13)

Let us say that U is 2-critical if it satisfies condition (3.12), and is 3-criticalif it satisfies condition (3.13). It turns out that the family N a,b,c

r is 2-criticalfor all r ∈ c + 1, . . . , c + b, in fact, since Lc(N a,b,c

r , p) is increasing in r, byTheorem 3.1.4 it follows that

log Lc

(N a,b,c

r , p)≥ log Lc

(N a,b,c

c+1 , p)≥ Ω

(p−1/2

),

and by applying the general upper bound given by Proposition 3.1.8 below,we also have

log Lc

(N a,b,c

r , p)≤ log Lc

(N a,b,c

c+b , p)≤ O

(p−b(log 1

p )2)

.

On the other hand, van Enter-Fey family N a,b,ca+b+c is 3-critical by Theorem

3.1.3; it is natural to conjecture that this is the case for all r taking values inc + b + 1, . . . , c + b + a.


A general upper bound

In this short section, we show that the critical length is at most singly expo-nential in the case r ≤ c + b, as we claimed above. Consider N a,b,c

r -bootstrappercolation.

Proposition 3.1.8. Fix r ∈ c + 1, . . . , c + b. There exists Γ = Γ(c) > 0 suchthat, if L > Lc(N b,c

r , p)Γ, then Pp

(〈A〉N a,b,c

r= [L]3

)→ 1, as p→ 0. Thus,

log Lc

(N a,b,c

r , p)= O

(log Lc(N b,c

r , p))= O

(p−(r−c)(log p)2

).

In the case that r ∈ c + 1, . . . , c + b, we can use dimensional reductionby means of a renormalization argument, to show that filling the whole [L]3

is at most as hard as filling L disjoint copies of [L]2 orthogonal to the e1-direction.

To do so in this regime, we will compare the family N a,b,cr with the two-

dimensional familyN b,cr consisting of all subsets of size r of the set Nb,c given

by (3.1). It turns out thatN b,cr is critical if and only if r belongs to this regime,

and in this case S(N b,cr ) = ±e1,±e2. The clue step is to refine the 1-

statement in Theorem 3.1.1, by using standard renormalization techniques.

Lemma 3.1.9 (Renormalization). UnderN b,cr -bootstrap percolation with r ∈ c+

1, . . . , c + b, there exists a constant Γ′ > 0 depending on c such that,

Pp

(〈A〉N b,c

r= [L]2

)≥ 1− exp

(−L1/2

), (3.14)

for all p small enough and L > Lc(N b,cr , p)Γ′ .

Proof. We are going to show that Pp(〈A〉N b,cr

= [L2]2) ≥ 1 − exp(−L). In

fact, tile [L2]2 with L2 disjoint copies of [L]2 in the obvious way, and considereach such copy as a single vertex in the new rescaled grid, so that each copyin [L2]2 corresponds to a vertex in [L]2. Define q := Pp(〈A〉N b,c

r= [L]2)

and δ := 1− q. Now, we couple the process with classical site percolation,declaring each copy of [L]2 in the rescaled grid to be open with probability q(or closed with probability δ) if it is internally filled.

By Theorem 3.1.1, there exists a constant Γ > 0 depending on c such that,if p is small enough and L > Lc(N b,c

r , p)Γ, then δ can be made arbitrarilysmall (in particular, less than the critical probability for site percolation), soit follows from standard results in percolation (see Theorem B.4.6), that if δis sufficiently small, then the probability of existing a component (of closedvertices) of diameter at least L/2 is smaller than exp(−L).

Finally, note that the latter event contains the event 〈A〉N b,cr6= [L2]2,

since every healthy component of diameter at most L/2 can be determinis-ticaly infected by using the rules of N b,c

r ; in fact, the only way for a healthycomponent to survive is by having a crossing either left to right or top tobottom in the rescaled grid. Therefore, by taking Γ′ = 2Γ we are done.


Now, we prove the general upper bound.

Proof of Proposition 3.1.8. Decompose [L]3 as L consecutive copies of [L]2 allof them orthogonal to the e1-direction, and call those copies Ri := i × [L]2

(see Figure 3.3).

FIGURE 3.3: [L]3 = R1 ∪ R2 ∪ · · · ∪ RL. We identify verticeswith cubes of sidelength 1.

Now, we couple the original process with the reduced two-dimensionalprocesses; if for each i ∈ 1, . . . , L, 〈A ∩ Ri〉N b,c

r= Ri in the N b,c

r -bootstrap

process, then [L]3 is internally filled. Therefore, by Lemma 3.1.9 we have

Pp

(〈A〉N a,b,c

r= [L]3

)≥ Pp

(L⋂

i=1

〈A ∩ Ri〉N b,cr

= Ri)

=L

∏i=1

Pp

(〈A ∩ Ri〉N b,c

r= Ri

)≥[1− exp

(−L1/2

)]L−−→p→0

1,

if L > exp(

Γ′ p−(r−c)(log p)2·1c>b)

.

Remark 3.1.10. The alert reader may have noticed that this proposition, inparticular, already gives us the upper bound in Case VI of our main Theorem3.1.4: c > a + b. It also covers the upper bound in Case V of Theorem 3.1.5:c ≥ a + b.

To deal with the upper bounds in the remaining cases, we will first studythe induced 2-dimensional processes; this is the content of next section.


3.1.3 Induced processes

To prove upper bounds, it is enough to give one possible way of growingfrom A step by step until we fill the whole of [L]3.

Definition 3.1.11. A rectangle is a set of the form R = [lx]× [ly]× [lz] ⊂ Z3.We say that a rectangle R is internally filled if R ⊂ 〈A ∩ R〉N a,b,c

r, and denote

this event by I•(R).

Throughout this thesis, we will use the fact that

e−q ≥ 1− q ≥ e−2q,

for all q ∈ (0, 23). Now, we deal with the induced 2-dimensional processes on

the faces, which will be one of the tools needed to deduce the upper boundsin Theorems 3.1.4 and 3.1.5. As we will see, the threshold function is some-how determined by the ‘easiest’ growth direction (as it is in most of cases inthe literature), which is the e3-direction.

The following is one of the main lemmas, it gives us the best lower boundfor the probability of growing one step along the e3-direction. This first stepis essential to determine the right exponents in Theorem 3.1.4.

Lemma 3.1.12 (Easiest growth). Under N a,b,cc+s -bootstrap percolation with s ∈

1, 2 and a ≥ s, if lx, ly, lz ≥ c, then

Pp

(I•([lx]× [ly]× [lz + 1]

) ∣∣∣I• ([lx]× [ly]× [lz]) )≥ 1− e−pslx ly/s.

If R1 := [lx]× [ly]× [lz] is completely infected, we just need to infect theright-most face [lx] × [ly] × lz + 1, and since we have c already infectedvertices in R1, then it is enough to find s infected vertices in some copy of theset Na,b × lz + 1, where Na,b is the 2-dimensional neighbourhood given by(3.1). In particular, it is enough to find s infected vertices in diagonal, as isshown in Figure 3.4.

FIGURE 3.4: When s = 1, it is enough one single vertex. Whens = 2, it is enough two vertices in diagonal.


Definition 3.1.13. We define an s-pattern as

1. a single vertex if s = 1;

2. two vertices in diagonal (i.e. a copy of (1, 1, 1), (2, 2, 1)) if s = 2.

In general, underN a,b,cr -bootstrap percolation, this induced 2-dimensional

process along the e3-direction is aN a,br−c-bootstrap process, and the probability

of the event I•([lx]× [ly]) under N a,bs -bootstrap percolation is at least proba-

bility of I•([lx]× [ly]) under N s,ss -bootstrap percolation for s ≤ a. Therefore,

Lemma 3.1.12 is a consequence of the following result corresponding to theinduced (along e3) supercritical two-dimensional setting.

Lemma 3.1.14 (Supercritical induced process). Under N s,ss -bootstrap percola-

tion with s ∈ 1, 2, if lx, ly ≥ s, then

Pp(

I•([lx]× [ly]

))≥ 1− e−pslx ly/s.

Proof. To get the rectangle R = [lx] × [ly] internally filled, it is enough tohave one s-pattern S in A ∩ R (right-most face in Figure 3.4). There are lxly/sdisjoint copies of such Ss (call them S1, · · · , Slx ly/s), so by independence,

Pp(

I•([lx]× [ly]

))≥ Pp

(⋃iSi ⊂ A ∩ R

)≥ 1−∏

i

(1−Pp (Si ⊂ A)

)≥ 1−

(1− p|S|

)lx ly/s

≥ 1− e−pslx ly/s,

since |S| = s.

Lemma 3.1.12 tells us the cost of growing along the e3-direction, and weare also interested in computing the cost of growing along the e1 and e2(harder) directions. To do so, we first consider the regime

r ≤ a + b,

this implies that given any rectangle R, all three induced 2-dimensional pro-cesses in the faces of R, namely, N a,b

r−c, Na,cr−b and N b,c

r−a, are supercritical. Thiscase is the easiest one because it is enough to find a copy of some ‘pattern’with constant size in all faces, no matter the size of R.

Lemma 3.1.15 (Regime r ≤ a + b). Fix h, w ≥ c. If p is small enough, then

Pp

(I•([h + 1]2 × [w])|I•([h]2 × [w])

)≥(

1− e−(c2)−1 p(

c2)wh

)2

.


Proof. Let us assume that [h]2 × [w] is internally filled. Once R1 = [h]2 × [w]is completely full, to get R2 = [h + 1]2 × [w] internally filled it is enough tohave one copy of ∆a in A ∩ (h + 1 × [h]× [w]) (front face), and one copyof ∆b in A ∩ ([h] × h + 1 × [w]) (top face), where ∆t is the discrete right-angled triangle whose legs are [r− t]× 1 and 1 × [r− t], for t = a, b (seeFigure 3.5).

FIGURE 3.5: An s-pattern on the right-most side (s = 1), onecopy of ∆a on the front side, and one copy of ∆b on the top side.

Since |∆t| = (r− t)(r− t+ 1)/2 and a ≥ 2, then |∆b| ≤ |∆a| ≤ (c2). Hence,

by independence between the front and top faces,

Pp(I•(R2)|I•(R1)) ≥(

1− e−|∆a|−1 p|∆a |wh) (

1− e−|∆b|−1 p|∆b |wh)

≥(

1− e−(c2)−1 p(

c2)wh

)2

.

Next, we cover all cases a + b < r ≤ a + c, where, the induced N a,br−c

process is still supercritical, but the induced processes N a,cr−b and N b,c

r−a arecritical.

Lemma 3.1.16 (Regime a+ b < r ≤ a+ c). ConsiderN a,b,cr -bootstrap percolation

with a + b < r ≤ a + c, and fix integers h, w ≥ c. If p is small enough, then

Pp

(I•([h + 1]2 × [w])|I•([h]2 × [w])

)≥(

1− e−1

r−a pr−aw)r (

1− e−1α pαw

)2h,

whereα := r− (a + b).


Proof. Let us assume that [h]2 × [w] is internally filled. Once [h]2 × [w] iscompletely full, to fill [h + 1]2× [w] it is enough to have the occurrence of theevents Fh and F′h defined as follows: Fh as (growing along the e1-direction)there exist r − a adjacent vertices in A ∩ (h + 1 × 1 × [w]), r − (a + 1)adjacent vertices in A ∩ (h + 1 × 2 × [w]),. . . , r − (a + b − 1) adjacentvertices in A∩ (h + 1× b× [w]), and for each i = b + 1, . . . , h, there existα = r− (a + b) adjacent vertices in A∩ (h + 1× i× [w]) (see Figure 3.6).

FIGURE 3.6: A 2-pattern on the right-most side (s = 2). Theother sides require α vertices in each of the lines along the e3-

direction (α = 2).

Observe that to guarantee the existence of k ∈ α, . . . , r − a adjacentvertices in A ∩ (h + 1 × r− a + 1− k × [w]) it is enough to have

h + 1 × r− a + 1− k × [k(t− 1) + 1, kt] ⊂ A,

for some t = 1, . . . , w/k, therefore

Pp(Fh) ≥r−a

∏k=α+1

(1−

(1− pk

)w/k) h

∏i=b+1

(1−

(1− pα

)w/α)

≥(

1− e−1

r−a pr−aw)b (

1− e−1α pαw

)h−b.

F′h is the analogue of the event Fh when growing along the e2-direction: Thereexist r− b adjacent vertices in A∩ (1 × h + 1 × [w]), r− (b + 1) adjacentvertices in A∩ (2×h+ 1× [w]),. . . , α+ 1 adjacent vertices in A∩ (a×h + 1 × [w]), and for each i = a + 1, . . . , h, there exist α adjacent vertices inA ∩ (i × h + 1 × [w]). Analogously we estimate

Pp(F′h) ≥(

1− e−1

r−b pr−bw)a (

1− e−1α pαw

)h−a.


Therefore, the probability of filling [h + 1]2 × [w] is at least

Pp(Fh)Pp(F′h) ≥(

1− e−1

r−a pr−aw)a+b (

1− e−1α pαw

)2h−(a+b).

Remark 3.1.17. In the regime a + c < r ≤ a + b + c, all three the induced2-dimensional processes N a,b

r−c, Na,cr−b and N b,c

r−a are critical.

As a warm up for understanding the arguments that will be used to provethe upper bounds, we finish this section by proving a statement that wasmentioned above.

Proposition 3.1.18. The critical length Lc(N a,b,cr , p) is polynomial for all r ≤ c.

Proof. It is straightforward to obtain a universal lower bound, in the sensethat it does not depend on the parameters, in fact, it is enough to boundLc(N a,b,c

1 , p), since Lc(N a,b,cr , p) is increasing in r. Set L := p−c0 , where c0 <

1/3; if there is percolation then A is necessarily non empty and this event hasprobability

1− (1− p)L3 ≤ 1− e−2pL3 ≤ 1− e−2p1−3c0 −−→p→0

0.

To prove a polynomial upper bound, set L := p−C0 , where C0 > 0 is someconstant, then the rectangle R0 := [a]× [b]× [c] can deterministically growto R1 := [a] × [b] × [L] by using the rule r′e3 : −r′ ∈ [r] ∈ N a,b,c

r , thenR1 can grow to R2 := [a] × [L]2 in the following way: if r ≤ b the growthis deterministic again, otherwise it is enough to find one copy of [a]× 1 ×[r− b] inside A ∩ ([a]× k × [L]), for all k = b + 1, . . . , L, this happens withprobability at least

L

∏k=b+1

(1−

(1− pa(r−b)

)L/(r−b))≥(

1− e−pa(r−b)L/(r−b))L

=(

1− e−pa(r−b)−C0 /(r−b))p−C0

≥ exp(−2e−pa(r−b)−C0 /(r−b)p−C0

)−−→p→0

1,

if C0 > a(r − b), this means that I•(R2) occurs with high probability, andfinally R2 can grow to [L]3 with high probability if C0 > (r − a)2/2, byfollowing the same argument (when r > a it is enough to find a copy of1 × [r− a]2 inside A ∩ k × [L]2 for all k = a + 1, . . . , L). We have shownthat R0 can grow to [L]3 with high probability and it remains to show thatwith high probability it occurs the event F defined as follows: there exists acopy of R0 somewhere inside [L]3 such that R0 ⊂ A. In fact, by consideringthe (roughly) L3/abc disjoint copies of R0 we have for 3C0 > abc


Pp(Fc) ≤(1−Pp(R0 ⊂ A)

)L3/abc

=(

1− pabc)L3/abc

≤ exp(−pabcL3/abc

)−−→p→0

0.

This gives us Lc(N a,b,cr , p) ≤ p−C0 , if C0 > maxa(r− b), (r− a)2/2, abc/3,

and we are finished.

3.1.4 Outline of the proof

As we said in Remark 3.1.10, we have proved the upper bounds of Theorem3.1.4 for case VI: c > a + b. Now, we deal with the remaining bounds.

In Section 3.2 we prove all upper bounds of the critical lengths in Theorem3.1.4 in increasing order of technicality, namely, III, IV, I, V and finally II.They are all proved basically in the following way: first we calculate the costof growing from a rectangle to any bigger rectangle, second we show that arectangle R of a well chosen size can grow with high probability (this means,with probability going to 1 as p goes to 0) and fill the whole of [L]3 and,finally we prove two properties of R, with high probability we can find aninternally filled (see Definition 3.1.11) copy of R somewhere in [L]3, and R isinternally filled with probability larger than the inverse of the correspondingcritical length (up to a constant factor in the exponent).

In Section 3.3, we prove the lower bounds of Theorem 3.1.4 that can be ob-tained by using standard arguments, which follow from Aizenman-Lebowitz-type lemmas. To do so, it is required to consider the notion of internallyspanned, which approximates the notion of internally spanned. This coversthe cases c < a + b.

In Section 3.4, we move to the lower bounds with c ≥ b + a, by intro-ducing an algorithm that we call the beams process, which will allow us tocontrol the size of the components that can be created in the intermediatesteps of the bootstrap dynamics, the trick will be to cover such componentswith beams (think of those large metal beams used for constructing build-ings). All initially infected sites are beams, and at every step we mergebeams that are within some constant distance, to create a bigger one, thenrepeat this algorithm and stop it at some finite time; each beam created dur-ing the process we call covered. When we observe the induced process alongthe e3-direction, it looks like subcritical two-dimensional bootstrap percola-tion, thus, we can couple the process and apply the exponential decay prop-erty (Theorem 2.3.11) to bound the probability of a beam been covered. Wecomplete the proof by combining these ideas with a different Aizenman-Lebowitz-type lemma concerning covered beams (those appearing in theprocess), instead of internally spanned rectangles.

In Sections 3.5 and 3.6 we prove the remaining upper bounds correspond-ing to Theorems 3.1.5 and 3.1.6.


3.2 Upper bounds for r ∈ c + 1, c + 2Recall that we will consider all upper bounds in Theorem 3.1.4 in the follow-ing order: III, IV, I, V, II.

From now until the end of Section 3.2, we set

s := r− c ∈ 1, 2. (3.15)

3.2.1 Case III: c ∈ b + 1, . . . , a + b− sIn this section we consider the families

N a,b,cc+s ,

with c ∈ b + 1, . . . , a + b− s (here a > s, otherwise this case does not exist).As usual in bootstrap percolation, to deal with the upper bounds, we in-

deed prove a stronger sentence; namely, we will always obtain a result likethe following.

Proposition 3.2.1. Fix a value c ∈ b + 1, . . . , a + b − s and consider N a,b,cc+s -

bootstrap percolation. There exists a constant Γ > 0 depending on c such that, ifL = exp

(Γp−s/2(log 1

p )3/2)

, then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.16)

As an standard strategy, we first study the way of growing.

Lemma 3.2.2. Fix h, w ≥ c. If p is small enough, then

1. Pp(

I•([h]2 × [w + 1])|I•([h]2 × [w]))≥ 1− e−

1s psh2

.

2. Pp(

I•([h + 1]2 × [w])|I•([h]2 × [w]))≥(

1− e−(c2)−1 p(

c2)wh

)2.

Proof. Let us assume that [h]2 × [w] is internally filled.

1. Apply Lemma 3.1.12 with lx = ly = h.

2. Apply Lemma 3.1.15.

The second step is to determine the size of a rectangle such that, once it isinternally filled, then it can grow until [L]3 with high probability.

Lemma 3.2.3. Set L = exp(

Γp−s2 (log 1

p )32

), h = cp−

s2 (log 1

p )12 , and

R1 := [h]2 × [c].

Conditionally on I•(R1), the probability that [L]3 is internally filled goes to 1, asp→ 0.

3.2. Upper bounds for r ∈ c + 1, c + 2 37

Proof. Consider the rectangles R2 ⊂ R3 ⊂ R4 ⊂ R5 := [L]3 containing R1,defined by

R2 := [h]2 × [c2p−(c2)+

s2 (log 1

p )12 ],

R3 := [h2]2 × [c2p−(c2)+

s2 (log 1

p )12 ],

R4 := [h2]2 × [L].

Note that

Pp(I•([L]3)|I•(R1)) ≥4

∏k=1

Pp(I•(Rk+1)|I•(Rk)).

We apply Lemma 3.2.2 to conclude that

Pp(I•(R2)|I•(R1)) ≥(

1− e−12 psh2

)c2 p−(c2)+

s2 (log 1

p )12

=

(1− e−

c22 log 1

p

)p−(c2)

≥ e−2pc22 −(

c2) → 1,

and

Pp(I•(R3)|I•(R2)) ≥(

1− e−(c2)−1 p(

c2)·c2 p−(

c2)+

s2 (log 1

p )12 ·h)2h2

≥(

1− e−2c log 1p)2h2

≥ exp(−4h2p2c

)→ 1,

We apply the lemma again to get

Pp(I•(R4)|I•(R3)) ≥(

1− e−1s psh4

)L

≥ exp(−2Le−

1s psh4

)→ 1,

since psh4 ≥ p−s Γp−s2 (log 1

p )32 , and also

Pp(I•(R5)|I•(R4)) ≥(

1− e−(c2)−1 p(

c2)·L·h2

)2L

≥ exp(−4Le−pc2

L)→ 1.

We conclude that Pp(I•([L]3)|I•(R1))→ 1, as p→ 0.

Finally, we are ready to show the upper bound.


Proof of Proposition 3.2.1. Set L = exp(

Γp−s2 (log 1

p )32

), where Γ is a constant

to be chosen. Consider the rectangle

R :=[cp−

s2 (log 1

p )12

]2× [c] ⊂ [L]3,

and the events FL := There exists an internally filled copy of R in [L]3, andGL := 〈A ∪ R〉 = [L]3. It follows that

Pp

(I•([L]3)

)≥ Pp(FL)Pp(GL|I•(R)),

and by the previous lemma, Pp(GL|I•(R)) ≥ Pp(I•([L]3)|I•(R)) → 1, asp→ 0. Therefore, it remains to show that Pp(FL)→ 1 too.

In fact, we claim that there exists a constant C′ > 0 such that

Pp(I•(R)) ≥ exp(−C′p−

s2 (log 1

p )32

), (3.17)

so using the fact that there are roughly L3/|R| disjoint (therefore indepen-dent) copies of R (which we label Q1, . . . , QL3/|R|), and |R| ≤ p−3, (3.17) im-mediately gives

Pp(FcL) ≤ Pp

(⋂i

I•(Qi)c

)≤[1−Pp(I•(R))

]L3/|R|

≤[

1− e−C′p−

s2 (log 1

p )32

]p3L3

≤ exp

(−e

3 log L−3 log(1/p)−C′p−s2 (log 1

p )32

).

