Universidade Federal de Minas Gerais
Instituto de Ciencias Exatas
Departamento de Matematica
On the phase transition for some percolation models in
random environments
Marcos Vinıcius Araujo Sa
Advisors:
Remy de Paiva SanchisMarcelo Richard Hilario
Support: CAPES
Belo Horizonte - MG
November, 2019
Resumo
Nesta tese nos consideramos dois modelos de percolacao em ambientes aleatoriose estamos interessados em seus fenomenos de transicao de fase.
O primeiro modelo de percolacao estudado e na rede cubica apresentando desor-dem colunar. Este modelo e definido em dois passos: primeiro as colunas verticaisde Z3 sao removidas independentemente com probabilidade 1 − ρ e, no segundopasso, os elos conectando sıtios na sub-rede remanescente sao declarados abertoscom probabilidade p de modo independente. Nosso resultado mostra que existeδ > 0 tal que o ponto crıtico pc(ρ) < 1/2 − δ para todo ρ > ρc, onde ρc denota oponto crıtico da percolacao de sıtios em Z2.
O segundo modelo e na rede quadrada esticada horizontalmente, que consistede uma versao generalizada de Z2
+ obtida ao se esticar a distancia entre suas col-unas, segundo uma variavel aleatoria positiva ξ. Neste modelo a probabilidade deum elo ser declarado aberto decaira exponencialmente segundo seu comprimento.Nosso resultado mostra a existencia da transicao de fase quando E(ξη) <∞, paraalgum η > 1, e a ausencia quando E(ξη) =∞, para algum η < 1.
PALAVRAS-CHAVE: Percolacao, transicao de fase, ambientes aleatorios, renor-malizacao multiescala.
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Abstract
In this thesis we consider two percolation models in random environments andwe are interested in their phase transition phenomenon.
The first percolation model we study is defined on the cubic lattice featuringcolumnar disorder. This model is defined in two steps: first the vertical columnsof Z3 are removed independently with probability 1 − ρ and, in the second step,the bonds connecting sites in the remaining sub-lattice are declared open withprobability p, independently. Our result shows that there exists δ > 0 such thatpc(ρ) < 1/2−δ for any ρ > ρc, where ρc denotes the critical point of site percolationin Z2.
The second model is defined on a horizontally stretched square lattice, whichis a generalized version of Z2
+ obtained by stretching the distances between itscolumns according to a positive random variable ξ. In this model the probabilityof a bond being declared open will decay exponentially according to its length.Our result shows the existence of a phase transition when E(ξη) < ∞, for someη > 1, and the absence of phase transition when E(ξη) =∞ for some η < 1.
KEY-WORDS: Percolation, phase transition, random environments, renormaliza-tion, multiscale analysis.
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Contents
1 Introduction 21.1 Basic notation and definitions . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Percolation model . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Motivation: a brief history of percolation theory in random envi-ronments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Models and main results . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Percolation model with columnar disorder . . . . . . . . . . 71.3.2 Percolation model on the stretched square lattice . . . . . . 10
2 A non-trivial bound for the critical threshold of a percolationmodel with columnar disorder 122.1 Block argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Proof of Lemma 2.1: (i) . . . . . . . . . . . . . . . . . . . . 142.1.2 Proof of Lemma 2.1: (ii) and (iii) . . . . . . . . . . . . . . 17
2.2 Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Enhancements along vertical columns . . . . . . . . . . . . . 192.2.2 The coupling scheme . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Percolation on a horizontally stretched square lattice 253.1 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Definition and notation . . . . . . . . . . . . . . . . . . . . . 283.2.2 Loss of memory . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.3 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 The multiscale scheme . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 Environments . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 1
Introduction
In Section 1.1, the notation that will be used throughout the text is introducedand some important definitions are given. In Section 1.2, a brief summary on thehistory of percolation theory will be presented focusing on models that motivatedthis thesis. Such models present inhomogeneities which are introduced by meansof a random environment. In Section 1.3 the models considered in this thesis willbe introduced and the main results obtained will be stated.
1.1 Basic notation and definitions
In this section some of the notation and definitions used throughout this thesiswill be introduced.
1.1.1 Graphs
A graph G is an ordered pair (V (G), E(G)) consisting of a set V (G) of verticesand a set E(G) ⊆
(V (G)
2
)of edges, where(V (G)
2
)={{v, w} ⊆ V (G); v 6= w
}is set of all (unordered) pairs of vertices in G. When there is no risk of ambiguity,we will abuse notation and not distinguish the graph G and its set of verticesV (G). We say that G is an infinite graph if |V (G)| = ∞, where |A| denote thecardinality of set A. Given G and v, w ∈ G, we write v ∼ w if v is a neighbor ofw, i.e. {v, w} ∈ E(G). In this case, v and w are called the endpoints of the edge{v, w}.
In specific situations throughout the text, other structures similar to graphs
will be mentioned: oriented graphs and multigraphs. An oriented graph−→G =
2
(V (G),
−→E (G)
)is a graph in which an orientation is assigned to each one of its
edges. A multigraph G is a graph in which the existence of loops (edges with onlyone endpoint) and multiple edges (more than one edge with the same endpoints)is permitted. From now on, all the definitions in this sections will be stated forgraphs, however they can be easily extended to oriented graphs and multigraphs.
Given the graphs G and H, H is called a subgraph of G if
V (H) ⊆ V (G) and E(H) ⊆ E(G).
Now let A ⊆ V (G). The subgraph of G induced by A is the subgraph whose setof vertices is A and whose set of edges consists of all edges in E(G) with bothendpoints in A. The boundary of A is
∂A = {w ∈ G \ A; there exists v ∈ A such that v ∼ w}.
For the sake of brevity, we do not distinguish the set {v} from the element v, anddenote ∂v the set of nearest neighbors of the vertex v. We say that the graph Gis locally finite if |∂v| <∞ for every vertex v.
A path γ (in A ⊆ G) is a sequence of distinct vertices (v0, v1, · · · , vn) (in A)such that vi−1 ∼ vi for any 1 ≤ i ≤ n. We say that such path γ starts at v0
and ends in vn. The graph G is said to be connected if for every pair of verticesv, w ∈ G there is a path starting at v and ending at w.
Now for d ∈ {1, 2, · · · }, consider the graph