Since log L = Γp−s2 (log 1

p )32 , by taking Γ > C′/3 we conclude Pp(FL)→ 1, as

p→ 0. To finish, it is only left to prove inequality (3.17).In fact, note that a way to make R be internally filled is the following: start

with [c]3 ⊂ A, and then grow from Rk = [k]2 × [c] to Rk+1, for

k = c, . . . , m := cp−s2 (log 1

p )12 .

This gives us


Pp (I•(R)) ≥ Pp([c]3 ⊂ A)m

∏k=c

Pp (I•(Rk+1)|I•(Rk))

≥ pc3m

∏k=c

(1− e−(

c2)−1 p(

c2)ck)2

≥ pc3m

∏k=c

(1− e−2p(

c2))2

≥ pc3+c2m

≥ exp(−C′p−

s2 (log 1

p )32

),

for C′ > c3, as we claimed.

3.2.2 Case IV: c = a + b− 1 and s = 2

In this section we consider the families

N a,b,a+b−1a+b+1 ,

corresponding to the case r = c + s = a + b − 1 + s and s = 2. Note that,since we are only considering the cases r ≤ a + c, then we are assuming

a ≥ s.

We will show the following.

Proposition 3.2.4. Consider N a,b,a+b−1a+b+1 -bootstrap percolation. There exists a con-

stant Γ = Γ(a, b) > 0 such that, if L = exp(

Γp−1(log 1p )

2)

, then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.18)

We first study the way of growing.

Lemma 3.2.5. Fix integers h, w ≥ c. If p is small enough, then

1. Pp(I•([h]2 × [w + 1])|I•([h]2 × [w])) ≥ 1− e−12 p2h2

.

2. Pp(I•([h+ 1]2× [w])|I•([h]2× [w])) ≥(

1− e−1

b+1 pb+1w)c

(1− e−pw)C1h,

withC1 =

(1

a + 1+

1b + 1

).


1. Follows by the same reasoning used in the proof of Lemma 3.1.12.


2. Follows by the same reasoning used in the proof of Lemma 3.1.16, witha slightly modification in the definition of the events Fh and F′h (seeFigure 3.7).

FIGURE 3.7: a = 2, b = 3, c = 4, s = 2 and α = r− (a + b) = 1.

For instance, define Fh as there exist b + 1 adjacent vertices in A∩ (h +1 × 1 × [w]), b adjacent vertices in A ∩ (h + 1 × 2 × [w]),. . . , 2adjacent vertices in A∩ (h+ 1×b× [w]) and, for each i = 2, . . . , b h+1

b+1cone vertex in A ∩ (h + 1 × (b + 1)i− 1 × [w]).

The definition of F′h and the rest of the proof is analogous.

Remark 3.2.6. The improvement of the constant C1 in the second part of thislemma (instead of 2, given by Lemma 3.1.16) also holds in case V: c = a + b,when s = 1. In general it always holds when α = r− (a + b) = 1

Next we show that if a rectangle R ⊂ [L]3 of a well chosen size is internallyfilled, then it can grow and fill [L]3 with high probability, for L larger than thecritical length (up to a constant factor in the exponent).


Γp−1(log 1p )

2)

, h = rp−1(log 1p )

12 , r = c + 2, and

R1 := [h]2 ×[

p−1 log 1p

].

Conditionally on I•(R1), the probability that [L]3 is internally filled goes to 1 asp→ 0.

Proof. Consider the rectangles R2 ⊂ R3 ⊂ R4 ⊂ R5 := [L]3 containing R1,defined by R2 := [h]2 × [p−(r+1)], R3 := [h2s]2 × [p−(r+1)], and R4 := [h2s]2 ×[L], with r := b + 1. As before,

Pp(I•([L]3)|I•(R1)) ≥4

∏k=1



We apply Lemma 3.2.5 with r := r− a and α := r− (a + b) to conclude that

Pp(I•(R2)|I•(R1)) ≥(

1− e−12 psh2

)p−(r+1)

=

(1− e−

r22 log 1

p

)p−(r+1)

≥ e−2pr22 −(r+1)

→ 1,

and

Pp(I•(R3)|I•(R2)) ≥[(

1− e−1r pr p−(r+1)

)c (1− e−

1α pα p−(r+1)

)2h2s]h2s

=(

1− e−p−1/r)ch2s (

1− e−p−(b+1)/α)2h4s

≥ exp(−2ch2se−p−1/r

)exp

(−4h4se−p−(b+1)

)→ 1,

since h2s ≤ h4s = r4s p−2s2(log 1

p )2s = O(p−3s2

). We apply the lemma again toget

Pp(I•(R4)|I•(R3)) ≥(

1− e−1s psh4s

)L

≥ exp(−2Le−

1s psh4s

)→ 1,

since psh4s log L, and also

Pp(I•(R5)|I•(R4)) ≥(

1− e−1r pr L

)cL (1− e−

1α pαL

)2L2

≥ exp(−2cLe−

1r pr L

)exp

(−4L2e−

1α pαL

)→ 1.

Therefore Pp(I•([L]3)|I•(R1))→ 1, as p→ 0.

Now, we prove the upper bound for the critical length.


Γp−1(log 1p )

2)

, where Γ is a constantto be chosen. Consider r = c + s (here s = 2) and the rectangle

R :=[rp−

s2 (log 1

p )12

]2×[

p−1 log 1p

]⊂ [L]3.

By following the same reasoning used to derive Proposition 3.2.1 in last sec-tion, by taking Γ > C′/3, it is enough to show that it is possible to choose aconstant C′ > 0 satisfying

Pp(I•(R)) ≥ exp(−C′p−1(log 1

p )2)

. (3.19)


In fact, start with Rc := [c]2 ×[

p−1 log 1p

]⊂ A, and then grow from Rk =

[k]2 ×[

p−1 log 1p

]to Rk+1, for k = c, . . . , m := rp−

s2 (log 1

p )12

Pp (I•(R)) ≥ Pp(Rc ⊂ A)m

∏k=c


≥ p|Rc|m

∏k=c

(1− e−

1b+1 pb+1c

)c(

1− e−pp−1 log 1

p)2k

≥ p|Rc|pC′1m(1− p)C′′1 m2

≥ p|Rc|+C′1m exp(−2r2C′′1 pp−2 log 1

p

)≥ exp

(−C′p−1(log 1

p )2)

,

with all C′s being positive constants depending on s and c.

3.2.3 Case I: c = b = a

To obtain the remaining upper bounds, we will use the value of an integralwhich was computed by Holroyd in [37]. Fix a positive integer d and con-sider the function fd : (0, ∞) → (0, ∞) given by fd(z) = − log(1 − e−zd

).When d = 1, ∫ ∞

0z f1(z2) dz =

12

∫ ∞

0f1(z) dz =

π2

12.

We will also use the integrability of fd for all d ≥ 2, note that∫ ∞

0fd(z) dz ≤

∫ 1/2

0(− log(zd/2)) dz +

∫ 1

1/2fd(z) dz +

∫ ∞

1f1(z) dz < ∞.

(3.20)In this section we consider the isotropic families

N c,c,cc+s .

We will prove the following.

Proposition 3.2.8. Consider N c,c,cc+s -bootstrap percolation. There exists a universal

constant Γ > 0 such that, if L = exp(Γp−s/2), then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.21)

The way of growing is simple.

Lemma 3.2.9. If h ≥ c and p is small enough, then

Pp

(I•([h + 1]3)|I•([h]3)

)≥(

1− e−1s psh2

)3.


Proof. All faces of [h]3 have the same difficulty to grow, they grow as theeasiest growth direction, so we can apply Lemma 3.1.12 on each face and useindependence.

Next, we determine the size of a rectangle such that, once it is internallyfilled then it can grow until [L]3 with high probability.

Lemma 3.2.10. Set L = exp(Γp−s/2) and R1 := [p−s]3. Conditionally on I•(R1),the probability that [L]3 is internally filled goes to 1 as p→ 0.

Proof. By Lemma 3.2.9 we have

Pp

(I•([L]3)|I•(R1)

)≥

L

∏h=p−s

Pp

(I•([h + 1]3)|I•([h]3)

)≥

L

∏h=p−s

(1− e−

1s psh2

)3

≥(

1− e−1s p−s

)3L

≥ exp(−6Le−

1s p−s

)→ 1,

as p→ 0, since L = exp(Γp−s/2).

Now, we are ready to show the upper bound.

Proof of Proposition 3.2.8. Set L = exp(Γp−s/2), where Γ is a constant to bechosen. Consider the rectangle

R := [p−s]3 ⊂ [L]3.

As before, by taking Γ > C′/3, it is enough to show that it is possible tochoose a constant C′ > 0, such that

Pp(I•(R)) ≥ exp(−C′p−s/2). (3.22)

In fact, by Lemma 3.2.9 we have


Pp(I•(R)) ≥ Pp([c]3 ⊂ A)p−s−1

∏h=c

Pp(I•([h + 1]3)|I•([h]3))

≥ pc3p−2

∏h=c

(1− e−

12 p2h2

)3

≥ pc3exp

(3

∞

∑k=1

log(1− e−12 psh2

)

)

≥ pc3exp

(4∫ ∞

0log(1− e−

12 psz2

) dz)

≥ exp(−c3 log 1

p

)exp

(Cp−s/2

∫ ∞

0log(1− e−y2

) dy)

≥ exp(−C′p−s/2

),

for some constants C, C′ > 0, and the approximation of the sum by the inte-gral holds since p→ 0.

3.2.4 Case V: c = a + b


N a,b,a+ba+b+s ,

corresponding to the case r = c + s = a + b + s. We will show the following.

Proposition 3.2.11. Consider N a,b,a+ba+b+s -bootstrap percolation. There exists a con-

stant Γ > 0 depending on a and b such that, if L = exp(Γp−s), then

Pp(I•([L]3))→ 1, as p→ 0. (3.23)

We first study the way of growing.

Lemma 3.2.12. Fix integers h, w ≥ c. If p is small enough, then

1. Pp(

I•([h]2 × [w + 1])|I•([h]2 × [w]))≥ 1− e−

1s psh2

.

2. Pp(

I•([h + 1]2 × [w])|I•([h]2 × [w]))≥(

1− e−1

r−a pr−aw)c (

1− e−1s psw

)2h.


1. Apply Lemma 3.1.12.

2. Apply Lemma 3.1.16, and note that α = r− c = s.


Next, we show that if a rectangle R ⊂ [L]3 of a well chosen size is inter-nally filled, then it can grow and fill [L]3 with high probability, for L largerthan the critical length (up to a constant factor in the exponent).

Lemma 3.2.13. Set L = exp(Γp−s) and fix ε > 0, h = p−s/2−ε, and

R1 := [h]2 × [h2].


Proof. Consider the rectangles R2 ⊂ R3 ⊂ R4 ⊂ R5 := [L]3 containing R1,defined by R2 := [h]2 × [p−(r−a+1)], R3 := [h2]2 × [p−(r−a+1)], andR4 := [h2]2 × [L], then

Pp(I•([L]3)|I•(R1)) ≥4

∏k=1



Pp(I•(R2)|I•(R1)) ≥(

1− e−1s psh2

)p−(r−a+1)

=(

1− e−p−2ε)p−(r−a+1)

≥ exp(−2p−(r−a+1)e−p−2ε

)→ 1,

and, following the same proof of Lemma 3.2.7 with r = r − a, we concludethat Pp(I•([L]3)|I•(R1))→ 1, as p→ 0; we use that

psh4 = ps p−2s−4ε Γp−s.

Now, we prove the upper bound for the critical length.

Proof of Proposition 3.2.11. Set L = exp(Γp−s), where Γ is a constant to be cho-sen. Fix ε ∈ (0, 1/8), and consider the rectangle

R :=[

p−s/2−ε]2× [p−s−2ε] ⊂ [L]3.

As usual, by taking Γ > C′/3, it is only left to show that there exists a constantC′ > 0 such that

Pp(I•(R)) ≥ exp(−C′p−s), (3.24)

To do so, set h = p−ε, then for every k = 1, . . . , m := p−s/2 set

hk = kh, wk = h2k, Rk = [hk]

2 × [wk], and R′k = [hk]2 × [wk+1].

Note that Rm = R, hk+1 = hk + h, and wk+1 = wk + (2k + 1)h2, so by Lemma3.2.12 we have


Pp(I•(Rm)) ≥ Pp(R1 ⊂ A)m−1

∏k=1

Pp(I•(R′k)|I•(Rk))Pp(I•(Rk+1)|I•(R′k))

≥ p|R1|m

∏k=1

(1− e−

1s psh2

k

)(2k+1)h2 [(1− e−

1r−a pr−awk+1

)c (1− e−

1s pswk+1

)2hk]h

≥ p|R1|m

∏k=1

(1− e−2pr−a

)ch (1− e−2ps

)h2 (1− e−

1s psh2

k

)4kh2

≥ p|R1|p(r−a)chm psh2mm

∏k=1

(1− e−

1s psk2h2

)4kh2

.

For p small, the last product is at least

∞

∏k=1

(1− e−

1s psh2k2

)4h2k≥ exp

(−4p−s

∞

∑k=1

psh2k f1(1s psh2k2)

)

≥ exp(−C1p−s

∫ ∞

0z f1(z2) dz

),

where f1(z) = − log(1− e−z) and C1 = C1(s) > 0. Thus,

Pp(I•(Rm)) ≥ exp(−C′p−s),

for some constant C′ = C′(s, f1) > 0.

Remark 3.2.14. By using Remark 3.2.6 and being careful with computations,we can show that underN a,b,a+b

a+b+1 -bootstrap percolation (s = 1), if L = exp(Γp−1),with

Γ > Γ(a, b) :=(

1a + 1

+1

b + 1+ 2)

π2

36,

then [L]3 is internally filled with high probability, as p → 0. We believe thatthis is the right constant in the exponent.

3.2.5 Case II: c = b > a

In this section we cover the last case c = b > a. Consider the families

N a,c,cc+s .

The corresponding upper bound goes as follows.

Proposition 3.2.15. ConsiderN a,c,cc+s -bootstrap percolation with c > a. There exists

a universal constant Γ > 0 such that, if L = exp(

Γp−s/2√

log 1p

), then

Pp(I•([L]3))→ 1, as p→ 0. (3.25)

We follow the same steps.


Lemma 3.2.16. Fix l, w ≥ c. If p is small enough, then

1. Pp(I•([l]× [w + 1]2)|I•([l]× [w]2)) ≥(

1− e−1s pslw

)2.

2. Pp(I•([l + 1]× [w]2)|I•([l]× [w]2)) ≥ 1− e−2

c(c+1) pc(c+1)/2w2.

Proof. Similar to the proof of Lemmas 3.1.12 and 3.2.2. In this case, to growalong the e2-direction is as easy as grow along the e3-direction (see Figure3.8).

FIGURE 3.8: One copy of the right-angled triangle ∆a insidel + 1× [w]2, and one s-pattern on the top and right-most sides

(s = 1).


Γp−s/2√

log 1p

), l = p−s/2(log 1

p )− 1

2 , and

R1 := [l]×[2sΓp−s log 1

p

]2.


Proof. Consider the rectangle R2 := [l]× [L]2 ⊂ [L]3 containing R1, then

Pp(I•([L]3)|I•(R1)) ≥ Pp(I•(R2)|I•(R1))Pp(I•([L]3)|I•(R2)).



Pp(I•(R2)|I•(R1)) ≥(

1− e− 1

s psl·2sΓp−s log 1p)2L

≥(

1− e−2Γp−s/2√

log 1p

)2L

≥ exp(−4Le−2Γp−s/2

√log 1

p

)→ 1,

and Pp(I•([L]3)|I•(R2)) ≥ exp(−2Le−pc2

L2)→ 1, as p→ 0.

Finally, we follow the same proof for the upper bound in the previoussection, with an extra argument in the end.


Γp−12

√log 1

p

), where Γ is a constant

to be chosen. Consider the rectangle

R :=[

p−s2 (log 1

p )− 1

2

]×[2sΓp−s log 1

p

]2⊂ [L]3.

Once again, it is enough to show that there exists a constant C′ > 0 such that

Pp(I•(R)) ≥ exp(−C′p−

s2

√log 1

p

), (3.26)

In fact, fix ε ∈ (0, 1/4) and set l = p−ε(log 1p )− 1

2 , then for every k = 1, . . . , m :=

p−s2+ε set

lk = kl, wk = lk log 1p , Rk = [lk]× [wk]

2, and R′k = [lk]× [wk+1]2.

This time Rm = [lm]× [wm]2 =[

p−s2

/√log 1

p

]×[

p−s2

√log 1

p

]2⊂ R, so we

have to show

• Pp(I•(Rm)) ≥ exp(−C1p−

s2

√log 1

p

), and

• Pp(I•(R)|I•(Rm)) ≥ exp(−8e−1p−

s2

√log 1

p

),

for some constant C1 > 0, then set C′ = C1 + 8e−1.In fact, note that lk+1 = lk + l and wk+1 = wk + w1, so by Lemma 3.2.16

we have


Pp(I•(Rm)) ≥ Pp(R1 ⊂ A)m−1

∏k=1

Pp(I•(R′k)|I•(Rk))Pp(I•(Rk+1)|I•(R′k))

≥ p|R1|m

∏k=1

(1− e−

1s pslkwk

)2w1(

1− e−2

c(c+1) pc(c+1)/2w2k

)l

≥ p|R1|m

∏k=1

(1− e−

1s psl2

k log 1p)2w1 (

pc(c+1)/2)l

≥ p|R1|pc(c+1)

2 lmm

∏k=1

(1− e−

1s ps−2εk2

)2w1.

The last product is at least

∞

∏k=1

(1− e−

1s ps−2εk2

)2w1= exp

(∞

∑k=1

2w1 log(

1− e−1s ps−2εk2

))

≥ exp((2 + ε)w1

∫ ∞

0log(

1− e−1s ps−2εz2

)dz)

≥ exp(−C′1p−

s2

√log 1

p

),

for some constant C′1 = C′1(s, f2) > 0. Then, for C1 := (1 + ε)C′1

Pp(I•(Rm)) ≥ exp(−C1p−

s2

√log 1

p

).

Finally, once Rm = [lm] × [wm]2 is internally filled, to fill R it only remainsto grow from wm = p−

s2

√log 1

p to 2sΓp−s log 1p , so we can use Lemma 3.2.16

part 1, to get

Pp(I•(R)|I•(Rm)) ≥2sΓp−s log 1

p

∏w=wm

(1− e−

1s pslmw

)2

≥ exp

(−4

∞

∑w=wm

e−(

ps/2/√

log 1p

)w)

,

since e−plmw = e−(

ps/2/√

log 1p

)w ∈ (0, 2

3) for all w ≥ wm. Set ξ := e−(

ps/2/√

log 1p

)and observe that wm log ξ = −1, so the last sum is at most

∞

∑w=− 1

log ξ

ξw =ξ−1/ log ξ

1− ξ=

e−1

1− e−1/(

p−s/2√

log 1p

) ≤ 2e−1p−s/2√

log 1p .

In summary, Γ > (C1 + 8e−1)/3 works.


3.3 Lower bounds via components process

In this section we only prove the lower bounds corresponding to the casesr ≤ a + b, because the proof is an application of the components process (seeDefinition 3.3.4 below), a variation of an algorithm introduced in [17]. Wewill cover the cases c ≥ a + b in Section 3.4.

Remark 3.3.1. From now on, when U = N a,b,cr we will omit the subscript in

the closure and simply write 〈·〉 instead of 〈·〉N a,b,cr

.

Recall that we are assuming r ≥ c + 1, so that for every A ⊂ [L]3, we alsohave 〈A〉 ⊂ [L]3. Aizenman and Lebowitz [1] obtained the matching lowerbound for the family N 1,1,1

2 by using the so-called rectangles process, and theyexploit the fact that for that model, the closure 〈A〉 is a union of rectangleswhich are separated by distance at least 2.

Since the threshold for the family N 1,1,13 was also determined in [17], we

will always assumec ≥ 2.

In this case, the closure 〈A〉 could not be a union of rectangles anymore, be-cause r ≥ 3. Thus, we need to introduce a notion about rectangles which isan approximation to being internally filled, and this notion requires a strongconcept of connectedness; we define both concepts in the following. Recallthe set Na,b,c (the anisotropic neighbourhood of the origin) given by Defini-tion 3.1.2.

Definition 3.3.2. Consider the graph G = (V, E) with vertex set [L]3 andedge set E = (x, y) : x − y ∈ Na,b,c. We say that a set S ⊂ [L]3 is stronglyconnected if it is connected in the graph G.

Definition 3.3.3. We say that the rectangle R ⊂ [L]3 is internally spanned byA, if there exists an strongly connected set S ⊂ 〈A ∩ R〉 such that R is thesmallest rectangle containing S. We denote this event by I×(R).

Note that when a rectangle is internally filled then it is also internallyspanned, therefore, it will be enough to show that the probability that [L]3 isinternally spanned goes to 0 as p → 0, for L less than the determined criticallength.

3.3.1 The components process

The following algorithm is an adaptation of an idea introduced by Cerf andCirillo in [17]. We will use it to show an Aizenman-Lebowitz-type lemma(see for instance, Lemma 2.1.8), which says that when a rectangle is internallyspanned, then it contains internally spanned rectangles of all intermediatesizes (see Lemmas 3.3.9 and 3.3.11 below).

Definition 3.3.4 (The components process). Let A = x1, . . . , x|A| ⊂ [L]3,and set R := S1, . . . , S|A|, where Si = xi for each i = 1, . . . , |A|. Fixr ≥ c + 1 and repeat the following steps until STOP:

3.3. Lower bounds via components process 51

1. If there exists a family of n ∈ 2, . . . , r sets Si1 , . . . , Sin ⊂ R such that

〈Si1 ∪ · · · ∪ Sin〉

is strongly connected, then choose a minimal such family, remove itfromR, and replace by 〈Si1 ∪ · · · ∪ Sin〉.

2. If there do not exist such a family of sets inR, then STOP.

Remark 3.3.5. We highlight two properties that are due to the way the algo-rithm evolves:

• At any stage of the component process, any set S = 〈Si1 ∪ · · · ∪ Sin〉added to the collectionR satisfies S = 〈A ∩ S〉 = 〈S〉. In particular, thesmallest rectangle containing S is internally spanned.

• Since G is finite, the process stops in finite time; consider the final col-lectionR′ and set V(R′) = ⋃

S∈R′S. We claim that V(R′) = 〈A〉.

In fact, clearly A ⊂ V(R′) ⊂ 〈A〉, and to prove that 〈A〉 ⊂ V(R′) weargue by contradiction. Suppose this is not the case, since A ⊂ V(R′),there would exist a vertex x ∈ 〈A〉 \ V(R′) having t ≥ r neighboursx1, . . . , xt ∈ V(R′), let us say that xi ∈ S′i for some sets S′i ∈ R′ fori = 1, . . . , t.

Since S′i = 〈S′i〉, each set S′i can have at most r − 1 of such neighboursso we can choose S′1 6= S′2. In particular 〈S′1 ∪ · · · ∪ S′r〉 is strongly con-nected via x and 〈S′1 ∪ · · · ∪ S′r〉 /∈ R′; this contradicts the definition ofR′.

Since we will only prove lower bounds for the cases r ∈ c + 1, c + 2,until the end of Section 3.4 we set

s := r− c ∈ 1, 2. (3.27)

3.3.2 Case I: c = b = a

Let us state the precise statement we want to show.

Proposition 3.3.6. Under N a,a,aa+s -bootstrap percolation, there is a constant γ > 0

depending on a such that, for L < exp(γp−s/2),

Pp(I×([L]3))→ 0, as p→ 0. (3.28)

The following is an analogue of the Aizenman-Lebowitz Lemma in [1],we prove it for all parameters a, b, c and r ∈ c + 1, . . . , a + b + c, whichcorrespond to the regime where N a,b,c

r is critical.

Lemma 3.3.7. Consider N a,b,cr -bootstrap percolation with r ≥ c + 1. If [L]3 is in-

ternally spanned then, for every k ≤ L there exists an internally spanned rectangleQ = [lx]× [ly]× [lz] ⊂ [L]3 satisfying k ≤ (lx + ly + lz)/3 ≤ rk.


Proof. First of all, note that when S is a strongly connected subset of [L]3

such that 〈A ∩ S〉 = S, then the smallest rectangle containing S is internallyspanned by A. Let S be the first set that appears in the components processsuch that, the smallest rectangle containing S, which we denote by

Q = Q(S) := [lx]× [ly]× [lz],

satisfies (lx + ly + lz)/3 ≥ k (such a set exists since V(R′) = 〈A〉 and [L]3 isinternally spanned). Since Q is internally spanned, it is only left to show thatthe sum lx + ly + lz is at most 3rk.

In fact, we know that S = 〈Si1 ∪ · · · ∪ Sin〉 for some sets Sit such that, foreact t = 1, . . . , n, the smallest rectangle Rt := [lx,t]× [ly,t]× [lz,t] containing Sitsatisfies lx,t + ly,t + lz,t ≤ 3k− 1. Since S is strongly connected and minimal,the new semi-perimeter is

lx + ly + lz ≤ 2 maxt∈[n]

lx,t + ly,t + lz,t

+ r

≤ r(3k− 1) + r= 3rk.

(3.29)

We are ready to prove the lower bound.

Proof of Proposition 3.3.6. Take L < exp(γp−s/2), where γ > 0 is some smallconstant to be chosen. Let us show that Pp(I×([L]3)) goes to 0, as p→ 0.

Fix δ > 0 and setk = δp−s/2.

If [L]3 is internally spanned, by Lemma 3.3.9 the following event occurs: thereexists an internally spanned rectangle Q = [lx]× [ly]× [lz] ⊂ [L]3 satisfyingk ≤ (lx + ly + lz)/3 ≤ rk. By symmetry, we can assume lz ≥ k.

By connectedness, there is no gap of size sr along the e3-direction. Moreprecisely, every copy of the slab [lx] × [ly] × [sr] must contain at least s ele-ments of A, within distance r if s = 2. Consider only the lz/sr disjoint slabs;since lxly = O(k2), if δ is small, the probability of this event is at most

(O(pslxly))lz/sr ≤ (O(psk2))k/sr

= (O(δ))k/sr

≤ e−k.

Finally, denote byRk the collection of rectangles [lx]× [ly]× [lz] ⊂ [L]3 satis-fying k ≤ (lx + ly + lz)/3 ≤ rk, it follows, by union bound that

Pp(I×([L]3)) ≤ ∑Q∈Rk

Pp(I×(Q))

≤ |Rk|e−k

≤ L4e−δp−s/2 → 0,


as p→ 0, for 4γ < δ, and we are finished.

3.3.3 Case II: c = b > a

In this section we will prove the following lower bound.

Proposition 3.3.8. Under N a,c,cc+s -bootstrap percolation, there is a constant γ > 0

depending on c such that, for L < exp(

γp−s2

√log 1

p

),

Pp(I×([L]3))→ 0, as p→ 0. (3.30)

The following is another variation of the Aizenman-Lebowitz Lemma in[1].

Lemma 3.3.9. ConsiderN a,b,cr -bootstrap percolation with r ≥ c+ 1. If [L]3 is inter-

nally spanned then, for every h, k ≤ L there exists an internally spanned rectangleQ = [lx]× [ly]× [lz] ⊂ [L]3 satisfying (ly + lz)/2 ≤ rk, and either

(a) lx ≥ h, or

(b) lx < h and (ly + lz)/2 ≥ k.

Proof. Let S be the first set that appears in the components process such that,the smallest rectangle Q := [lx]× [ly]× [lz] containing S, satisfies either lx ≥ hor (ly + lz)/2 ≥ k (such a set exists since V(R′) = 〈A〉 and [L]3 is internallyspanned). Since Q is internally spanned, it is only left to show that the semi-perimeter (ly + lz)/2 is at most rk.

In fact, we know that S = 〈Si1 ∪ · · · ∪ Sin〉 for some sets Sit such that, foreact t = 1, . . . , n, the smallest rectangle Rt := [lx,t]× [ly,t]× [lz,t] containingSit satisfies (ly,t + lz,t)/2 ≤ k− 1

2 . Since S is connected and minimal, the newsemi-perimeter is

ly + lz2≤ 2 max

t∈[n]

ly,t + lz,t

2

+

r2

≤ r(k− 12) +

r2

≤ rk.

(3.31)

Proof of Proposition 3.3.8. Take L < exp(γp−s2

√log 1

p ), where γ > 0 is some

small constant to be chosen. Let us show that Pp(I×([L]3)) goes to 0, asp→ 0.

Fix δ > 0 and set

h = δp−s2 (log 1

p )− 1

2 , k = p−s2

√log 1

p .

If [L]3 is internally spanned, by Lemma 3.3.9 the following event occurs:There exists an internally spanned rectangle Q = [lx] × [ly] × [lz] ⊂ [L]3

satisfying (ly + lz)/2 ≤ rk, and either


(a) lx ≥ h, or

(b) lx < h and (ly + lz)/2 ≥ k.

In the case that Q satisfies (b), we have that lx < h and, either ly or lz is atleast k, by symmetry (c = b), we can assume lz ≥ k.

Since there is no gap of size sr along the e3-direction, every copy of theslab [lx] × [ly] × [sr] must contain at least s elements of A within constantdistance. Consider only the lz/sr disjoint slabs; since lxly = O(hk), if δ issmall, the probability of this event is at most

(O(pslxly))lz/sr ≤ (O(pshk))k/sr

= (O(δ))k/sr

≤ e−k.

In the case that Q satisfies (a), we use the fact that a ≤ c− 1 = r − (s + 1),thus, there is no gap of size (s + 1)r along the e1-direction, and every copy ofthe slab [(s + 1)r]× [ly]× [lz] must contain at least s + 1 elements of A withinconstant distance. Since lx ≥ h, the probability of this event is at most(

O(ps+1lylz))lx/(s+1)r

≤(

O(ps+1k2))h/(s+1)r

≤(

O(ps+1p−s log 1p ))h/(s+1)r

≤ pΩ(h)

≤ e−Ω(δk).

Therefore, the probability that Q is internally spanned is at most

max

e−k, e−Ω(δk)≤ e−c(δ)k,

for some small constant c(δ) > 0. Finally, denote by Rh,k the collection ofrectangles [lx]× [ly]× [lz] ⊂ [L]3 satisfying either (a) or (b) above, it follows,by union bound that

Pp(I×([L]3)) ≤ ∑Q∈Rh,k

Pp(I×(Q))

≤ |Rh,k|e−c(δ)k

≤ L7e−c(δ)p−

s2

√log 1

p → 0,

as p→ 0, for 7γ < c(δ), and we are finished.

3.3.4 Case III: c ∈ b + 1, . . . , a + b− sIn this section we prove the last case that can be done via the componentsprocess.


Proposition 3.3.10. Fix c ∈ b + 1, . . . , a + b− s. Under N a,b,cc+s -bootstrap per-

colation, there is a constant γ = γ(c) > 0 such that, for L < exp(γp−s2 (log 1

p )32 ),

Pp(I×([L]3))→ 0, as p→ 0. (3.32)

The corresponding analogue of the Aizenman-Lebowitz Lemma goes asfollows.

Lemma 3.3.11. Consider N a,b,cr -bootstrap percolation with r ≥ c + 1. If [L]3 is

internally spanned then, for every h, k ≤ L there exists an internally spanned rect-angle Q = [lx]× [ly]× [lz] ⊂ [L]3 satisfying (lx + ly)/2 ≤ rh, and either

(a) lz ≥ k, or

(b) lz < k and (lx + ly)/2 ≥ h.

The proof of this lemma is very similar to that of Lemma 3.3.9, thereforewe omit and proceed to the proof of the lower bound. This time we need touse the FKG inequality (see Theorem B.4.2) to upper bound the probabilityof crossing along the e3-direction.

Proposition 3.3.10. Take L < exp(γp−s2 (log 1

p )32 ), where γ > 0 is some small

constant. Fix δ > 0 and set

h = δp−s2 (log 1

p )12 , k = p−(s+1)/2.

If [L]3 is internally spanned, by Lemma 3.3.11, there is an internally spannedrectangle Q = [lx]× [ly]× [lz] satisfying (lx + ly)/2 ≤ rh, and either (a) or (b)happens in Lemma 3.3.11.

In the case that Q satisfies (a), we have lz ≥ k and lxly = O(h2). Asbefore, every copy of the slab S := [lx] × [ly] × [sr] must contain at least selements of A within constant distance. If s = 2 we work a little bit more: letus enumerate as A1, . . . , AN the collection of all pairs of vertices in S that arewithin distance r. Note that N = Θ(lxly) = O(h2) and the event

Fi := Ai ⊂ A

is increasing for each i ∈ [N], hence, by FKG inequality (see Corollary B.4.3)

P

(N⋂

i=1

Fci

)≥

N

∏i=1

[1−P(Fi)]

= [1− p2]O(h2)

≥ exp(−Ω(p2h2)

).

Now, by considering the lz/sr disjoint slabs; if δ is small, the probability ofthe former event is at most


(1−P

(N⋃

i=1

Fi

))lz/sr

≤(

1− e−Ω(psh2))k/sr

=(

1− e−Ω(δ2 log 1p ))p−(s+1)/2/sr

=(

1− pΩ(δ2))p−s/2−1/2/sr

≤ e−p−s/2−1/3.

In the case that Q satisfies (b), we can assume w.l.o.g. that ly ≥ h anduse the fact that b ≤ c − 1 = r − (s + 1). This time there is no gap alongthe e2-direction, so, every copy of the slab [lx]× [(s + 1)r]× [lz] must containat least s + 1 elements of A within constant distance. The probability of thisevent is at most

(O(ps+1lxlz)

)ly/(s+1)r≤(

O(ps+1hk))h/(s+1)r

≤(

O(ps+1p−s−1/2 log 1p ))h/(s+1)r

≤ pΩ(h)

≤ e−Ω(h log 1p ).

Therefore, the probability that Q is internally spanned is at most

e−c(δ)p−

s2 (log 1

p )32,

for some small constant c(δ) > 0. By adapting the end of the proof of Propo-sition 3.3.8, we can show that Pp(I×([L]3)) goes to 0, as p→ 0.

3.4 Lower bounds via beams process

To deal with the cases c ≥ a + b we introduce a new variation of the com-ponents process, which we call the beams process; this time, the union boundover rectangles is not useful anymore. Therefore, instead of covering the in-fected vertices step by step with rectangles, we cover them with beams sothat, when we observe this induced process along the e3-direction it lookslike subcritical two-dimensional bootstrap percolation.

3.4.1 The beams process

The lower bound in the case c = a+ b will be proved via a coupling with sub-critical two-dimensional families, by using a modification of the techniquesused in the previous section.

3.4. Lower bounds via beams process 57

Recall the family N a,bm given by the collection of all subsets of size m of

Na,b = a′e1 : ±a′ ∈ [a] ∪ b′e2 : ±b′ ∈ [b]. (3.33)

Observe that S(N a,bm ) = S1 if and only if m ≥ a + b + 1, in particular, our

exponential decay result (Theorem 2.3.11) holds for these families.In this section we set

m := a + b + 1. (3.34)

Definition 3.4.1. A beam is a finite set S ⊂ Z3 of the form S = H× [w], whereH ⊂ Z2 is connected and 〈H〉N a,b

m= H.

Remark 3.4.2. Given a finite set S ⊂ Z3, we denote by B(S) the beam con-taining S which is constructed in the following way: by translating S if neces-sary, we can assume that the smallest rectangle containing S is R× [w], withR := [h1]× [h2]. Consider the set H0 given by

H0 := x ∈ R : (x × [w]) ∩ S 6= ∅,

then we take H := 〈H0〉N a,bm

, and set B(S) := H × [w]. Note that H ⊂ R.

Example 3.4.3. In the picture below we show a set S consisting of all graycubes. If we consider the beam B(S) with respect to the subcritical familyN 1,2

4 , then the length w of B(S) (along the e3-direction) is determined by theleft-most cube and the right-most cubes. In this picture, S is a union of 4disjoint subsets; one of them is the 1× 1× 1 cube to the left, which will grow(horizontally) to become (or fill) a 1× 1× w shape.

An analogous situation will happen to each of the other 3 disjoint subsets.Finally, the dashed lines to the right indicate that those spaces will be filledtoo in the closure 〈H0〉N 1,2

4(see Figure 3.9 below).

It will be important for us to have an upper bound on the number ofbeams of a given size, which are contained in [L]3. The following lemma isjust another consequence of Proposition 2.3.10.

Lemma 3.4.4 (Counting beams). Let Bn1,n2 be the collection of all beams of theform H × [w] ⊂ [L]3 satisfying w ≤ n1 and |H| ≤ n2. Then

|Bn1,n2 | ≤ n1L3(3e)n2 .


FIGURE 3.9: A beam w.r.t. the subcritical family N 1,24 .

Proof. In fact, the number of copies inside [L] of the segment [w] is at mostL, therefore, the number of segments inside [L] with at most n1 vertices, is atmost ∑n1

w=1 L = n1L.Now we give an upper bound for the number of H’s. Denote Hh the

collection of all connected sets H ⊂ [L]2 such that |H| = h, so we can write

h|Hh| = ∑H∈Hh

|H| = ∑x∈[L]2

∑H∈Hh

1x ∈ H = ∑x∈[L]2

cs(x),

where cs(x) is the number of connected subsets of [L]2 with size h + 1, con-taining a fixed point x. To each of such sets we can associate an spanning treein an injective fashion, so by Proposition 2.3.10, |Hh| ≤ L2(3e)h−1. It followsthat the number of H’s is at most

n2

∑h=1|Hh| ≤ L2

n2

∑h=1

(3e)h−1 ≤ L2(3e)n2 .

We want to control the process of infection by covering all possible in-fected sites with beams, we do that step by step in order to get a nice controlover the sizes. The following algorithm is a variation of the components pro-cess. We will use it to show an Aizenman-Lebowitz-type lemma which saysthat when [L]3 is internally filled, then it contains covered beams of all inter-mediate sizes (see Lemma 3.4.8 below).

Definition 3.4.5 (The beams process). Let A = x1, . . . , x|A| ⊂ [L]3, and setB := S1, . . . , S|A|, where Si = xi for each i = 1, . . . , |A|. Fix r ≥ c + 1 andrepeat the following steps until STOP:

1. If there exists a family of n ∈ 2, . . . , r beams Si1 , . . . , Sin ⊂ B suchthat

〈Si1 ∪ · · · ∪ Sin〉is strongly connected, then choose a minimal such family, remove itfrom B, and replace by B(〈Si1 ∪ · · · ∪ Sin〉).

2. If there do not exist such a family of sets in B, then STOP.


Definition 3.4.6. We call any beam S = B(〈Si1 ∪ · · · ∪ Sin〉) ⊂ [L]3 added tothe collection B a covered beam, and denote the event that S is covered by

IV(S).

Again, there are two properties that are due to the way the algorithmevolves:

• Any covered beam S satisfies 〈A ∩ S〉 ⊂ 〈S〉 = S.

• The process stops in finite time, thus, we can consider the final col-lection B′ and set V(B′) :=

⋃S∈B′

S. By using the same arguments of

Remark 3.3.5, it follows that 〈A〉 ⊂ V(B′) (just replace the sets S′i by thebeams B(S′i)).

3.4.2 Case V: c = a + b

In this section we prove the following.

Proposition 3.4.7. Under N a,b,a+ba+b+s -bootstrap percolation, there exists a constant

γ = γ(a, b) > 0 such that, for L < exp(γp−s),

Pp[I•([L]3)]→ 0, as p→ 0. (3.35)

The beams process and Lemma 2.3.9 allow us to prove a beams versionof the Aizenman-Lebowitz Lemma for this case. Let λ > 0 be the constant inLemma 2.3.9 associated to the subcritical two-dimensional family N a,b

m . It ispossible to convince ourselves that λ ≥ m, by following the construction inthe proof of Proposition 2.3.4 (or, just take maxλ, m instead of λ).


internally filled, then for every k = λ, . . . , L, there is a covered beam S = H × [w]satisfying w, |H| ≤ 2λk, and either w ≥ k or |H| ≥ k.

Proof. Let S = H × [w] be the first beam that appears in the beam processsatisfying either w ≥ k or |H| ≥ k (such a set exists since V(B′) = [L]3).Then, it is enough to show that w ≤ rk and |H| ≤ 2λk.

We know that S = B(〈Si1 ∪ · · · ∪ Sin〉) for some beams Sit = Ht × [wt]such that 〈Si1 ∪ · · · ∪ Sin〉 is connected and minimal. Moreover, by definitionof S, wt ≤ k− 1, so

w ≤ 2 maxt∈[n]wt+ r ≤ r(k− 1) + r ≤ rk. (3.36)

Analogously, |Ht| ≤ k− 1, and note that H = 〈H1∪ · · · ∪Hs〉N a,bm

(see Remark3.4.2), which is connected, so by Lemma 2.3.9,

|H| ≤ λ · 2 maxt∈[n]|Ht| ≤ 2λ(k− 1) ≤ 2λk. (3.37)


Now, let us prove the lower bound in the case c = a + b.

Proof of Proposition 3.4.7. Take L < exp(γp−s), where γ > 0 is some smallconstant. Let us show that Pp(I•([L]3)) goes to 0, as p→ 0. Fix ε > 0.

If [L]3 is internally filled, by Lemma 3.4.8 there exists a covered beam S =H × [w] ⊂ [L]3 satisfying w, |H| ≤ ε/ps, and moreover, either w ≥ ε/2λps or|H| ≥ ε/2λps, hence, by union bound, Pp[I•([L]3)] is at most

∑S∈B ε

ps , εps

(Pp[IV(S) ∩ w ≥ ε/2λps] + Pp[IV(S) ∩ |H| ≥ ε/2λps]).

To bound the first term, we use the fact that H × [w] is covered; thisimplies that there is no gap of size sr along the e3-direction. Therefore, byconsidering the w/sr disjoint slabs (and using the FKG inequality CorollaryB.4.3, if s = 2), if ε is small, then there exists some c1 = c1(ε, r) > 0 such that

Pp[IV(H × [w]) ∩ w ≥ ε/2λps] ≤(

1− e−Ω(ps|H|))w/sr

=(

1− e−Ω(ε))ε/2srλps

≤ e−c1/ps.

To bound the second term, for each S ∈ B εps , ε

psconsider the set

A′ :=

x ∈ [L]2 : |(x × [w]) ∩ 〈A ∩ S〉|O(1) ≥ s

,

where the subindex O(1) in the cardinality symbol means that the verticesparticipating in the intersection are within constant distance, if s = 2.

In other words, x ∈ A′ if and only if there exist y1, ys ∈ x × [w] suchthat ‖y1 − ys‖ = O(1), and either y1, y2 ∈ A or y1 ∈ S got infected by usingat least

r− s + 1 = a + b + 1 = m

infected neighbours in y + Na,b, where Na,b is given by (3.33). Now, by ap-plying Markov’s inequality,

Pp(|A ∩ (x × [w])|O(1) ≥ s) = O(wps) ≤ ε.

Therefore, by monotonicity we can couple the process in [L]2 × [w] havinginitial infected set A, with N a,b

m -bootstrap percolation on [L]2 × 1 ⊂ Z2

and initial infected set ε-random (call it Aε), in this case Pp(x ∈ A′) ≤Pε

(x ∈ 〈Aε〉N a,b

m

).

In particular, under N a,bm there should exists a connected component of

size at least |H| ≥ ε/2λps inside [L]2. On the other hand, there are at mostL2 possible ways to place the origin in H, so by Theorem 2.3.11, if K denotesthe cluster of 0, then


Pp[IV(S) ∩ |H| ≥ ε/2λps] ≤ Pε[|H| ≥ ε/2λps ∩ ∃x ∈ H : x = 0]≤ ∑

x∈[L]2Pε(|K| > ε/2λps ∩ x = 0)

≤ L2Pε(|K| ≥ ε/2λps)

≤ e2γ/pse−Cε/2λps

= e−(Cε/2λ−2γ)/ps,

where C = − 160β2 log ε ∼ − log ε and, we choose ε > 0 such that Cε > 0 and

γ < Cε/4λ at first. By Lemma 3.4.4 we conclude that

Pp[I•([L]3)] ≤ ∑S∈B ε

ps , εps

e−c1/ps+ e−(Cε/2λ−2γ)/ps

≤ ε

ps L3(3e)ε/pse−c2/ps

≤ e4γ/pseε log(3e)/ps

e−c2/ps → 0,

for c2, γ > 0 small enough.

3.4.3 The coarse beams process

In this section we study the last case c ≥ a + b + 1. The lower bound willbe proved by using a coupling with subcritical two-dimensional bootstrappercolation again, as we did in the previous section, however, this time weinfect squares instead of single vertices.

The trick now, is to consider the following coarser process.

Definition 3.4.9 (Coarse bootstrap percolation). Assume that b + 1 dividesL and we partition [L]2 as L2/(b + 1)2 copies of := [b + 1]2 in the ob-vious way and, think about as a single vertex in the new scaled grid[L/(b + 1)]2. Given a two-dimensional family U , suppose we have somefully infected copies of ∈ [L/(b + 1)]2 and denote this initially infected setby A, then, we define coarse U -bootstrap percolation to be the result of ap-plying U -bootstrap percolation to the new rescaled vertices. We denote theclosure of this process as 〈A〉b.

To avoid trivialities, we assume that b + 1 divides L. Consider

m := a + b + 1 < c + s = r.

Definition 3.4.10. A coarse beam is a finite set S ⊂ Z3 of the form S = H× [w],where H ⊂ Z2 is connected and 〈H〉b = H under coarse N a,b

m -bootstrappercolation.


Remark 3.4.11. Given a connected finite set S ⊂ [L]2 × [L], we partition[L]2 as in Definition 3.4.9 and, denote as Bb(S) the coarse beam containingS which is constructed in the (coarse) analogous way, as we did in Remark3.4.2. Note that every coarse beam is a beam in the sense of the previoussection.

The following algorithm is a refinement of that one in Definition 3.4.5.

Definition 3.4.12 (The coarse beams process). Let A = x1, . . . , x|A| ⊂ [L]3,and set B := S1, . . . , S|A|, where Si = xi for each i = 1, . . . , |A|. Fixr ≥ c + 1 and repeat the following steps until STOP:

1. If there exists a family of n ∈ 2, . . . , r beams Si1 , . . . , Sin ⊂ B suchthat

〈Si1 ∪ · · · ∪ Sin〉is strongly connected, and 〈Si1 ∪ · · · ∪ Sin〉 6= Si1 ∪ · · · ∪ Sin , then choosea minimal such family, remove it from B, and replace by the coarsebeam Bb(〈Si1 ∪ · · · ∪ Sin〉).

2. If there do not exist such a family of sets in B, then STOP.

Let us call any beam S = Bb(〈Si1 ∪ · · · ∪ Sin〉) added to the collection B acovered beam, and denote the event S is covered by

IVb (S).

The two highlighted usual properties are preserved for this algorithm too:

• Any covered beam S satisfies 〈A ∩ S〉 ⊂ 〈S〉 = S.

• There is a final collection B′ and we can set V(B′) :=⋃

S∈B′S. Then, we

also have 〈A〉 ⊂ V(B′).

3.4.4 Case VI: c > a + b

In this section we prove the lower bound corresponding to the last case. Toavoid cumbersome notation, we will assume w.l.o.g. that s = 1.

Proposition 3.4.13. UnderN a,b,cc+1 -bootstrap percolation with c > a+ b, there exists

a constant γ = γ(c) > 0 such that, for L < exp(γp−1(log p)2),

Pp[I•([L]3)]→ 0, as p→ 0. (3.38)

Recall from Lemma 2.3.9 the constant λ > 0 associated to the familyN a,bm .

We state the analogue of Lemma 3.4.8 for the coarse beams setting withoutproof because the arguments are exactly the same. We obtain a slightly dif-ferent constant because the number of vertices of the form in a coarse beamH equals |H|/(b + 1)2.



internally filled then, for every h, k = λ, . . . , L, there exists a covered (coarse) beamS = H × [w] ⊂ [L]3 satisfying w ≤ rk, |H| ≤ 2(b + 1)2λh, and either w ≥ k or|H| ≥ h.

Finally, we prove the lower bound in the remaining case.

Proof of Propositon 3.4.13. Take L < exp(γp−1(log p)2), where γ > 0 is somesmall constant. Let us show that Pp(I•([L]3)) goes to 0, as p → 0. Fix δ > 0and set

h = δp−1 log 1p , k = p−

32

If [L]3 is internally filled, by Lemma 3.4.14 there exists a covered beam S =H × [w] ⊂ [L]3 satisfying w ≤ k, |H| ≤ (b + 1)2h, and either w ≥ k/2λor |H| ≥ h/2λ (as we said, the cardinality of H viewing S as a beam equal(b + 1)2|H| viewing S as a coarse beam), hence Pp[I•([L]3)] is at most

∑S∈Bk,(b+1)2h

(Pp[IVb (S) ∩ w ≥ k/2λ] + Pp[IV

b (S) ∩ |H| ≥ h/2λ]).

To bound the first term, we use the fact that A∩ (H×rk + 1, · · · , rk + r) 6=∅ for all k = 0, . . . , w/r − 1, since H × [w] is covered. Therefore, for somec1 > 0,

Pp[IVb (H × [w]) ∩ w ≥ k/2λ] ≤ (1− (1− p)rh)w/r

≤ (1− e−2rε log 1p )k/2rλ

= (1− p2rε)k/2rλ

≤ e−p2rε− 32 /2rλ

= e−c1 p−1(log 1p )

2.

To bound the second term we use the fact that r = c + 1 ≥ a + b + 2. Moreprecisely, if [L]3 is internally filled, then every copy of [b + 1]2 × [L] shouldcontain at least 2 vertices of A within some constant distance (otherwise,there is no way to infect such a copy).

Then, given S = H × [w] ∈ Bk,(b+1)2h consider the set A′ consisting of allcopies of ⊂ [L]2 (as in Definition 3.4.9) such that the rectangle × [w] ⊂S contains at least 2 vertices of A within distance r. By union bound, theprobability of finding such vertices is at most

∑x∈×[w]

∑0<‖y−x‖≤r

Pp(x, y ∈ A) ≤ Cwp2 ≤ p13 .

Therefore, by monotonicity we can couple the process in [L]2 × [w] havinginitial infected set A, with coarse N a,b

m -bootstrap percolation on [L/(b + 1)]2

and initial infected set ε-random with ε = ε(p) := p13 (call it Aε), in this case

Pp( ∈ A′) ≤ Pε ( ∈ 〈Aε〉b).


In particular, under N a,bm (coarse) there should exist a connected compo-

nent of size at least |H| ≥ h/2λ inside [L]2. On the other hand, there are atmost L2 possible ways to place the origin in H, so by Theorem 2.3.11, if Kdenotes the (coarse) cluster of 0, then

Pp[IVb (S) ∩ |H| ≥ h/2λ] ≤ Pε[|H| ≥ h/2λ ∩ ∃x ∈ H : = 0]

≤ ∑⊂[L]2

Pε(|K| > h/2λ ∩ = 0)

≤ L2Pε(|K| ≥ h/2λ)

≤ e2γp−1(log 1p )

2e−Ch/2λ

= e−(c′−2γ)p−1(log 1

p )2,

for some constant c′ = c′(β, λ) > 0 (recall C ∼ − log p by Theorem 2.3.11).Take γ < c′/2 at first; by Lemma 3.4.4 we conclude that

Pp[I•([L]3)] ≤ ∑S∈Bk,(b+1)2h

[e−c1 p−1(log 1

p )2+ e−(c

′−2γ)p−1(log 1p )

2]≤ kL3(3e)(b+1)2he−c3 p−1(log 1

p )2

≤ e4γp−1(log 1p )

2e−c3 p−1(log 1

p )2→ 0,

for c3, γ > 0 small enough and, we are finished.

3.5 Upper bounds for r ∈ c + 3, . . . , c + aIn this section we prove Theorem 3.1.5. When s = r − c > 2, the way ofgrowing is not simple anymore. Now, for each s ≥ 3, we need to set the rightdefinition of an s-pattern, which will be some configuration we need to findin order to grow one step along the easiest direction, and at the same timeminimises the cost of growing.

As we did in the cases s = 1, 2, it is enough to give the best possible lowerbound of Pp(I•([l] × [k])) under N s,s

s -bootstrap percolation. We will workon this 2-dimensional problem first, concerning supercritical families.

Heuristics

Let us start by thinking a little bit about the minimal size of a set that candeterministically grow and infect the whole of Z2. Given s ≥ 3, consider thesubset Bs of [s]2 consisting of the diagonal Ds := (x, s− x) : x ∈ [s− 1] andall the vertices below Ds at even distance of Ds, i.e.

Bs = Ds ∪ (x, y) ∈ [s]2 : s− (x + y) is positive and odd. (3.39)

3.5. Upper bounds for r ∈ c + 3, . . . , c + a 65

Note that |Bs| =⌊

s+12

⌋ ⌈s+1

2

⌉, and 〈Bs〉N s,s

s= Z2. We refer to Bs as the s-board

(see Figure 3.10 below)Under N s,s

s -bootstrap percolation, it is enough to find a copy of Bs inside[l]× [k] to infect the whole of [l]× [k] (or Z2), this gives us

Pp (I•([l]× [k])) ≥ 1− exp(−p|Bs|kl/s2

), (3.40)

which, in the symmetric case I (c = b = a), would give an upper bound (bysetting k = l and adapting the proof of Proposition 3.2.8),

log Lc(N c,c,cc+s , p) = O

(p−|Bs|/2

).

But this is very far from best possible, here is a way to improve it in the cases = 3 (note that |B3| = 4). We can also grow by finding some i ≤ l − 2 suchthat, in the column i × [k] there are 3 adjacent infected vertices, and in thedouble column i + 1, i + 2 × [k] there is an infected 2-pattern (see Figure3.10).

FIGURE 3.10: To the left, a copy of the 3-board. To the right, a3-pattern.

The cost in this case is

Pp(I•([l]× [k])) ≥ 1− exp(−Ω(p5k2l)

), for k ≤ εp−2, (3.41)

and the upper bound for the simplest case c = b = a = s = 3 becomes (aftersome computations)

log Lc(N 3,3,36 , p) = O

(p−5/3

). (3.42)

We believe the latter is the configuration with the least cost; in this case, wesay that a 3-pattern is any copy of the union of 3 adjacent vertices in 1× [k]and 2 adjacent vertices in 2 × [k] (see Definition 3.5.1).

There is a key point about the 3-pattern; the number of infected verticesto find in the last column, namely 2, is bigger than the exponent 5/3, this fact


is necessary to ensure that (3.41) holds for k = l = p−5/3, because after thissize we can grow easier.

With this point in mind, what should be the good definition for a 4-pattern, corresponding to s = 4?

If we choose a configuration given by 4 adjacent vertices in 1 × [k], 3adjacent vertices in 2× [k], and one 2-pattern in 3, 4× [k], then we obtaina cost of 1− exp

(−Ω(p9k3l)

), for k ≤ εp−2, so that we would like to deduce

that for c = b = a = s = 4, log Lc(N 4,4,48 , p) = O

(p−9/4); however, this is not

true, the problem here is that 2 < 9/4.On the other hand, just the choice of 4 adjacent vertices in 1 × [k] and

3 adjacent vertices in 2 × [k] will give us that log Lc(N 4,4,48 , p) = O(p−7/3),

and the last number of adjacent vertices, namely 3, is bigger than 7/3. Again,this fact will ensure that a rectangle with sidelength p−7/3 (even sidelengthp−3) can be internally filled with probability at least exp

(−C′p−7/3), and

after that size it is easy to grow.

The s-patterns, s ≥ 3

Now, we formalize the concepts mentioned above and prove the stated in-equalities.

Definition 3.5.1. An s-pattern is a union of t + 1 sets of vertices:

S0 ∪ S1 ∪ · · · ∪ St,

where t = ts < s is the biggest integer satisfying

|S0|+ |S1|+ · · ·+ |St|t + 2

< s− t, (3.43)

and for each i = 1, . . . , t, Si ⊂ i + 1 ×Z is a copy of 1 × [s− i] (so that|Si| = s− i) in the following restricted sense (recall that e2 = (0, 1))

Si = m(s− i)e2 + i + 1 × [s− i], for some integer m ≥ 0.

We also define

αs :=s + (s− 1) + · · ·+ (s− t)

t + 2. (3.44)

Remark 3.5.2. The restrictions above are made to guarantee that when twos-patterns intersect in some column i, this necessarily implies that they coin-cide in column i. This fact and independence imply that the probability ofexisting a set Si inside i + 1 × [k] is at least

1− exp(−Ω

(p|Si|k

)). (3.45)

Note that inequality (3.43) holds for t = 1, since s + (s − 1) ≤ 3(s − 1),thus, ts is well defined for all s ≥ 3; let us find its value. Such a maximumcan be written as

ts := maxt ∈ [s− 1] : αs < s− t,


and αs < s− t if and only if t2 + 3t− 2s < 0, thus

ts =

⌈√9 + 8s− 5

2

⌉. (3.46)

Finally, we replace to obtain

αs =(ts + 1)s− (ts + 1)ts/2

ts + 2=

(⌈√9+8s−3

2

⌉+ 1) (

s−⌈√

9+8s−32

⌉/2)

⌈√9+8s−3

2

⌉+ 2

,

which is the same as (3.7).

Remark 3.5.3. By the maximality of t = ts, it follows that s− (t + 1) is lessthan (s(t + 2)− (t + 1)(t + 2)/2)/(t + 3), so

(s− (t + 1))(t + 3) ≤ s(t + 1) + s− (t + 1)t/2− (t + 1),

hence, we conclude that

s− t− 1 ≤ αs < s− t.

Thus, αs is integer if and only if αs = s− t− 1, which occurs if and only if

αs =(t + 1)(t + 4)

2.

The next step is to provide a lower bound for the probability of the eventI•([l]× [k]), moreover, we will only need such bounds for values k ≥ Ω (p−αs).Therefore, the strategy will be to consider the values of k in the different inter-vals

[εp−(s−t−1), εp−(s−t)

),[εp−(s−t), εp−(s−t+1)

), . . . ,

[εp−(s−1), εp−s

), and

[εp−s, ∞). This is the main lemma for s ≥ 3.

Lemma 3.5.4 (Supercritical induced process). Fix m ∈ [s]. UnderN s,ss -bootstrap

percolation, there exists δ > 0 such that, if k = Ω(p−m) then

Pp(I•([l]× [k])) ≥ 1−(

1− δs

∏i=m+1

[1− exp

(−Ω

(kpi))])l/s

. (3.47)

If moreover, k ≤ (2/3)p−(m+1), then

Pp(I•([l]× [k])) ≥ 1− exp(−Ω

(ks−mlp∑s

i=m+1 i))

. (3.48)

Proof. Partition the rectangle R = [l] × [k] into l/s copies of R′ = [s] × [k],and note that R is internally filled if we can find s restricted (in the sense ofDefinition 3.5.1) sets

S0 ∪ S1 ∪ · · · ∪ Ss−1


in the rectangle R′ (or any of its disjoint copies), so by Remark 3.5.2 it followsthat

Pp(I•(R)) ≥ 1−(

1−s

∏i=1

[1− exp

(−Ω

(kpi))])l/s

.

Now, if k = Ω(p−m) then

m

∏i=1

[1− exp

(−Ω

(kpi))]≥ δ,

this proves (3.47). Finally, if k ≤ (2/3)p−(m+1), then for every i ≥ m + 1 wehave kpi ≤ 2/3, hence

s

∏i=m+1

[1− exp

(−Ω

(kpi))]≥ Ω

(s

∏i=m+1

kpi

)= Ω

(ks−m p∑s

i=m+1 i)

,

and (3.48) follows by applying 1− q ≥ e−2q for q small.

We have so far obtained the lower bound for the supercritical processinduced along the easiest direction. Now, we proceed to prove all upperbounds of Theorem 3.1.5 in the same order of difficulty. However, most of theproofs are analogous versions of the previous cases s ∈ 1, 2 correspondingto Theorem 3.1.4, thus, we will sketch some of them and only point out whatare the new ideas.

3.5.1 Cases III and IV

Recall that the case V: c ≥ a+ b was covered by Proposition 3.1.8 (see Remark3.1.10).

Next, we move to the remaining upper bounds in Theorem 3.1.5 and be-gin with the cases that do not involve the integration of the functions fd (see(3.20)). From now on we set

s := r− c ∈ 3, . . . , a. (3.49)

Case III: c ∈ b + 1, . . . , a + b− s


N a,b,cc+s ,

with c ∈ Is := b + 1, . . . , a + b− s (here a > s, otherwise this case does notexist). We have to prove the following.

Proposition 3.5.5. Fix c ∈ Is and consider N a,b,cc+s -bootstrap percolation. There

exists a constant Γ = Γ(c) > 0 such that, if L = exp(

Γp−αs(log 1p )

(t+3)/(t+2))

,then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.50)


Note that we still have r ≤ a + b, so that nothing changes along the e1 ande2 (harder) directions, and by Lemma 3.2.2 we know that

Pp

(I•([h + 1]2 × [w])|I•([h]2 × [w])

)≥(

1− e−(c2)−1 p(

c2)wh

)2

(3.51)

We also know the cost of growing along the easiest direction.

Corollary 3.5.6 (Supercritical process). Consider N s,ss -bootstrap percolation.

(a) If p−(s−t−1) ≤ k < εp−(s−t) (in particular, if p−αs ≤ k < p−αs−δ and0 < δ < s− t− αs), then

Pp(I•([k]2)) ≥ 1− exp(−Ω

(pαs(t+2)kt+2

)). (3.52)

(b) If k ≥ p−s, then

Pp(I•([k]2)) ≥ 1− exp (−Ω (k)) . (3.53)

Proof. We use Lemma 3.5.4. For (a) take m = s− t− 1 and l = k (recall thats− t− 1 ≤ αs < s− t). For (b) consider m = s.

The second step, as usual, is the following.


Γp−αs(log 1p )

(t+3)/(t+2))

, h = Cs p−αs(log 1p )

1/(t+2)

for some large constant Cs, and

R1 := [h]2 × [c].

Conditionally on I•(R1), the probability that [L]3 is internally filled goes to 1, asp→ 0.

Proof. Consider the rectangles R2 ⊂ R3 ⊂ R4 ⊂ R5 := [L]3 containing R1,defined by

R2 := [h]2 × [p−(c2)+αs−δ],

R3 := [p−s]2 × [p−(c2)+αs−δ],

R4 := [p−s]2 × [L].

As usual,

Pp(I•([L]3)|I•(R1)) ≥4

∏k=1


By applying (3.51) and Corollary 3.5.6 (a) we obtain


Pp(I•(R2)|I•(R1)) ≥(

1− e−Ω(pαs(t+2)ht+2))p−(

c2)+αs−δ

=

(1− e−Ω

(Ct+2

s log 1p

))p−(c2)+αs−δ

≥(

1− pCs)p−(

c2)+αs−δ

→ 1,

if Cs is large, and

Pp(I•(R3)|I•(R2)) ≥(

1− e−(c2)−1 p(

c2)·p−(

c2)+αs−δ·h

)2p−s

≥(

1− e−p−δ/2)2p−s

→ 1,

and now we can use item (b) to get

Pp(I•(R4)|I•(R3)) ≥(

1− e−Ω(p−s))L→ 1,

since αs < s. Finally, by (3.51) it is clear that Pp(I•(R5)|I•(R4))→ 1.

The proof of Proposition 3.5.5 is straightforward.


Γp−αs(log 1p )

(t+3)/(t+2))

, where Γ is aconstant to be chosen. Consider the rectangle

R :=[Cs p−αs(log 1

p )1/(t+2)

]2× [c] ⊂ [L]3.

As usual, it is enough to show that there exists a constant C′ > 0 such that

Pp(I•(R)) ≥ exp(−C′p−αs(log 1

p )(t+3)/(t+2)

), (3.54)

We fill R in the same way as before: start with [c]3 ⊂ A, and then grow fromRk = [k]2 × [c] to Rk+1, for k = c, . . . , m := Cs p−αs(log 1

p )1/(t+2)

Pp (I•(R)) ≥ pc3m

∏k=c

(1− e−Ω(p(

c2)k))2

≥ pc3+c2m

≥ exp(−C′p−αs(log 1

p )(t+3)/(t+2)

),

for C′ > c3, and we are done.


Case IV: c = a + b− s + m and m ∈ [s− 1]


N a,b,a+b−s+ma+b+m ,

corresponding to the case r = c + s = a + b + m with m ∈ [s − 1]. We areonly considering the cases r ≤ a + c, thus, we assume a ≥ s. Set

M := maxαs, m.

We will show the following.

Proposition 3.5.8. Consider N a,b,a+b−s+ma+b+m -bootstrap percolation. There exists a

constant Γ = Γ(a, b) > 0 such that, if L = exp(

Γp−M(log 1p )

2)

, then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.55)

By Lemma 3.1.16 we have

Pp(I•([h + 1]2 × [w])|I•([h]2 × [w])) ≥(

1− e−Ω(pb+mw))c (

1− e−Ω(pmw))2h

(3.56)Now, we show that if a rectangle R ⊂ [L]3 of a well chosen size is inter-

nally filled, then it can grow and fill [L]3 with high probability.


Γp−M(log 1p )

2)

, h = Cs p−αs(log 1p )

1/(t+2) for somelarge constant Cs, and

R1 := [h]2 ×[C′s p−m log 1

p

].


Proof. Consider the rectangles R2 ⊂ R3 ⊂ R4 ⊂ R5 := [L]3 containing R1,defined by R2 := [h]2 × [p−(r+1)], R3 := [p−2s]2 × [p−(r+1)], with r := b + m,and R4 := [p−2s]2× [L]. The rest of the proof is as always; we apply Corollary3.5.6 to deduce that

Pp(I•(R2)|I•(R1)) ≥(

1− e−Ω(pαs(t+2)ht+2))p−(r+1)

=(

1− pCs)p−(r+1)

→ 1,

if Cs is large. The rest of the proof is straightforward by using (3.56).

Now, we prove the upper bound for the critical length. Note that M <2αs.



Γp−M(log 1p )

2)

, where Γ is a constant

to be chosen. Set w := m(2αs −M)p−m log 1p and consider the rectangle

R :=[Cs p−αs(log 1

p )1/(t+2)

]2× [w] ⊂ [L]3.

We need to show that there exists C′ > 0 satisfying

Pp(I•(R)) ≥ exp(−C′p−M(log 1

p )2)

. (3.57)

In fact, start with Rc := [c]2 × [w] ⊂ A, and then grow from Rk = [k]2 × [w]to Rk+1, for

k = c, . . . , K := Cs p−αs(log 1p )

1/(t+2)

to obtain

Pp (I•(R)) ≥ Pp(Rc ⊂ A)K

∏k=c


≥ p|Rc|K

∏k=c

(1− e−Ω(pb+mw)

)c (1− e−

1m pmw

)2h

≥ p|Rc|pC′1K(

1− p2αs−M)C′′1 K2

≥ e−(c2w+C′1K) log 1

p exp(−Ω(p2αs−M p−2αs(log 1

p )2/(t+2))

)≥ exp

(−C′p−M(log 1

p )2)

,

with all C′s being positive constants depending on s and c.

3.5.2 Cases I and II

Finally, we deal with the cases that involve integration. We first consider theisotropic case.

Case I: c = b = a and r ∈ c + 3, . . . , 2c

In this section, we consider N c,c,cc+s -bootstrap percolation with 3 ≤ s ≤ c. We

will prove the following.

Proposition 3.5.10. ConsiderN c,c,cc+s -bootstrap percolation. There exists a universal

constant Γ > 0 such that, if L = exp (Γp−αs), then

Pp

(I•([L]3)

)→ 1, as p→ 0. (3.58)


The induced process in all three directions is coupled by N s,ss -bootstrap

percolation, and recall that (Corollary 3.5.6) for k ≥ p−s

Pp(I•([k]2)) ≥ 1− exp (−Ω (k)) . (3.59)

This time we need to use the full strength of Lemma 3.5.4, which correspondsto all sizes k ≥ p−(s−t).

Corollary 3.5.11. ConsiderN s,ss -bootstrap percolation and m ∈ s− t, . . . , s− 1.

If εp−m ≤ k < εp−(m+1), then

Pp(I•([k]2)) ≥ 1− exp(−Ω

(p−δ(s,m)/2

)), (3.60)

where δ(s, m) := −m2 + (2s + 3)m− s(s + 1) > 0.

Proof. By setting l = k and applying Lemma 3.5.4 we get

Pp(I•([k]2)) ≥ 1− exp(−Ω

((εp−m)s−m+1p∑s

i=m+1 i))

= 1− exp(−Ω

(p−sm+m2+m ps(s+1)/2−m(m+1)/2

))= 1− exp

(−Ω

(p−δ(s,m)/2

)).

The inequality δ(s, m) > 0 is equivalent to

2s + 3−√

9 + 8s < 2m,

which holds whenever m ≥ s− t.

Now, we can set the size of a rectangle that will grow w.h.p.

Lemma 3.5.12. Set L = exp(Γp−αs) and R1 := [εp−(s−t)]3. Conditionally onI•(R1), the probability that [L]3 is internally filled goes to 1 as p→ 0.

Proof. By (3.59) and Corollary 3.5.11 we have

Pp

(I•([L]3)|I•(R1)

)≥(

1− e−Ω(p−s))L s−1

∏m=s−t

p−(m+1)

∏h=εp−m

Pp(I•([h + 1]3)|I•([h]3))

≥ exp(−2Le−Ω(p−s)

) s−1

∏m=s−t

(1− e−Ω(p−δ)

)p−(m+1)

,

where δ = δ(s, m)/2 > 0, and every factor goes to 1, as p→ 0.

Now, we are ready to show the upper bound.

Proof of Proposition 3.5.10. Set L = exp(Γp−αs), where Γ is a constant to bechosen. Consider the rectangle

R := [εp−(s−t)]3 ⊂ [L]3.


As before, we only need to show that there is a constant C′ > 0, such that

Pp(I•(R)) ≥ exp(−C′p−αs). (3.61)

Recall thats− t− 1 ≤ αs < s− t.

It is enough to consider the (hardest) case s− t− 1 = αs, since we can use thesame idea to deduce the case s− t− 1 < αs (indeed, fewer steps are needed).

Incredibly, when αs = s− t− 1 we can apply Lemma 3.5.4 one more timefor m = s− t− 2 to get the lower bound needed to obtain right exponents.More precisely, since αs is integer, we know that

s =(t + 1)(t + 4)

2,

thus, under N s,ss -bootstrap percolation, if εp−(s−t−2) = K1 ≤ k < K2 =

εp−(s−t−1),

Pp(I•([k]2)) ≥ 1− exp(−Ω

(ks−m+1p∑s

i=s−t−1 i))

= 1− exp(−Ω

(kt+3p

12 (t+1)(t+2)(t+3)

))= 1− exp

(−Ω

(kt+3pαs(t+3)

)).

While, by Corollary 3.5.6 we already computed the following matching ratioin the exponents: if εp−(s−t−1) = K2 ≤ k < K3 = εp−(s−t),

Pp(I•([k]2)) ≥ 1− exp(−Ω

(kt+2pαs(t+2)

)).

On the other hand, by the discussion we had in the heuristics, we know thatthe is a lower bound which holds for all values of k, namely,

Pp(I•([k]2)) ≥ 1− exp(−Ω(p|Bs|k2)

)≥ pCs ,

for some large constant Cs > 0. This implies for R1 := [p−(s−t−2)]3 that

Pp(I•(R1)) ≥ Pp([c]3 ⊂ A)K1

∏h=c

p3Cs ≥ pc3+3Cs p−(s−t−2).


Finally, by setting R2 := [p−(s−t−1)]3 we obtain

Pp(I•(R)) ≥ Pp(I•(R1))Pp(I•(R2)|I•(R1))Pp(I•(R)|I•(R2))

≥ Pp(I•(R1))K2

∏k=K1

(1− e−Ω(kt+3 pαs(t+3))

)3 K3

∏k=K2

(1− e−Ω(kt+2 pαs(t+2))

)3

≥ Pp(I•(R1)) exp(

3∫ ∞

0log(1− e−Ω(zt+3 pαs(t+3))) dz

)× exp

(3∫ ∞

0log(1− e−Ω(zt+2 pαs(t+2))) dz

)≥ e−p−αs+1/2

exp(

Cp−αs

∫ ∞

0log(1− e−yt+3

) dy)

× exp(

Cp−αs

∫ ∞

0log(1− e−yt+2

) dy)

≥ exp(−C′p−αs

),

for some constants C, C′ > 0.

Case II: c = b > a

In this section we cover the last case c = b > a. Consider the families

N a,c,cc+s .

The corresponding upper bound goes as follows.

Proposition 3.5.13. ConsiderN a,c,cc+s -bootstrap percolation with c > a. There exists

a universal constant Γ > 0 such that, if L = exp(

Γp−αs(log 1p )

(t+1)/(t+2))

, then

Pp(I•([L]3))→ 1, as p→ 0. (3.62)

The proof is a combination of all ideas we have already seen, thus, wewill only sketch it.

Sketch of the proof. By following the proof of Corollary 3.5.11, we can see thatunder N s,s

s -bootstrap percolation, if m ≥ s − t, then there exists a constantγ > 0 such that, for all εp−m ≤ w < εp−(m+1) and w1−γ ≤ l ≤ w,

Pp(I•([l]× [w])) ≥ 1− exp(−Ω

(p−δ/2

)), (3.63)

where δ = δ(s, m, γ) := −m2 + (2s + 3− γ)m− s(s + 1) > 0 (apply continu-ity as γ→ 0). This implies that the rectangle

R =[

p−αs(log 1p )−1/(t+2)

]×[εp−(r−c)

]2

is internally filled with probability at least


Pp(I•(R)) ≥ exp(−C′p−αs(log 1p )

(t+1)/(t+2)), (3.64)

and R can grow with high probability.

3.6 Upper bounds for r ∈ a + c + 1, . . . , b + cLet us start by considering the case a = 1.

Lemma 3.6.1. Consider N 1,c,cr -bootstrap percolation with 1 + c < r ≤ 2c, fix

integers l, w ≥ c and set R = [l]× [w]2. If p is small enough, then

Pp

(I•([l]× [w + 1]2)|I•(R)

)≥(

1− e−Ω(pr−clw))2 (

1− e−Ω(pαw))2l

,

whereα := r− c− 1.

Proof. To grow from w to w + 1 (along the e3-direction, say) it is enough tofind r− c (vertically) adjacent infected vertices in the area lw, and α adjacentvertices on each vertical line. We conclude by applying FKG inequality.

Fixs := r− c ∈ 2, . . . , b.

Theorem 3.6.2. As p→ 0,

log Lc

(N 1,c,c

r , p)= O

(p−s+1/2

)(3.65)

The case c > b will be covered later in the next theorem.

Proof: Case a = 1, c = b. It is enough to show that

R := [p−1/2]× [W]2

(where W = eΓp−(r−c−1)) is internally filled with probability at least

exp(−C′p−s+1/2

),

for some constant C′ > 0. In fact, start with R0 := [p−1/2]× [c]2 and applythe previous lemma with l = p−1/2 to get

Pp(I•(R)) ≥ p|R0|W

∏w=c

(1− e−Ω(ps−1/2w)

)2 (1− e−Ω(pαw)

)2p−1/2

≥ exp(

C∫ ∞

0

[log(

1− e−Ω(ps−1/2z))+ p−1/2 log

(1− e−Ω(pαz)

)]dz)

≥ exp(−Ω

(p−α−1/2

∫ ∞

0[− log

(1− e−z)]dz

)),

and we are finished, since −α− 1/2 = −s + 1/2.

3.6. Upper bounds for r ∈ a + c + 1, . . . , b + c 77

For a > 1 things get more complicated, for example, when trying to provethe lemma for a = 2, c = b = 3, we will need to find 2 objects, namely,3 vertical vertices besides 2 vertical vertices (instead of the single object: 2vertices in the area lw). Now, we deal with the general case: fix

s := r− c ∈ a + 1, . . . , b.

We want to show that for r ∈ a + c + 1, b + c, as p→ 0,

log Lc

(N a,b,c

r , p)=

O(

p−(s−a+αa))

if c = b,

O(

p−s log 1p

)if c > b,

(3.66)

The following lemma is a calculus exercise, however it will be importantto determine the exponent of the critical length. Moreover, it somehow justi-fies that the upper bounds in the case r ∈ c + 3, . . . , a + c are best possible.

Lemma 3.6.3. Consider the function g : [a]→ Q given by

g(t) =t

t + 1

(a− t− 1

2

). (3.67)

Then maxt∈[a] g(t) = αa.

Proof. See Appendix A.

When a + c < r ≤ b + c, the induced process along the e1-direction,namely N a,b

r−c, is critical, and a rectangle R := [l] × [W]2 can easily growalong this direction only when W eΓp−(r−c−a)

. Therefore, for c = b, ourstrategy will be to find an optimal value of l and keep it fixed from the be-ginning. Then we will grow along the e2 and e3 directions until we obtain asize W p−(r−c), since after that point we can grow with high probability.

The following is an analogue of Lemma 3.5.4.

Lemma 3.6.4 (Regime a + c < r ≤ b + c). Consider N a,b,cr -bootstrap percolation

with a + b < r ≤ b + c, fix integers l, h, w ≥ c and set R = [l]× [h]× [w]. If p issmall enough, then

(i) for α := r− b− a,

Pp (I•([l]× [h + 1]× [w])|I•(R)) ≥(

1− e−Ω(pr−bw))a (

1− e−Ω(pαw))l

,

(ii) set Rw = [l]× [h]× [w + 1]. For t ∈ [a] and εp−s+t < h ≤ εp−s+t−1,

Pp (I•(Rw)|I•(R)) ≥(

1− exp[−Ω

(ps− t−1

2 l1/th)t]) (

1− e−Ω(ps−ah))l

.

Proof. (i) Similar to the proof of Lemma 3.1.16.


(ii) Suppose that R is completely occupied and partition the two-dimensionalrectangle R′w := [l]× [h]×w+ 1 (so that Rw = R∪R′w) into l/a copiesof R′ = [a]× [h]× w + 1. Note that

Pp (I•(Rw)|I•(R)) ≥ Pp(F1 ∩ F2),

where, F1 is the event that inside some of those copies of R′ we can(vertically) find s adjacent vertices followed by s− 1 vertices followedby · · · followed by s − a + 1 vertices, and F2 is the event that for alli = 1, . . . , l, we can find s− a adjacent vertices in i × [h]× w + 1.It follows that

Pp(F2) ≥(

1− e−Ω(ps−ah))l

,

while

Pp(F1) ≥ 1−(

1−a

∏i=1

(1− e−Ω(ps−i+1h))

)l/a

and following the reasoning used in the proof of Lemma 3.5.4, for everyt ∈ [a] and εp−s+t < h ≤ εp−s+t−1 we deduce

Pp(F2) ≥ 1− exp(−Ω

(p∑t

i=1(s+1−i)lht))

≥ 1− exp(−Ω

(ps+1− t+1

2 l1/th)t)

.

Finally, since F1 and F2 are increasing, we conclude by applying FKG.

Corollary 3.6.5. Consider N a,c,cr -bootstrap percolation with a + c < r ≤ 2c, fix

integers l, w ≥ c and set R = [l]× [w]2. If t ∈ [a] and εp−s+t < w ≤ εp−s+t−1,then for p is small enough,

Pp

(I•([l]× [w + 1]2)|I•(R)

)≥(

1− exp[−Ω

(ps− t−1

2 l1/tw)t])2 (

1− e−Ω(ps−aw))2l

.

Proof. By symmetry, when b = c and h = w, we can improve the bound initem (i) of the previous lemma to the same bound in item (ii).

Finally, we prove (3.66).

Proof of Theorem 3.1.6. 1. Case c = b. It is enough to show that

R := [l]× [W]2

(where W = eΓp−(s−a), and l = l(p) will be optimized) is internally filled

with probability at least

exp(−Ω(p−(s−a+αa))

).

3.6. Upper bounds for r ∈ a + c + 1, . . . , b + c 79

Moreover, by Lemma 3.6.4 (with b = c), we can see that it is enough toshow that

Rs := [l]× [p−s]2

is internally filled with probability at least exp(−Ω(p−(s−a+αa))

).

In fact, start with R0 := [l] × [c]2 and apply the previous corollary toget

Pp(I•(Rs)) ≥ p|R0|p−s

∏w=c

(1− e−Ω(ps−aw)

)2l p−(s−a+1)

∏w=c

(1− exp

[−Ω

(ps− a−1

2 l1/aw)a])2

×a−1

∏t=1

p−(s−t+1)

∏w=p−(s−t)

(1− exp

[−Ω

(ps− t−1

2 l1/tw)t])2

≥ exp(

C∫ ∞

0l log

(1− e−Ω(ps−az)

)dz)

×mint∈[a]

p−s

∏w=c

(1− exp

[−Ω

(ps− t−1

2 l1/tw)t])2

≥ exp(−C′lp−s+a) exp

C′∫ ∞

0log

1− e−Ω

(ps− t−1

2 l1/tw)t dz

≥ exp

(−Ω

[lp−s+a +

(ps− t−1

2 l1/t)−1

]),

where t ∈ [a] minimizes the last product. Moreover, we need to require

lp−s+a = Θ((

ps− t−12 l1/t

)−1)= Θ

(p−s+ t−1

2 l−1/t)

,

so that we can choose l1+1/t = p−a+ t−12 , or

l = p−t

t+1

(a− t−1

2

).

Now, note that t should maximize the function g, so by Lemma 3.6.3we conclude g(t) = αa, and

Pp(I•(Rs)) ≥ exp(−Ω

[lp−s+a]) = exp

(−Ω

[p−αa p−s+a]) ,

as desired.

2. Case c > b. Set L = exp(

Γp−s log 1p

)and

R = [l]× [p−s]× [L].

Note that L exp(

p−(r−c−a)(log 1p )

2)

.


By Lemma 3.6.4, R can grow w.h.p. and fill the whole of [L]3, and R isinternally filled with probability at least

exp(−Ω(p−s log 1

p ))

,

since we can start with [l]3 completely occupied (l a large constant),then grow at the same time along the e2 and e3-directions until

R0 = [l]× [p−s]2,

and finally grow along the e3-direction until R.

81

Chapter 4

Applications

4.1 Universality results for kinetically constrainedspin models in two dimensions

We devote this section to the discussion in some detail of recent universal-ity results obtained by Martinelli, Morris and Toninelli [46] for kineticallyconstrained models (KCM), which are reversible interacting particle systemson Zd with continuous time Markov dynamics (see Section B.5) of Glaubertype. These models have been introduced in physical literature to modelliquid-glass transition and more generally the slow “glassy dynamics” whichoccurs in different systems (see e.g. [56]). In particular, they were devised tomimic the fact that the motion of a molecule in a dense liquid can be inhibitedby the presence of too many surrounding molecules. That explains why, inall physical models, the constraints specify the maximal number of particleson certain sites around a given one in order to allow creation/destruction onthe latter.

The authors of [46] considered two-dimensional KCM with update ruleU , and focused on proving universality results for the mean infection time ofthe origin, in the same spirit as those established in the setting of U -bootstrappercolation (see Theorem 4.1.9 below).

KCM can be viewed as a natural non-monotone and stochastic counter-part of U -bootstrap percolation. The following definition is a special case ofan even more general class of kinetically constrained spin models introducedby Cancrini, Martinelli, Roberto and Toninelli [15].

Definition 4.1.1. Let U be a finite collection of finite subsets of Zd \ 0, andlet q ∈ (0, 1). The U -KCM (or simply KCM) on Zd with density q is definedas follows:

• Each vertex has an independent exponential clock which rings ran-domly at rate 1.

• If the clock at vertex v rings at (continuous) time t ≥ 0, and there ex-ists X ∈ U such that the set v + X is completely empty (or 0), then itsoccupation variable is updated to be empty (or 0) with rate q and to beoccupied (or 1) with rate 1− q.

We will follow the notation used in [46], therefore, exceptionally in thissection, we set

q→ 0, and p := 1− q.

82 Chapter 4. Applications

We let µ denote the Bernoulli product measure with density p

µ = ⊗x∈Z2Ber(p),

and denote the expectation operator w.r.t. this measure by Eµ(·). However,we keep the notation Pq for the product measure⊗x∈Z2Ber(q). We denote byσt(v) the occupation variable at v at time t, and will focus on the followingnatural parameter of the model, a quantity used to characterize the dynamicsof the KCM process.

Definition 4.1.2. The infection time of 0 is the time at which the origin is firstinfected

τ0 := inf

t ≥ 0 : σt(0) = 0

. (4.1)

This stopping time is also called the mean infection time or persistence time.The key question is to determine the divergence of the time scale Eµ(τ0) asq ↓ qc(Zd,U ), where µ is the invariant measure of the process and qc(Zd,U )is the critical probability of the U -bootstrap process on Zd (see Definition2.1.1)

qc(Zd,U ) := infq : Pq(〈A〉U = Zd) = 1. (4.2)

Remark 4.1.3. When d = 2, if U is not subcritical, then qc(Z2,U ) = 0, byTheorem 2.1.3.

4.1.1 The model

We consider the probability space (S, µ), where S = 0, 1Z2. Given σ ∈ S

and x ∈ Z2, we denote by σ(x) ∈ 0, 1 the occupation variable at x, and wewill say that x is “empty" (or “infected") if σ(x) = 0.

Given a two-dimensional update family U = X1, . . . , Xm, the corre-sponding KCM is the Markov process on S associated to the generator L (seeSection B.5.2) that acts on local functions f as

(L f )(σ) = ∑x∈Z2

cx(σ)(µx( f )− f

)(σ), (4.3)

where µx( f ) denotes the expectation of f w.r.t. the variable σ(x), and cx is theindicator function of the event that there exists an update rule X ∈ U suchthat σ(y) = 0 for every y ∈ X + x.

Cancrini, Martinelli, Roberto and Toninelli [15] showed that L is the gen-erator of a reversible Markov process on S, with reversible invariant measureµ, since each update set belongs to Zd \ 0, so the constraints never dependon the state of the to-be-updated site. Moreover, the process started at µ isstationary.

The results of [15] prove that Eµ(τ0) and the relaxation time Trel(q;U ) (seeDefinition 4.1.17) are finite for q > qc

(Zd,U

)and infinite for q < qc

(Zd,U

).

As we said above, the key question is thus to determine the divergence of thetime scales Trel(q;U ) and Eµ(τ0) as q ↓ qc(Zd,U ). We will now illustrate wellknown results in the form of examples, which show that such scalings can bevery different, depending on the nature and properties of the family U .

4.1. Universality results for kinetically constrained spin models in twodimensions 83

Definition 4.1.4. We define the d-dimensional family N dk to be the collection

of all subsets of size k of the set ±e1, . . . ,±ed.

When k ∈ [d], N dk it not subcritical, so that qc(Zd,N d

k ) = 0. On the otherhand, qc(Zd,N d

k ) = 1 for all k > d (see Theorem 1.1.1).

Example 4.1.5 (Nearest neighbours). The KCM corresponding to U = N dk

is called the k-facilitated model (FA-kf), and was introduced by Friedricksonand Andersen [31]. It turns out that for this family, the mean infection timeEµ(τ0) diverges as 1/qΘ(1) when k = 1 [15, 54], and as

exp(k−1) Θ(

q−1/(d−k+1))

when 2 ≤ k ≤ d [48].

Example 4.1.6 (Supercritical model). Another KCM which is extensively stud-ied, was introduced by Jäckle and Eisinger [40], and is called the East model.Its update family is

E d =−e1, , . . . , −ed

.

Its name is justified by the following fact: for d = 1, a vertex can update ifand only if it is the neighbour “to the east" of an empty site. Since this familyis supercritical, qc(Zd, E d) = 0. For d = 1, it is shown that Eµ(τ0) diverges as

exp((

1 + o(1)2 log 2

)(log

1q

)2)as q ↓ 0 (see [21, 2, 15]). Later, the authors of [20] proved a similar scaling inany dimension d ≥ 1.

Motivated by the large diversity of possible scalings of the mean infectiontime, the authors of [46] asked for a very natural “universality” question:

Question 4.1.7. Is it possible to group all possible update families U into distinctclasses, in such a way that all members of the same class induce the same divergenceof the mean infection time as q approaches from above the critical value qc(Zd,U )?

Such a general question has not been addressed so far, even in the physicsliterature: physicists lack a general criterion to predict the different scalings.However, from Chapter 2, we know that this question has being solved fortwo-dimensional U -bootstrap percolation. The update families U were clas-sified by Bollobás, Smith and Uzzell [11] into three universality classes: su-percritical, critical and subcritical, according to a simple geometric criterion.

The authors of [46] took an important step towards establishing a similaruniversality picture for two-dimensional KCM with supercritical or criticalupdate family U . Using a geometric criterion too, they proposed a classifica-tion of the two-dimensional update families into universality classes.

In [48] it was proved that

Eµ(τ0) = Ω(TU ), (4.4)


where TU is defined in the following.

Definition 4.1.8. The typical infection time at density q of an update family Uis defined to be

TU = Tq,U := inf

t ≥ 0 : Pq(0 ∈ At

)≥ 1/2

,

where At is the infected set at time t under U -bootstrap percolation, it wasdefined in (2.1).

4.1.2 Universality results

The techniques developed to prove Theorems 2.1.3 and 2.2.5 in Chapter 2about supercritical and critical update families, also work to determine thescaling as q → 0 of the typical time TU it takes to infect the origin. Moreprecisely, Theorems 2.1.3 and 2.2.5 are equivalent to the following results interms of TU .

Theorem 4.1.9. Let U be a two-dimensional update family. Then, as q→ 0,

a) if U is supercritical thenTU = q−Θ(1);

b) if U is critical and balanced with difficulty α, then

TU = exp(Θ(q−α)

);

c) if U is critical and unbalanced with difficulty α, then

TU = exp(

Θ(q−α[log(1/q)]2

)).

Now, we define the universality classes for KCM with a supercritical orcritical update family, as introduced in [46]. We begin with the supercriticalcase.

Definition 4.1.10. A supercritical two-dimensional update family U is saidto be supercritical rooted if there exist two non-opposite stable directions in S1.Otherwise it is called supercritical unrooted.

Their first main result, provides an upper bound on Eµ(τ0) for every su-percritical two-dimensional update family that is (by the results of [44]) sharpup to the implicit constant factor in the exponent.

Theorem 4.1.11 (Supercritical KCM). Let U be a supercritical two-dimensionalupdate family. Then, as q→ 0,

a) if U is unrooted

Eµ(τ0) ≤ q−O(1) = exp(

O(

log TU))

,

4.1. Universality results for kinetically constrained spin models in twodimensions 85

b) if U is rooted,

Eµ(τ0) ≤ exp(

O(

log q−1)2)= exp

(O(

log TU)2)

.

On the other hand, it was proved in [44] that the upper bounds in Theo-rem 4.1.11 are best possible up to the implicit constant factor in the exponentfor all supercritical update families (note that this follows from (4.4) for un-rooted models).

Now, we consider the critical update families. In this setting the distinc-tion between critical unrooted and critical rooted is more subtle, the upperbounds are given in terms of the difficulty α = α(U ) (recall Definition 2.2.2),and it is necessary to introduce a new parameter β = β(U ), which also playsan important role. Let C denote the collection of open semicircles of S1.

Definition 4.1.12. The bilateral difficulty is defined by

β = β(U ) := minC∈C

maxu∈C

max

α(u), α(−u)

. (4.5)

Remark 4.1.13. For a critical update family the difficulty is finite, but thebilateral difficulty may be infinite, for example, the Duarte model is given bythe critical family

D = −e1, e2, −e1,−e2, e2,−e2,

and β(D) is infinite. In general, α ≤ β ≤ ∞, and β can be infinite even for su-percritical update families (for example, one can embed the one-dimensionalEast model in two dimensions).

Definition 4.1.14. A critical update family U with difficulty α and bilateraldifficulty β is said to be α-rooted if β ≥ 2α. Otherwise it is said to be β-unrooted.

The following is the main result in [46].

Theorem 4.1.15 (Critical KCM). Let U be a critical two-dimensional update familywith difficulty α and bilateral difficulty β. Then, as q→ 0,

a) if U is α-rooted

Eµ(τ0) ≤ exp(

O(

q−2α(

log q−1)4))

= exp(

O(

log TU)2)

;

b) if U is β-unrooted

Eµ(τ0) ≤ exp(

O(

q−β(

log q−1)3))

= exp(

O(

log TU)β/α

).

Note that when U is unbalanced, then the upper and lower bounds givenby Theorems 4.1.15 and 4.1.9 differ by only a single factor of log(1/q) (in theexponent), and the authors of [46] suspect that in this case the lower bound


is correct. The authors of [44] proved a matching lower bound for the Duartemodel (up to a constant factor in the exponent). For all other critical mod-els, however, the best known lower bound is that given by Theorem 4.1.9and (4.4). Moreover, the authors of [46] conjectured the following.

Conjecture 4.1.16. Let U be an α-unrooted, unbalanced, critical two-dimensionalupdate family with difficulty α. Then, as q→ 0,

Eµ(τ0) = exp(

Θ(

q−α(

log q−1)2))

.

We remark that an example of an update family satisfying the conditionsof Conjecture 4.1.16 is the anisotropic model N 1,2

3 .We finish this section by recalling a powerful tool used to prove upper

bounds for the mean infection time.

Definition 4.1.17. We say that C > 0 is a Poincaré constant for a given KCMif, for all local functions f , we have

Var( f ) ≤ CD( f ), (4.6)

where D( f ) = ∑x µ(cxVarx( f )

)is the KCM Dirichlet form of f associated to

L. If there exists a finite Poincaré constant we then define

Trel(q,U ) := inf

C > 0 : C is a Poincaré constant for the KCM

.

Otherwise we say that the relaxation time is infinite.

The most important connection between Eµ(τ0) for the stationary KCMprocess (with µ as initial distribution) and Trel(q,U ) is as follows (see [14]):

Eµ(τ0) ≤Trel(q,U )

q∀ q ∈ (0, 1). (4.7)

The upper bounds of Theorems 4.1.11 and 4.1.15 also hold for Trel(q,U ).Indeed, the authors of [46] first established upper bounds for the relaxationtime, then derived the upper bounds on Eµ(τ0) by using (4.7).

4.2 Fixation for 2-dimensional U -voter dynamics

Given some spin dynamics on Zd, the critical probability for fixation is theinfimum over p ∈ [0, 1] such that fixation at + occurs when the initial statesfor the vertices are chosen independently to be + with probability p and tobe − with probability 1− p. In this last section, we will allow an abuse ofnotation: we will write Pp for the joint distribution of the initial spins in state+ (namely, ⊗x∈ZdBer(p)), and the dynamics realizations.

For the zero-temperature Glauber dynamics of the Ising model, Fontes,Schonmann and Sidoravicius [30] showed that pIs

c (Zd) < 1 (Theorem 1.3.2).

Since, by symmetry between + and −, pIsc (Z

d) ≥ 1/2, we can restate this bysaying that there exists a phase transition.

4.2. Fixation for 2-dimensional U -voter dynamics 87

Recently, Morris [49] generalized these dynamics by considering any up-date family U and defining the U -Ising dynamics (see Section 4.2.1); he con-jectured that for the critical families, this model also exhibits a phase tran-sition. We will focus on the U -voter dynamics (see Section 4.2.2), and showthat in this case, for a wide class of such families, we have a phase transition.

4.2.1 Motivation: The U -Ising dynamics

Let U = X1, . . . , Xm be an arbitrary finite family of finite subsets of Zd \0. Given a configuration in +,−Zd

, we say that X ∈ U disagrees withvertex v ∈ Zd if each vertex in v + X has the opposite state to that of v. TheU -Ising dynamics on Zd with states + and − were introduced by Morris [49]as follows:

• Every v ∈ Zd has an independent exponential random clock with rate1.

• When the clock at vertex v rings, if there exists X ∈ U which disagreeswith v then v flips its state. Otherwise nothing happens.

FIGURE 4.1: The clock at vertex v rings at time τk, then thestate of v is updated by using the two-dimensional family U =X1, X2, X3, where X1 = −e1, e2 is marked with ×, X2 =−2e1,−e2 marked with

√, and X3 = e1, (2, 2), (−2,−2)

marked with ?.

Special cases of these dynamics have been extensively studied, for example,consider the family N d

r defined as the collection of all subsets of size ≥ r of±e1, . . . ,±ed; when U = N d

d this process coincides with the so called zero-temperature Glauber dynamics of the Ising model (sometimes called Metropolisdynamics), see, for example [45].

We define the critical probability for the U -Ising dynamics to be

pIsc (Z

d,U ) := inf

p : Pp(U -Ising dynamics fixate at +) = 1

.

Note that pIsc (Z

d,N dd ) < 1 by Theorem 1.3.2, and the family N d

d is critical.Morris [49] conjectured that for all critical models there exists a phase transi-tion.


Conjecture 4.2.1. For every critical d-dimensional family U , it holds that

pIsc (Z

d,U ) < 1.

For d = 2, we would like to combine techniques of [13] and [30] to proveConjecture 4.2.1, however, it is not straightforward. One of the main difficul-ties of this conjecture is that for the U -Ising dynamics we do not know how toprove that droplets can be eroded in polynomial time (see Definition 4.2.5).For this reason we instead focus on the U -voter model where, as we will see,there is an additional bias in favor of the leading state that we will be able toexploit (see Proposition 4.2.14).

4.2.2 The U -voter dynamics: Main result

Definition 4.2.2. Let U = X1, . . . , Xm be an arbitrary finite family of finitesubsets of Zd \ 0. The U -voter dynamics on Zd with states + and − aredefined as follows:

(a) Every v ∈ Zd has an independent exponential random clock with rate1.

(b) When the clock at v rings at (continuous) time t ≥ 0, the vertex v choosesX ∈ U uniformly at random. If the set v+ X is entirely in state ∗ (where∗ ∈ +,−), then the state of v becomes ∗. Otherwise nothing happens.

For example, when U = ULV := x : x ∈ U for some finite setU ⊂ Zd \ 0, in (b) the vertex v chooses some x ∈ U independently withprobability 1/|U|, and then vertex v immediately adopts the same state asx. This is usually called a linear voter model (see Section B.5.2); of particularinterest is the case where U consists of all 2d unit vectors in Zd. For relatedresults see [43] an references therein.

The generator V of this Markov process acts on local functions f as

V f (σ) = ∑v∈Zd

rv(σ)

m[ f (σv)− f (σ)],

where rv(σ) denotes the number of rules disagreeing with vertex v whenthe current configuration is σ, and σv is the configuration obtained from σ byflipping the state of vertex v. Observe that we have symmetry with respect tothe interchange of the roles of −s and +s for these dynamics, and the systemis monotone, namely, rv(σ) is increasing in σ when σ(v) = − and decreasingin σ when σ(v) = +.

We are interested in the long-term behavior of this system, starting froma randomly chosen initial state, and ask whether the dynamics fixate or not.

Let pvotc (Zd,U ) be the critical probability of the U -voter dynamics on Zd

pvotc (Zd,U ) := inf

p : Pp(U -voter dynamics fixate at +) = 1

. (4.8)


We remark that the families ULV described above are not critical and, in fact,their dynamics do not fixate at + (unless p = 1). For instance, if U consistsof all 2d unit vectors and d ≥ 2, then almost surely∫ t

0 1σs(v) = − dst

→ 1− p,

as t → ∞ (see [22]); but if fixation at + occurred then this ratio should con-verge to 0. However, the critical families have a behavior quite different fromthat of ULV.

Remark 4.2.3. We do not think that fixation at + occurs for families whichare not critical, for instance,

• The families ULV are supercritical and the corresponding dynamics donot fixate (see, for example, [22]).

• The family N dd+1 is subcritical and for the dynamics corresponding to

this family, any translate of 1, 2d that is entirely − at time t = 0 willremain − forever. It could be the case that some vertices will fixate at+ and others at −.

From now on, we set d = 2. To state our result we need some extradefinitions.

Definition 4.2.4. Let D be a T -droplet (see Definition 2.1.6). We say that D isa (T , L)-droplet if its diameter is at most L.

We will always consider subsets T ⊂ S such that D is finite, for instance,when U is critical we can choose at least one such T (see Lemma 4.2.12 be-low). Suppose that every vertex in a (T , L)-droplet D is in state −, and ev-ery vertex outside D is frozen in state + (see Figure 4.2). When we run theU -voter dynamics, one might expect D to become entirely filled with + inpolynomial time in L.

Definition 4.2.5. Let D be a (T , L)-droplet. Assume we start the process withD entirely occupied by states −, and all other states are +. The droplet erosiontime T(D) is the first time when D is fully +.

The droplet erosion time is well defined, because T ⊂ S , so the statesoutside D will never flip (see Figure 4.2), and eventually every state in Dwill become + forever. In probabilistic terms, by recurrence of finite stateirreducible Markov chains it follows that T(D) < ∞ almost surely.

Definition 4.2.6. We say that U is poly-eroding if we can choose a constantc > 1 and a finite set T ⊂ S , such that any (T , L)-droplet D satisfies

Pp(T(D) > Lc) ≤ e−L, (4.9)

for all L large enough. We say moreover that U is (c, T )-eroding.


FIGURE 4.2: 4 stable directions determining a (T , L)-droplet.

The authors of [30] proved that N 22 is (2 + ε,S)-eroding, for any fixed

constant ε > 0 and S = S(N 22 ) = ±e1,±e2. Indeed, they proved that

Pp(T(D) > CL2) ≤ e−γL,

for some positive constants C and γ, and all L large enough. Moreover,numerical simulations suggest the following conjecture to be true, which isquite obvious but hard to prove.

Conjecture 4.2.7. Every critical family is poly-eroding.

Our main theorem is that in dimension d = 2, there exists a phase transi-tion for some critical families (Conjecture 4.2.7 would imply it for all criticalfamilies).

Theorem 4.2.8. If U is a poly-eroding critical two-dimensional family, then

pvotc (Z2,U ) < 1. (4.10)

The following are examples of poly-eroding critical families. Fix a 2-dimensional family U ′ satisfying

(I) ∀X ∈ U ′, X ∩H−e2 6= ∅ and 0 ∈Hull(X) (the convex hull of X),

and let V ,W be 1-dimensional families such that either

(II) 0 < ν− ≤ w+ and w− ≤ ν+, or

(II’) V = ∅ and w+w− > 0,

where ν∗ (resp. w∗) denotes the number of rules of V (resp. W) entirelycontained in Z∗ (Z+ = N and Z− = −N). Then, for every a ∈ Z+, thefollowing induced 2-dimensional family is poly-eroding:

Ua(U ′,V ,W) := U ′ ∪⋃

Y∈VR(a, Y) ∪

⋃Y∈W

R(−a, Y),

where R(a, Y) denotes the rule (a, 0) ∪ (0, y) : y ∈ Y.


An example satisfying (I) and (II) could be V = −3,−2,−1, 2,W = −4, 1, 5, −1, 3, and U ′ = ∅. Another examples are

U1(1,−1, 1, −1, 1, −1) = N 22 \ e1,−e1,

and the family

U1(1,−1,∅, 1, −1) = e2,−e2, −e1, e2, −e1,−e2,

which is the Duarte model (compare with (1.10)).Checking that these families are poly-eroding involves more notation,

thus, the proof will be omitted here, but we will be able to deduce it by usingProposition 4.2.14 and Lemma 4.2.17 below, just by drawing the rules. Oneof the main ingredients in the proof is the fact that every rule in Ua(U ′,V ,W)has at most 1 vertex living in the x-axis, namely, either (a, 0) or (−a, 0);basically for this reason, the same results will serve to conclude that givena1, . . . , ak ∈ Z+, the family

Ua1(U′1,V1,W1) ∪ · · · ∪ Uak(U

′k,Vk,Wk)

is poly-eroding, whenever U ′i ,Vi,Wi satisfy (I) and (II)/(II’) for each i ≤ k.

Outline of the proof

In order to prove Theorem 4.2.8, we will combine techniques of [11] and [30],indeed, we will be able to prove an stronger result, namely, that fixation oc-curs in time with a stretched exponential tail.

Theorem 4.2.9. There exist constants γ > 0 and p0 < 1 such that, for everyp > p0,

Pp[σt(0) = −] ≤ exp(−tγ), (4.11)

for all sufficiently large t.

Let us deduce Theorem 4.2.8 from Theorem 4.2.9.

Proof of Theorem 4.2.8. Fix t ≥ 0 and consider the events

F := σs(0) is constant for s ∈ [t− 1, t],

F′ := ∃s ∈ [t− 1, t] : σs(0) = −.Note that Pp(σt(0) = −) ≥ Pp(F′|F)Pp(F), and that Pp(F) ≥ e−1, so by thestrong Markov property (see Theorem B.5.5) it follows that

Pp(F′) ≤ ePp[σt(0) = −].

Now, by Theorem 4.2.9 and union bound (Theorem B.2.4), for p > p0,


Pp[∃s ≥ t : σs(0) = −] ≤∞

∑m=0

Pp[∃s ∈ [t + m, t + m + 1) : σs(0) = −]

≤∞

∑m=0

e exp(−(t + m + 1)γ)

≤ e∫ ∞

te−sγ

ds

≤ e−tγ/2,

if t is large enough. In particular, if we consider the events Fmm∈N givenby Fm := σs(0) = +, ∀s ≥ m, it follows that for m0 large

∑m≥m0

Pp(Fcm) ≤ ∑

m≥m0

exp(−mγ/2) < +∞.

Thus, by the Borel-Cantelli Lemma (Theorem B.2.5)

Pp[0 fixates at +] = Pp

[⋃i≥1

⋂m≥i

Fm

]= 1.

Hence, pvotc (Z2,U ) ≤ p0 < 1 and we are finished.

At this point, it only remains to show Theorem 4.2.9; the following is ansketch.

Sketch of Theorem 4.2.9. As that proof in [30], we use a multi-scale analysis;this consists of observing the process in some large boxes Bk at some timesTk which increase rapidly with k, and tiling Z2 with disjoint copies of Bk inthe obvious way. This is done by induction on k; T0 = 0 and suppose weare viewing the evolution inside the interval [Tk−1, Tk). In Bk we couple theprocess with a block-dynamics which favors the spins in state − (the − team),in the sense that, when there is some− in Bk at time Tk in the original processthen it is also true for the block-dynamics.

Inside Bk we allow the − team to ‘infect’ the + team via their own boot-strap process (meaning that just spins in state + are allowed to flip). Weprove that by time Tk, every droplet D ⊂ Bk full of−s has ‘relatively big’ sizewith small probability. In other words, such droplets satisfy |D| |Bk| withhigh probability.

Then, we prove that before such droplets D could be created, the + teaminside Bk will typically eliminate it. Moreover, we have to show that theprobability that the − team could receive any help from outside of Bk is alsosmall.

The inductive step goes as follows: at time Tk, if there is some − in Bk,we declare Bk to be a −, otherwise declare Bk to be +, and now, we observethe evolution in a new time interval [Tk, Tk+1). The next step, is to considera larger box Bk+1 consisting of several copies of Bk that we have declared tobe either − or +, and we start over again. By induction on k, we will show


that if q := 1− p is very close to 0, Theorem 4.2.9 holds for all times of theform t = Tk. Finally, by using one more coupling trick, we can extend thestatement for all t ≥ 0.

The rest of this chapter is devoted to the full proof of Theorem 4.2.9, andis organized as follows. In Section 4.2.3 we describe the 1-dimensional ap-proach, which is a restriction of our dynamics to a finite segment, and explainhow to deduce the main theorem from this setting. Moreover, we presentsome examples and technical difficulties that arise when trying to prove it inthe general setting of critical families. In Section 4.2.4 we couple the U -voterdynamics with a block-dynamics and show that the probability that thereexist −s spins in each block decreases sufficiently fast as time increases. InSection 4.2.5 we prove that the probability that a block is ‘influenced’ by non-neighbors blocks, before it becomes entirely +, is sufficiently small, and finally,put the pieces together.

4.2.3 The 1-dimensional approach

In this section we prove that once we can find an stable direction fair enoughthen we can prove that the family is poly-eroding by using a 1-dimensionalargument. We will consider a particular restricted evolution of the dynamicsin dimension 1 by freezing everything except a finite segment orthogonal tosome stable direction, then prove that such a segment can be eroded in poly-nomial time and show how things can be deduced from this 1-dimensionalsetting.

A fair stable direction

Fix y = (y1, y2) ∈ S a rational direction, it means either y2/y1 is rational ory1 = 0. For each L ∈ N, we let $ = $(y, L) be any fixed segment consistingof L consecutive vertices in the discrete line x ∈ Z2 : 〈x, y〉 = 0.

Suppose we freeze each vertex in H2y in state − and each vertex outside

H2y ∪ $ in state +, and at time t = 0 each vertex in $ has state −, then we let

the dynamics evolve only on $ (see Figure 4.3).

FIGURE 4.3: H2y entirely − and Z2 \ (H2

y ∪ $) entirely +

Given a configuration η ∈ +,−$ denote η+ (resp. η−) the set of verticesin η having + (resp. −) state.


Definition 4.2.10. 1. Fix y ∈ S , L ∈ N, and consider $ = $(y, L). Forevery t ≥ 0 we let ηt denote the configuration in +,−$ at time t inthese restricted 1-dimensional dynamics.

2. Say that y ∈ S is a fair direction if it is rational, and in the 1-dimensionaldynamics, for each L ∈N and t ≥ 0 the following holds

∑v∈η+

t

rv(ηt) ≤ ∑u∈η−t

ru(ηt). (4.12)

Denote by [−] (resp. [+]) the configuration in +,−$ where all verticesare in state − (resp. +), and observe that Condition (4.12) is trivial for t = 0since η0 = [−] and this gives LHS in (4.12) equals 0. Moreover, when t islarge ηt = [+] (the segment fixates at [+]) and rv([+]) = 0 for all v (sincey ∈ S), hence LHS = 0 too.

Note that Condition (4.12) is implied by the stronger condition

∑v∈η+

rv(η) ≤ ∑u∈η−

ru(η), for all η ∈ +,−$, (4.13)

which does not depend on the trajectory of the 1-dimensional dynamics.However, in general we do not know whether they are equivalent.

Our aim now is to show that the existence of a fair direction is a sufficientcondition for a family to be poly-eroding. Given a fair direction y, we areinterested in the segment erosion time

τ = τ(y, L) := inft : ηt = [+]. (4.14)

Here is the core of the 1-dimensional approach.

Proposition 4.2.11. If there is a fair direction y, then there is a constant c > 3 suchthat for L large enough we have P[τ > Lc−1] ≤ e−1.

We move the proof of this proposition to the next section. This result takesaccount of the constant c > 1 in Definition 4.2.6, so, it is left to choose a goodsubset of the stable set. This is the content of the next lemma.

Lemma 4.2.12. Given y ∈ S , there exists a finite set S4 ⊂ S such that y ∈ S4 and0 ∈ Hull(S4).

Before proving this lemma, we introduce some useful information aboutthe structure of S . Write [u, v] for the closed interval of directions between uand v (also (u, v) for the open interval). Say [u, v] is rational if both u and v arerational directions. Our choice of S4 will depend on the following lemma.

Lemma 4.2.13. The stable set S is a finite union of rational closed intervals of S1.

Proof. See [11].

With this tool, now we prove that in fact the set S4 in Lemma 4.2.12 canbe chosen of size 3 or 4 (thus, justifying the subindex 4).


Proof of Lemma 4.2.12. Let y ∈ S4 by definition. If −y ∈ S since U is criticalwe can choose x ∈ (y,−y) ∩ S , z ∈ (−y, y) ∩ S and set S4 = x, y,−y, z.

If −y /∈ S , then take x ∈ S in the open semicircle opposite to y, we cansuppose wlog that x ∈ (y,−y).

FIGURE 4.4: 3 or 4 stable directions

Moreover, since S is closed by Lemma 4.2.13, we can choose x such thatS ∩ [x,−y] = x (see Figure 4.4). Then select z ∈ S ∩ (x,−x) and observethat in fact z ∈ S ∩ (−y,−x), so define S4 = x, y, z. In both cases, 0 ∈Hull(S4).

By combining the previous results, we can prove that every family withfair directions is poly-eroding.

Proposition 4.2.14. If there is a fair direction, then there exist a constant c > 1,and a set S4 ⊂ S such that for every (S4, L)-droplet D and every t ≥ 0,

Pp[T(D) > tLc−1] ≤ Le−t, (4.15)

when L is large enough. In particular, U is (c + ε,S4)-eroding

Proof. To fix ideas, we can assume that y = (0, 1) is a fair direction. Considerthe set S4 containing y, given by Lemma 4.2.12 and any (S4, L)-droplet D.Note that D is finite because 0 ∈ Hull(S4), hence T(D) is well defined.

We couple the dynamics with the following one: We first allow to flip justthe vertices in the first (top) line of the droplet, then, when they are all instate +, we allow to flip just the vertices in the second line, and so on untilwe arrive at the bottom line. This coupled dynamic dominates the originalone by monotonicity, and since the height of the droplet is at most L, then itis enough to show that for all t ≥ 0,

Pp[Ttop > tLc−1] ≤ e−t, (4.16)

where Ttop is the time to erode the top line in the coupled dynamics; we finishthe proof by applying the union bound (Theorem B.2.4) over all rows of D.

Moreover we can assume that this top line has L vertices, since all lineshave at most L vertices and having less vertices only helps to erode faster.To this end, let us consider the 1-dimensional process ηt given in Definition4.2.10; because of the boundary conditions, it follows that Ttop ≤ τ in distri-bution, thus, by Proposition 4.2.11,

Pp[Ttop > Lc−1] ≤ P[τ > Lc−1] ≤ e−1.


Finally, by the Markov property (see Theorem B.5.3) it follows (4.16).

4.2.3.1 A martingale argument

In this section we prove Proposition 4.2.11. To do so, we use Markov’s in-equality (see Proposition B.2.3)

P[τ > s] ≤ E[τ]/s.

The first step is to show that, if we can find a function on +,−$ providinga bias in favor to the vertices in state + in the dynamics ηt (see Definition4.2.10), then we can bound E[τ] in terms of f and the extreme configurations[−] and [+].

Since $ = $(y, L) has L vertices can identify it with the initial segment [L],so we can write the generator for the 1-dimensional process ηt as

V f (η) =L

∑v=1

rv(η)

m[ f (ηv)− f (η)].

Lemma 4.2.15. Suppose there exists a function f : +,−L → R such thatV f (ηt) ≤ −1 for all t < τ, then

E[τ] ≤ f ([−])− f ([+]). (4.17)

Proof. Consider the martingale Mt = f (ηt)−∫ t

0 V f (ηs) ds (see Theorem B.5.10).By optional stopping (Theorem B.5.9) we have

f ([−]) = E[ f (η0)] = E[M0]

= E[Mτ]

= E

[f (ητ)−

∫ τ

0V f (ηs) ds

]≥ E[ f (ητ)] + E

[∫ τ

01 ds

]= f ([+]) + E[τ],

and the result follows.

In order to apply this result, the next step is to show that if y is a fairdirection then we can define an explicit function f such that the variationf ([−])− f ([+]) is polynomial in L.

Proposition 4.2.16. If y is a fair direction then there exists a function f satisfyingthe hypothesis in Lemma 4.2.15 such that RHS in (4.17) is O(L2).

Proof. Set h0 = 0 and for k = 0, 1, . . . , L− 1 consider the sequence

hk+1 = hk + (L− k)m.

Then, define the function f as follows: given η ∈ +,−L with k = k(η)vertices in state − we set f (η) = hk. Observe that


f ([−])− f ([+]) = hL − h0 =L−1

∑k=0

[hk+1 − hk]

=L−1

∑k=0

(L− k)m

= O(L2).

Moreover, given η = ηt with k = k(η) ≥ 1 vertices in state − we have

m[V f ](η) =L

∑v=1

rv(η)[ f (ηv)− f (η)]

= − ∑v∈η−

rv(η)[ f (η)− f (ηv)] + ∑v∈η+

rv(η)[ f (ηv)− f (η)]

= − ∑v∈η−

rv(η)[hk − hk−1] + ∑v∈η+

rv(η)[hk+1 − hk]

= − ∑v∈η−

rv(η)[m(L− (k− 1))] + ∑v∈η+

rv(η)[m(L− k)]

≤ − ∑v∈η−

rv(η)m(L− (k− 1)) + ∑v∈η−

rv(η)m(L− k)

= −

∑v∈η−

rv(η)

m

≤ −m.

So V f (η) ≤ −1 for all η 6= [+] and we are done.

Finally, we are ready to conclude.

Proof of Proposition 4.2.11. If there is a fair direction y then apply Proposition4.2.16 and then Lemma 4.2.15 to get

E[τ] = O(L2) ≤ e−1Lc−1,

for any constant c > 3, and for L large enough, so by applying Markov’sinequality we are finished.

4.2.3.2 Examples

It is easy to show that N 22 and Duarte families verify Condition (4.12). In

fact, y = e2 is a fair direction for both families, and all configurations ηt inthe 1-dimensional dynamics are of the form

ηt = [+, · · · ,+,−, · · · ,−,+, · · · ,+],


this means, a block of +s followed by a block of −s followed by anotherblock of +s; the right-most block of +s being empty for the Duarte family,and either block of +s (left or right) could be empty for N 2

2 .To fix ideas, let us consider the familyN 2

2 and assume that only the right-most block is empty, thus

∑v∈η+

t

rv(ηt) = 1, and 2 ≤ ∑u∈η−t

ru(ηt).

The other configurations and Duarte family can be checked in a similar way.Therefore, (4.12) holds.

Now, we give a criterion (that only depends on the family U ) which willallow us to check if there exists a fair direction just by drawing the rules inZ2. For y ∈ S consider the line ly := x ∈ Z2 : 〈x, y〉 = 0.

Lemma 4.2.17 (Drawing). Suppose there exists a rational direction y ∈ S suchthat

(a) Every X ∈ U has at most 1 vertex in ly.

(b) Fix a ∈ ly. For each X ∈ U such that X ⊂ Hy ∪ a there exists X′ ∈ Usuch that X′ ⊂H−y ∪ −a, and the map X 7→ X′ is injective.

Then y is a fair direction.

Observe that Condition (b) holds for symmetric families (i.e., when X ∈ Uimplies −X ∈ U ), because X′ = −X works.

Proof. In fact, fix L ∈ N, and consider any configuration in η ∈ +,−$. Wewill show that

∑v∈η+

rv(η) ≤ ∑u∈η−

ru(η). (4.18)

If (v, X) ∈ η+ × U is counted in LHS of (4.18), this is because v ∈ η+ and Xdisagrees with v, hence v + X is entirely −. Moreover, since y ∈ S , the setv + X must have a vertex u ∈ η−, and only 1 by (a), so v + X ⊂ Hy ∪ u orX ⊂Hy ∪ u− v.

Now, by (b) ∃X′ ∈ U with X′ ⊂ H−y ∪ v− u, or u + X′ ⊂ H−y ∪ v,thus, u + X′ in entirely +, meaning that X′ disagrees with u and (u, X′) ∈η−×U is counted in RHS of (4.18). Since the map (v, X) 7→ X′ is an injection,for any pair (v, X) that contributes 1 in LHS we can find a contribution of 1in RHS in a one to one way, so inequality (4.18) follows.

As we said above, by using this lemma and drawing the rules of the fam-ilies Ua(U ′,V ,W), it becomes evident that e2 is a fair direction. It looks likesuch families (or their rational rotations) are the only examples we can givesatisfying (a) and (b). However we are free to construct a lot of examples ofdifferent nature by using the following trivial observation.

Remark 4.2.18 (Adding rules). Infinitely many families with a fair directiony can be constructed from a single family U , just by properly adding new


rules X ⊂ H−y. Moreover, if U critical, then the new families can be chosencritical as well, we just need to add new rules carefully, without modifying(too much) the stable set (see Remark 4.2.19).

No fair directions

For a concrete example of critical families without fair directions consider thecollection of all subsets of size 3 of

±e1,±e2,±2e1,±2e2,

call it U3,8. It is easy to check that S(U3,8) = ±e1,±e2. To check that nodirection in S(U3,8) is fair, by symmetry it is enough to consider y = e2; inthe 1-dimensional setting, calculations show that configurations η of the form

η = [−,+,+,+,+,−,+,+,+,+,−,+,+,+,+,−, · · · ,+,+,+,+,−],

which alternate four +s and one −, do not satisfy (4.12). However, simu-lations indicate that we can erode the segment perpendicular to y in timeO(L2.2), which would mean that S(U3,8) is poly-eroding.

Another such family, which is special since its droplets are triangular, is

UB = (−1, 1), (−1,−1), (0, 1), (1, 1), (0,−1), (1,−1),

with S(UB) =−e1, 1√

2(1, 1), 1√

2(1,−1)

(see Figure 4.5).

FIGURE 4.5: A family without fair directions, its stable set anda failing configuration

This family only has 3 candidates to be y and all of them fail Condition(4.12). In fact, say we choose y = −e1 and take L = 2n + 1 for some fixed n.The configuration η given by

η+ = (0, 2k) : k = 0, ..., n,

yields ∑u∈η− ru(η) = n, while ∑v∈η+ rv(η) = 2n. An analogous situationhappens for the other 2 candidates.


On the other hand, simulations suggest that in fact we can erode the seg-ment perpendicular to y = −e1 in time O(L4/5).

As a last example, we illustrate how to construct a poly-eroding familyfrom UB, in the sense of Remark 4.2.18.

Remark 4.2.19. The direction y = −e1 is fair for the family

UB ∪ (−1, 2), (−1,−1),

this is just because we added a rule in the + side to compensate inequality(4.18) since the it has the same stable set as UB then it is critical and poly-eroding so our main theorem holds for this new family.

In general, if we consider any rule X0 ⊂ He1 , X0 6= (−1, 1), (−1,−1)such that there exist x, x′ ∈ X0 with x2 ≥ −x1 and x′2 ≤ x′1, we can check that−e1 is a fair direction for the family UB ∪ X0, and the latter has the samestable set as UB. Of course, we can construct infinitely many such families inthis way.

4.2.4 The process inside rectangles

We move to the proof of Theorem 4.2.9, let us fix a (c, T )-eroding criticalfamily U . The strategy starts by constructing a rapidly decreasing sequenceqkk with q0 = q = 1− p, and study the process in big space and time scales.More precisely, set l0 = 1, t0 = 0, and define for k ≥ 1 the sequences

qk := exp(−a/qk−1), (4.19)

lk :=⌊

1qk−1

⌋(1+2δ)c, tk :=

(1

qk−1

)(1+δ)c, (4.20)

Lk :=k

∏i=0

li, Tk :=k

∑i=0

ti, (4.21)

and the squaresBk := [Lk]

2. (4.22)

At time Tk we tile Z2 into copies of Bk in the obvious way, then, couple the U -voter dynamics with a block-dynamics (which is more ‘generous’ to the spinsin state −), defined as follows: for every k ≥ 0,

• at time Tk every copy of Bk is monochromatic and the U -voter dynamicsafresh; until it arrives at time Tk+1.

• as t is close to Tk+1, if there exists some copy of Bk inside some copy ofBk+1 which is in state −, then at time Tk+1 we declare the state of Bk+1to be −. Otherwise we declare it to be +.

Definition 4.2.20. Define qk as the probability that at time Tk the block Bk isin the state − in the block-dynamics.

Note that q0 = q0. The following is the core of the proof.


Proposition 4.2.21. If q is small enough, then qk ≤ qk.

We proceed by induction. Assume it holds for k and let us prove it fork + 1. Consider the evolution inside B′k+1 with + boundary conditions duringthe interval [Tk, Tk+1) and define the event

Fk+1 := ∃− ∈ Bk+1 as t Tk+1. (4.23)

In the next two subsections we will prove the

Lemma 4.2.22. Pp(Fk+1) ≤ qk+1/2.

4.2.4.1 Bootstrapping the vertices in state −

Set n = lk+1 and identify each vertex in [n]2 with a copy of [Lk]2. Consider U -

bootstrap percolation on [n]2 by setting the initially infected set A to consistof all vertices in state −. Then run the covering algorithm (see Definition2.1.7), by declaring the infected sites to be those in state − (thus, spins +become−), until we stop with a finite collection of droplets, say D1, . . . , Dz,each one entirely −. Consider the event

E =

diam(Di) ≤ ε2n1/(1+2δ)c, for all i = 1, . . . , z

, (4.24)

where ε > 0 is the constant given by Lemma 2.1.9.

Lemma 4.2.23. Pp(Ec) ≤ qk+1/4.

Proof. If some Di has diameter bigger than ε2n1/(1+2δ)c, then by Lemma 2.1.8there exists a covered droplet D with ε2n1/(1+2δ)c ≤ diam(D) ≤ 3ε2n1/(1+2δ)c.If Z denotes the number of such droplets when A is qk-random, then byMarkov’s inequality we have

Pp(Ec) ≤ Ep[Z] ≤3ε2n1/(1+2δ)c

∑s=ε2n1/(1+2δ)c

n3(

s2

εs

)qεs

k

≤3ε2n1/(1+2δ)c

∑s=ε2n1/(1+2δ)c

n3( esqk

ε

)εs

≤ n33ε2n1/(1+2δ)c

∑s=ε2n1/(1+2δ)c

(3eε)εs

≤ Cn3 exp(−ε2n1/(1+2δ)c

)≤

exp(−2an1/(1+2δ)c

)4

,

(here we used n1/(1+2δ)cqk ≤ 1 and picked a < ε2/2), by Lemma 2.1.9, sincethe number of droplets in Z2

n with diameter s is O(n2+1/(1+2δ)c), and eachhas area at most s2. Finally, note that exp(−2an1/(1+2δ)c) ≤ qk+1.


Lemma 4.2.24. If a small enough, then

Pp(Fk+1|E) ≤ qk+1/4 (4.25)

We move the proof of this statement to the next subsection. Now, it iseasy to deduce Lemma 4.2.22.

Proof of Lemma 4.2.22. Note that P(Fk+1) ≤ Pp(Fk+1|E) + Pp(Ec) and applyLemmas 4.2.23 and 4.2.24.

4.2.4.2 Erosion step

To get (4.25) we use the poly-eroding property of U . First, we recall an upperbound of Lk as a function of qk, which was computed in [30].

Lemma 4.2.25. If q is small enough then the sequence qkk is decreasing and forarbitrary δ > 0,

Lk ≤ (1/qk)δ , (4.26)

uniformly in k.

Proof. See equation (4.8) from [30].

We need to estimate the probability that starting at time Tk from a config-uration in Bk+1 where E holds and letting the system evolve with + boundaryconditions, some spin−will be present as t is close to Tk+1. An upper boundis obtained by starting the evolution at time Tk with − spins at all sites of thedroplets D1, . . . , Dz participating in E.

Proof of Lemma 4.2.24. If E occurs, by (4.26), for small q each droplet has di-ameter at most

ε2l1/(1+2δ)ck+1 · Lk ≤ ε2 1

qk

(1qk

)δ

≤(

1qk

)1+δ

,

hence, each Di is a(T , (1/qk)

1+δ)-droplet. Since U is (c, T )-eroding, if q

is small enough, for each i = 1, . . . , z, the probability that at time Tk+1 =

Tk + (1/qk)(1+δ)c there is any spin − inside Di is at most

Pp

[T(Di) >

(1qk

)(1+δ)c]≤ exp

(−(

1qk

)1+δ)

.

For small q we also have z ≤ |B′k+1| ≤ [(5/3)Lk+1]2 ≤ 1/qk+1, by (4.26)

again. Therefore

Pp[Fk+1|E] ≤1

qk+1exp

(−(

1qk

)1+δ)≤ qk+1

4,

for q small.


4.2.5 Wrapping up

In this last section, we finish the proof of Theorem 4.2.9.

Control of the outer influence: Proof of Proposition 4.2.21

Let B′k be the block with the same center as Bk but of sidelength 53 Lk.

Definition 4.2.26. We let the process P to be the U -voter dynamics run onlyon the induced graph Z2[B′k+1] with + boundary conditions.

The strategy now will be the following: we will prove that the probabilitythat by time Tk+1 the state of every vertex in Bk+1 in the original dynamicsdiffers from the process P is small. We will do this by arguing that the prob-ability of having some − inside Bk+1 with the help of some vertex outsideB′k+1 on time [Tk, Tk+1) is small. Then, by using Lemma 4.2.22, we will de-duce Proposition 4.2.21. Finally, we put all the pieces together and deduceTheorem 4.2.9.

Definition 4.2.27. We call a sequence (x1, s1), . . . , (xr, sr) of vertex-time pairs,where xi ∈ Z2 and si ≥ 0, a path of clock rings (and say that such a sequenceis a path from x1 to xr in time [s1, sr]) if

1. 0 < ‖xi+1− xi‖1 ≤ C for each i ∈ [r− 1], where C = maxX∈U‖x‖ : x ∈ X.

2. s1 < · · · < sr.

3. The clock of vertex xi rings at time si for each i ∈ [r].

The key point now is that if there does not exist a path of clock-rings fromx1 to xr in time [s1, sr], then the state of vertex xr at time sr is independent ofthe state of vertex x1 at time s1.

Lemma 4.2.28. If F′k+1 is the event that there exists a path of clock-rings from somevertex outside B′k+1 to some vertex inside Bk+1 in time [Tk, Tk+1], then

Pp(F′k+1) ≤ qk+1/2. (4.27)

With this estimate, we are ready to finish our induction step.

Proof of Proposition 4.2.21. If F′k+1 does not occur, then the state of every ver-tex in Bk+1 at time Tk+1 is the same in the U -voter dynamics as it is in theprocess P , since the boundary conditions cannot affect Bk+1. This givesqk+1 ≤ Pp(Fk+1) + Pp(F′k+1), and the result follows from Lemmas 4.2.22 and4.2.28.

It remains to show (4.27).

Proof of Lemma 4.2.28. If F′k+1 occurs, every such a path have length at least

rk :=⌊

13C

Lk+1

⌋≥ 1

6C

(1qk

)(1+2δ)c.


By (4.26), for each r ∈ N, the number of paths of length r starting on theboundary of B′k+1 is at most

O(

Lk+1(4C2)r)≤ 1

qk+1(4C2)r

Let Pk(r) be the probability that there exist times Tk ≤ s1 < · · · < sr ≤Tk+1 such that (x1, s1), . . . , (xr, sr) is a path of clock rings. Observe that Pk(r)does not depend on the choice of the path, since all clocks have the samedistribution. We have to bound Pk(r).

Set s0 = Tk and for every m ∈ [r] choose sm to be the first time the clockat xm rings after time sm−1. Let Gm be the event that sm − sm−1 ≤ 2tk+1/r,so Pp(Gm) = 1− exp(−2tk+1/r) ≤ 2tk+1/r, and the events Gm are indepen-dent, therefore

Pk(r) = Pp(sr ≤ Tk+1)

≤ Pp

(r

∑m=1

1Gm ≥ r/2

)

≤(

rr/2

)(2tk+1

r

)r/2

≤(

4etk+1

r

)r/2

.

Finally, observe that for r ≥ rk we haver

tk+1≥ 1

6C1

qδck

so

Pp(F′k+1) ≤∞

∑r=rk

1qk+1

(4C2)r(

4etk+1

r

)r/2

≤ 1qk+1

∞

∑r=rk

(O(qδc

k ))r/2

≤ 1qk+1

exp

[−Ω

(1qk

)(1+2δ)c]

≤ qk+1/2,

since c > 1.

All together now

In this last section we will finish the proof by showing that

Pp[σt(0) = −] ≤ exp(−tγ), (4.28)

for all t > 0.


Proof of Theorem 4.2.9. By Proposition 4.2.21, for all times Tk we already have

Pp(σTk(0) = −) ≤ qk ≤ qk = exp(−at1/(1+δ)c

k

),

and tk−1/tk = (qk−1/qk−2)(1+δ)c ≤ c′ for some constant c′ < 1, so, by defini-

tion, Tk ≤ (1− c′)−1tk, hence

Pp(σTk(0) = −) ≤ exp(−T1/(1+2δ)c

k

).

Therefore, Theorem 4.2.9 holds for all times of the form t = Tk, with anyγ ≤ 1/(1 + 2δ)c. To conclude that it holds for all t > 0, we use the samecoupling trick used in [30], which consists of comparing evolutions startedfrom product measures with different values of q.

Let us rewrite qk, tk, Tk as qk(q), tk(q), Tk(q) because they depend on theinitial q. We have shown that there exists some b > 0, such that for every0 < q ≤ b, when t = Tk(q) it holds that

Pp(σt(0) = −) ≤(−t1/(1+2δ)c

). (4.29)

Now write bk = qk(b), uk = tk(b) and Uk = Tk(b), and observe that bk de-creases (so uk increases) as k increases. For fixed k, consider the parameter qdecreasing continuously from b to b1, so the corresponding Tk(q) increasingcontinuously from Tk(b) to Tk(b1) = t1(b1) + · · ·+ tk(b1) = Uk+1 − u1.

By continuity of Tk(q) and the intermediate value theorem, any t outsideI :=

⋃k≥1

[Uk − u1, Uk) can be written as t = Tk(t)(q[t]), for some k(t) ≥ 1,

b1 < q[t] ≤ b. Observe that p = 1− q ≥ 1− b1 ≥ 1− q[t] =: p[t].Combining monotonicity and (4.29) we get

Pp(σt(0) = −) ≤ Pp[t](σt(0) = −) ≤ exp(−t1/(1+2δ)c

),

and now, we have shown that the theorem holds for all q < b1 and t /∈ I.Finally, suppose t ∈ [Uk − u1, Uk) for some k. The key observation is the

following: if σt(0) = − and the spin at the origin does not flip between timest and Uk then σUk(0) = −.

Pp[σt(0) = −] ≤ eu1Pp[σUk(0) = −]

≤ eu1 exp(−U1/(1+2δ)c

k

)≤ exp

(−t1/(1+3δ)c

),

since the probability that no flips occur at the origin from t to Uk is at leaste−u1 . Thus, γ = 1/(1 + 3δ)c works for all t ≥ 0.

107

Appendix A

Proof of Lemma 3.6.3

In this appendix we use calculus to compute the maxima of the function g inLemma 3.6.3.

Consider the extension f : [1, a] → R of the function g to the closedinterval [1, a]:

f (t) =t

t + 1

(a− t− 1

2

)=

(a + 1)tt + 1

− t2

. (A.1)

The first and second derivatives of f are

f ′(t) =a + 1

(t + 1)2 −12

,

f ′′(t) =−2(a + 1)(t + 1)3 < 0.

Therefore f is a concave function on [1, a], maximized at the only θ satisfyingf ′(θ) = 0;

f ′(θ) = 0 ⇐⇒ (θ + 1)2 = 2(a + 1) ⇐⇒ θ =√

2(a + 1)− 1.

thus, the restriction of f to the discrete set (t ∈ [a]) is maximized at θ, ifθ ∈ [a]. Now, assume that θ /∈ Z, so f is maximized at either

t := bθc

or t + 1. Computations give us

f (t) ≥ f (t + 1) ⇐⇒ t3 + 3t− 2a ≥ 0 ⇐⇒ t ≥√

9 + 8a− 32

.

Defineδ = δ(a) := 1

2(√

9 + 8a−√

8 + 8a),

and observe that

θ ≤ t + 12 − δ ⇐⇒ t ≥ δ + θ − 1

2 =√

9+8a−32 .

Since t < θ < t + 1, we deduce that

108 Appendix A. Proof of Lemma 3.6.3

• f (t) ≥ f (t + 1), for θ ∈ (t, t + 12 − δ], and

• f (t) < f (t + 1), for θ ∈ (t + 12 − δ, t + 1).

To conclude that maxt∈[a] f = αa (see (3.7)), it is enough to show that⌈√9 + 8a− 3

2

⌉=

t if θ ∈ (t, t + 1

2 − δ],t + 1 if θ ∈ (t + 1

2 − δ, t + 1).

In fact, since t = bθc = b√

2(a + 1)c − 1, then t− 1 <√

9+8a−32 ≤ t + 1, and

moreover,θ ≤ t + 1

2 − δ ⇐⇒√

9+8a−32 ≤ t.

109

Appendix B

Probability on graphs

We always use the standard notation [n] = 1, . . . , n ⊂ N, and denote thecardinality of any set S by |S|.

B.1 Graph theory

A graph G is a pair (V, E) where V is a set of elements, called vertices and Eis a subset of V2 := (u, v) ∈ V × V : u 6= v, called the set of edges. If V is

finite then |V2| =(|V|2

), where

(nk

)is the binomial coefficient:

(nk

):=

n!

k!(n− k)!, if n ≥ k

0, if n < k.

Given u, v ∈ V, we say that u and v are adjacent in G, and write u ∼ v, if(u, v) ∈ E. The neighbourhood of v is the set N(v) = u ∈ V : u ∼ v andthe degree of v is |N(v)|. A graph is called locally f inite if every vertex hasfinite degree. When W ⊂ V, the induced graph G[W] is the subgraph of Gwith vertex set W and edge set (u, v) ∈ E(G) : u, v ∈W.

A path in G is a sequence of distinct vertices (v0, v1, . . . , vn) with vi−1 ∼ vifor each i ∈ [n]. If moreover, vn ∼ v0, we say it is a cycle. G is connected iffor any two distinct vertices u, v ∈ V there exists a path (v0, v1, . . . , vn) withv0 = u and vn = v.

A tree T is a connected graph with no cycles, and we say that T is rootedif we fix a vertex v∗ ∈ V(T) which we call the root. Finally, a labelled tree is apair (T, ϕ), where T is a finite tree and ϕ : V(T)→ [|V(T)|] is a bijection.

Theorem B.1.1 (Cayley’s formula). For every positive integer n, the number oflabelled trees on n vertices is nn−2.

The following are some well known inequalities that we use.

1. For all q ∈ (0, 23),

e−2q ≤ 1− q ≤ e−q.

2. Binomial inequality (nk

)≤( en

k

)k

110 Appendix B. Probability on graphs

3. Stirling’s approximation√

2πnn+1/2 ≤ n!en

B.1.1 Asymptotic notation

Here it is some of the standard asymptotic notation that we use throughoutthis thesis. If f , g : (0, 1) → (0, ∞) are functions of p, then we will writef = O(g) if there exists a constant C > 0 (which does not depend on p) suchthat

f (p) ≤ Cg(p) (B.1)

for every sufficiently small p > 0. We will also write f = Ω(g) if g = O( f )and f = Θ(g) if both f = O(g) and f = Ω(g). When the domain of thefunctions f and g is [1, ∞) (instead of (0,1)), the notation f = O(g) becomesf (L) ≤ Cg(L) for every sufficiently large L, and we keep the same conven-tion for f = Ω(g) and f = Θ(g).

We also use the notation f = o(g) or f g to say that f (p)/g(p) goes to0 as p→ 0.

B.2 Probability and expectations

In this section, we introduce the abstract notions in probability theory, fo-cused in the terminology that we use in this thesis (for a reference, see [26]).

A probability space is a triple (Ω,F , P), where Ω is a set (states of nature),F is a σ-algebra on Ω (set of events), and P : F → [0, 1] is a probabilitymeasure, i.e.

• P(Ω) = 1, and

• if Fi ∈ F is a countable sequence of pairwise disjoint events, then

P

(⋃i

Fi

)= ∑

iP(Fi).

Given a mapping X : Ω → R, we always write the expression X ≤ xas a shorthand to denote the event ω ∈ Ω : X(ω) ≤ x. An analogousconvention is set for the expressions X = x and X > x. We say that X isa random variable if X ≤ x ∈ F , for all x ∈ R.

Since every random variable is a measurable function, we can performLebesgue integration. We call the integral of X w.r.t. P the expectation of X(when X is integrable), and it is denoted by E(X).

E(X) :=∫

ΩX dP. (B.2)

Note that the expectation operator E is linear on the space of integrable ran-dom variables. The variance of X is Var(X) := E[(X −E(X))2]. There is aneasy way to compute expectations in some cases.

B.2. Probability and expectations 111

Proposition B.2.1. If X is a non-negative random variable such that P(X > x) isa continuous function of x, then

E(X) =∫ ∞

0P(X > x) dx.

Example B.2.2. A random variable X : Ω → [0, ∞) has an exponential distri-bution with rate λ > 0, and we write X ∼ exp(λ), if

P(X > x) = e−λx,

for every x ≥ 0. For this random variable, E(X) = 1/λ and Var(X) = 1/λ2.

The following results are very useful in probability.

Proposition B.2.3 (Markov’s inequality). If X is a non-negative random variable,then for every t > 0

P(X ≥ t) ≤ E(X)/t. (B.3)

Proposition B.2.4 (Union bound). If Fi ∈ F for all i ∈N, then

P

(⋃i≥1

Fi

)≤ ∑

i≥1P(Fi). (B.4)

Theorem B.2.5 (Borel-Cantelli lemma). If ∑∞m=1 P(Fm) < ∞, then

P

(⋂i≥1

⋃m≥i

Fm

)= 0. (B.5)

Every random variable X : (Ω, P) → N induces a probability measureon N (N could be replaced by any countable subset of R), usually denotedby X P and called the distribution of X, which satisfies

X P(x) = P(X = x), for all x ∈N.

Probability measures on discrete sets are somehow easier to handle.

Discrete probability spaces

When Ω is a countable set, we do not need to use measure theory machinery,since we let F be the collection of all subsets of Ω, and a probability measureis the same as a mapping P : Ω→ [0, 1] such that

• ∑ω∈Ω P(ω) = 1, and

• if F ∈ F , then P(F) = ∑ω∈F P(ω).

When Ω is finite, the uniform measure on Ω is defined by P(ω) = 1/|Ω|, forall ω ∈ Ω.


Example B.2.6. A random variable X : Ω → [n] has a uniform distribution on[n], if X P is a uniform measure on [n], in other words, if

P(X = x) =1n

, for all x ∈ [n].

In this setting, the expectation of X becomes

E(X) = ∑ω∈Ω

X(ω)P(ω) = ∑x∈X(Ω)

xP(X = x). (B.6)

Example B.2.7. We say that a random variable X : Ω→ 0, 1 has a Bernoullidistribution with parameter p, and we write X ∼ Ber(p), if P(X = 1) = p andP(X = 0) = 1− p. For this random variable, E(X) = p.

B.3 Product measure

From measure theory, we know that given (Ωi,Fi, µi), i = 1, . . . , n probabil-ity spaces, and setting Ω = Ω1 × · · · ×Ωn, then there is a unique probabilitymeasure P on the σ-algebra ⊗n

i=1Fi generated by the rectangles (i.e. sets ofthe form F1 × · · · × Fn, Fi ∈ Fi), such that

P(F1 × · · · × Fn) =n

∏i=1

µi(Fi).

We call this measure the product measure and is denoted by P = ⊗ni=1µi.

Indeed, probability measures exist on infinite product spaces. Consider

RN = (ω1, ω2, . . . ) : ωi ∈ R.

Let Bn be the Borel σ-algebra in Rn, and equip RN with the product σ-algebraBN, which is generated by the finite dimensional rectangles.

Theorem B.3.1 (Kolmogorov’s extension theorem). Suppose we are given prob-ability measures µn on (Rn,Bn) that are consistent, that is,

µn+1((a1, b1]× · · · × (an, bn]×R) = µn((a1, b1]× · · · × (an, bn]).

Then there is a unique probability measure ⊗iµi on (RN,BN) with

⊗iµi(ω : ωi ∈ (ai, bi], 1 ≤ i ≤ n) = µn((a1, b1]× · · · × (an, bn]).

When Λ is a countable set, we identify it with either [n] (if |Λ| = n < ∞)or N (if |Λ| = ∞), and consider the probability measure on 0, 1Λ givenby the product measure ⊗i∈Λµi, where each µi is a Bernoulli measure Ber(p)with parameter p,

µi(1) = p = 1− µi(0).

B.4. Classical Bernoulli percolation 113

Usually, Λ is a set of vertices, or edges. We denote this measure by

Pp = ⊗i∈ΛBer(p). (B.7)

Independence

Given a probability space (Ω,F , P), a sequence of events F1, . . . , Fm ∈ F issaid to be independent if, for every ∅ 6= I ⊂ [m],

P

(⋂i∈I

Fi

)= ∏

i∈IP(Fi).

The random variables X1, . . . , Xm : Ω → R are said to be independent if, forevery x1, . . . , xm ∈ (−∞, ∞],

P(X1 ≤ x1, . . . , Xm ≤ xm) = ∏i

P(Xi ≤ xi).

Remark B.3.2. Consider the product measure Pp on Ω = 0, 1V(G) and twodisjoint sets S1, S2 ⊂ V(G). Suppose that there are two events F1 and F2 suchthat the occurrence of Fi only depends on the state of the vertices in Si fori = 1, 2, then F1 and F2 are independent. For example, for i = 1, 2, considerthe event Fi := all vertices in Si are in state 2− i, then

Pp(F1 ∩ F2) = Pp(F1)Pp(F2) = p|S1|(1− p)|S2|. (B.8)

If there are two randon variables X1, X2 such that the values of Xi only de-pends on the state of the vertices in Si, then X1 and X2 are independent.

B.4 Classical Bernoulli percolation

Percolation is a model for disordered medium, we can think that it refers tothe movement and filtering of fluids through porous materials, for example,when we make coffee. It is common to use terminology in percolation theorydiffers from that of graph theory, for instance, vertices and edges are calledsites and bonds, while components are called clusters.

Start with a countable graph G and let the set Λ to be either V(G) orE(G), and then consider the product measure Pp on 0, 1Λ. When we selectvertices, we speak of site percolation; when we select edges, bond percolation.This is the simplest studied model and is called Bernoulli percolation (for areference, see [10]).

The FKG inequality

The FKG inequality (a.k.a. Harris’ inequality) is a powerful tool used in per-colation, which states that certain (non independent) events are positivelycorrelated.


Consider the partial order in 0, 1Λ given by ω ≤ ω′ if ωλ ≤ ω′λ for allλ ∈ Λ.

Definition B.4.1. We say that an event F in Ω = 0, 1Λ is increasing if ω ∈ Fand ω ≤ ω′ implies ω′ ∈ F. We say that an event is decreasing if its comple-ment is increasing.

For example, given any set S ⊂ Λ, the event

F =⋂

λ∈S

ωλ = 1

is increasing.

Theorem B.4.2 (FKG inequality). If F1 and F2 are both either increasing or de-creasing events, then

Pp(F1 ∩ F2) ≥ Pp(F1)Pp(F2). (B.9)

Corollary B.4.3. If F1, . . . , Fn are increasing events, then

P

(n⋂

i=1

Fci

)≥

n

∏i=1

P(Fci ). (B.10)

B.4.1 The critical probability

We will only focus on site percolation on Zd. We say that the vertex x ∈ Zd

is open if it is in state 1, otherwise we say is is closed. We can think that theedges of Zd are pipes and fluid flow through vertex x only if x is open. Wedefine the (random) cluster of 0 to be

C := x ∈ Zd : there exists an open path from 0 to x.

A main quantity of interest is the percolation probability θ(p), which is theprobability that the origin belongs to an infinite open cluster.

θ(p) := Pp(|C| = ∞).

Note that θ(0) = 0 and θ(1) = 1, moreover, it is possible to show that θis a non-decreasing function of p, therefore there exists a phase transitioncorresponding to the value of p such that θ(p) starts to be positive. The criticalprobability pc(d) is defined by

pc(d) := supp : θ(p) = 0. (B.11)

It turns out that pc(1) = 1, so this case is not interesting. However, for higherdimensions, the situation is different.

Theorem B.4.4 (Phase transition). For every d ≥ 2,

0 < pc(d) < 1. (B.12)

B.5. Continuous time processes 115

The regime p < pc(d) is called the subcritical phase, while p > pc(d) iscalled the supercritical phase. Both regimes have been extensively studied(even at criticality, near pc), and now we state two key results correspond-ing to the former regime.

B.4.2 The subcritical regime

Let Bn be the ball of radius n with centre at 0 ∈ Zd (here, we are using thesupremum norm), and ∂Bn be the boundary of Bn, i.e. the set of all pointsthat are at distance n from 0. Denote the event that there exists an open pathjoining 0 to some vertex in ∂Bn by 0↔ ∂Bn.

Theorem B.4.5 (Exponential decay). If p < pc, there exists C(p) > 0 such that

Pp(0↔ ∂Bn) ≤ e−C(p)n, (B.13)

for all n ≥ 1.

In words, for p < pc, the radius of C has a tail which decays at least ex-ponentially fast. This result implies that Ep|C| < ∞ for p < pc (in particularPp(|C| < ∞) = 1). Indeed, a stronger statement holds: not only the radius ofC decays exponentially fast, but so does its volume |C|. That is the content ofthe next result.

Theorem B.4.6 (Exponential decay of the cluster size distribution). If p < pc,there exists λ(p) > 0 such that

Pp(|C| ≥ n) ≤ e−λ(p)n, (B.14)

for all n ≥ 1.

B.5 Continuous time processes

Let us start by recalling a basic notion. Given two events F1, F2, such thatP(F1) > 0, the conditional probability of F2 given F1 is

P(F2|F1) :=P(F1 ∩ F2)

P(F1). (B.15)

It turns out that P(·|F1) defines a new probability measure on Ω, which as-signs zero mass to events contained in Fc

1 . The conditional probability is veryuseful when the events Fi are related to random variables.

Consider a probability space (Ω,F , P), a σ-algebra F0 ⊂ F , and a ran-dom variable M with E|M| < ∞. The conditional expectation of M given F0,is a random variable Y satisfying

• Y is F0-measurable, and

•∫

F Y dP =∫

F M dP, for all F ∈ F0.


By using the Radon-Nikodym Theorem, it is possible to show that the condi-tional expectation exists and is unique (up to a zero-measure set). We denoteit by E(M|F0). Note that the extreme cases are

E(M|∅, Ω) = E(M), and E(M|F ) = M.

B.5.1 Markov processes

In this section we briefly overview the general theory of Markov processes,which are stochastic processes satisfying the following memoryless principle:conditional on the present state of the system, its future and past states areindependent (for a reference, see [41]).

Let (Ω,F ) and (S,M) be measurable spaces and consider a family ofmeasurable functions Xt : Ω → St (we call them variables) indexed byt ∈ [0, ∞), then, define the canonical filtration to be the family of σ-algebrasFtt, where Ft ⊂ F is the smallest σ-algebra on Ω such that all variablesXs, 0 ≤ s ≤ t, are (Ft,M)-measurable:

Ft := σ(Xs : 0 ≤ s ≤ t). (B.16)

In particular, Ft ⊂ Ft+s ⊂ F , for all s, t ≥ 0.A transition kernel from S into S is a mapping Q : S×M→ [0, 1] satisfying

1. For every F ∈ M, the mapping Q(·, F) isM-measurable, and

2. For every x ∈ S, the mapping Q(x, ·) is a probability measure on (S,M).

When S is countable, Q is called a transition matrix, since it can be specifiedby a |S| × |S| matrix Q(x, y)x,y∈S, whose entries (x, y) are all non-negativeand sum up to 1 in the second coordinate (sum over rows of Q).

In this case, we can (informally) say that a Markov process is a collectionof random variables Xtt≥0 such that for every s ≥ 0, the probability ofjumping from Xs = x to a new state y is Q(x, y); in particular, this probabilitydoes not depend on the states of Xs for r < s.

In the general setting, we will have a collection of kernels Qtt associatedto Xtt, and require the conditional probability of Xs+t knowing the pastXr : 0 ≤ r ≤ s before time s to be given by Qt(Xs, ·), and this conditionaldistribution should only depend on the present state Xs. Now, we formalizethis idea.

Remark B.5.1. Let B(S) be the vector space of all bounded measurable func-tions f : S→ R. Given f ∈ B(S), Q induces a new function Q f ∈ B(S) givenby

Q f (x) :=∫

f (·) dQ(x, ·).

Thus, we can think of Q : B(S)→ B(S) as a linear operator.

Definition B.5.2. Let Qtt be a collection of transition kernels on S indexedby t ∈ [0, ∞), satisfying


• For every x ∈ S, Q0(x, ·) assigns probability (or mass) 1 to x.

• (Chapman–Kolmogorov identity) For every s, t ≥ 0

Qt+s = QtQs.

• For every F ∈ F , the map (t, x) 7→ Qt(x, F) is B1 ⊗F -measurable.

A Markov process with transition kernels Qtt≥0 is a collection of variablesXt : Ω→ St such that, for every s, t ≥ 0 and f ∈ B(S),

E[ f (Xs+t)|Fs] = Qt f (Xs). (B.17)

Theorem B.5.3 (Markov property). Let Xtt≥0 be a Markov process and fixs ≥ 0. Then, for every t ≥ 0,

P(Xs+t ∈ ·|Fs) = Qt(Xs, ·) = P(Xs+t ∈ ·|Xs). (B.18)

Definition B.5.4. We say that a random variable τ : Ω → [0, ∞) is a (finite)stopping time if the event τ ≤ t ∈ Ft, for every t ≥ 0. In this case, we definethe random variable Xτ by Xτ(ω) := Xτ(ω)(ω). We also define σ-algebra ofthe past before τ by

Fτ := F ∈ F : ∀t ≥ 0, F ∩ τ ≤ t ∈ Ft.

Theorem B.5.5 (Strong Markov property). Let Xtt≥0 be a Markov process andτ be a stopping time. Then, for every t ≥ 0,

P(Xτ+t ∈ ·|Fτ) = Qt(Xτ, ·). (B.19)

From now on, we assume that S is a compact metric space andM is theBorel σ-algebra. Let C(S) be the set of all continuous functions f : S → R,and regard C(S) as a Banach space for the supremum norm

‖ f ‖ := supσ∈S| f (σ)|.

Definition B.5.6. Let Xtt be a Markov process with transition kernels Qtt≥0and set

D(L) :=

f ∈ C(S) :Qt f − f

tconverges in C(S) as t→ 0

.

We define the generator of Xtt to be the linear operator L : D(L) → C(S)given by

L f := limt→0

Qt f − ft

. (B.20)

It is well known that there is a one-to-one correspondence between gen-erators L and transition kernels Qtt satisfying the conditions in Definition


B.5.2. We will only focus on Markov processes corresponding to spin flip sys-tems; in that setting, the dynamics realizations are determined by the actionof the generator on local functions. This is the content of the next section.

B.5.2 Interacting particle systems

In this section we will follow the approach given by Liggett [42, 43]. Hence-forth, we will fix a countable set Λ and set

S = 0, 1Λ. (B.21)

In this case, we refer to a Markov process as an interacting particle system (IPS),and we denote the elements of S by σ = (σ(x))x∈Λ (instead of X). We endowthe single spin space 0, 1 with the discrete topology, and the state space Swith the corresponding product topology.

Definition B.5.7. A function f : S 7→ R is said to be a local if for each σ ∈ S,f (σ) depends on only finitely many of the variables σ(x). Given v ∈ Λ andσ ∈ S, we define σv ∈ S by

σv(x) =

σ(x), x 6= v,1− σ(x), x = v.

The dynamics of an IPS σtt are specified by the transition rates: a func-tion c(x, σ) : Λ× S → [0, ∞), which represents the rate at which the coordi-nate σ(x) flips from 0 to 1 or from 1 to 0 when the system is in state σ. It turnsout that if c(·, ·) is uniformly bounded, c(x, ·) is continuous for each x, and

supx∈Λ

∑v∈Λ

supσ∈S|c(x, σ)− c(x, σv)| < ∞,

then the operator L acting on local functions f as

L f (σ) = ∑x∈Λ

c(x, σ)[ f (σx)− f (σ)], (B.22)

is the generator of a unique IPS σtt.

Example B.5.8 (The voter model). Let Q(x, v)x,v∈Λ be a transition matrixon Λ. The voter model is the spin system with rates given by

c(x, σ) =

∑v∈Λ Q(x, v)σ(v), σ(x) = 0,∑v∈Λ Q(x, v)[1− σ(v)], σ(x) = 1.

There is a graphical representation of this model, which is another way ofdescribing the rates: a site x waits an exponential time with parameter 1, atwhich time it flips to the value it sees at that time at a site v which is chosenwith probability Q(x, v).


B.5.3 Martingales

In this section we review an stochastic process with the following property:fixed a time, the conditional expectation of any next value in the process,given all prior values, is equal to the present value.

Let Mt : Ω → Rt denote a family of random variables indexed by t ∈[0, ∞) (resp. t ∈N∪0), and FM

t t be the canonical filtration associated toMtt. We say that Mtt is a continuous time (resp. discrete time) martingaleif for all t, s ≥ 0,

1. E|Mt| < ∞, and

2. E(

Mt+s|FMt)= Mt.

Note that if Mtt∈[0,∞) is a continuous time martingale then Mnn∈N is adiscrete time martingale.

There is an important result about martingales called the optional stop-ping theorem, which has several versions (see e.g. [41]); one of them is thefollowing.

Theorem B.5.9 (Optional Stopping). Let Mnn≥0 be a discrete time martingaleand τ be a stopping time with E(τ) < ∞. If there exists a constant C such that|Mn −Mn−1| ≤ C for all n ∈N, then

E(Mτ) = E(M0). (B.23)

We finish by stating a result that relates IPS and martingales.

Theorem B.5.10. Fix η ∈ S and let σtt be an IPS with generator L such thatP(σ0 = η) = 1. For every f ∈ C(S),

Mt := f (σt)−∫ t

0L f (σs)ds (B.24)

is a martingale.

Indeed, this result holds for a wider class of Markov processes, undersuitable conditions (see e.g. [41]).

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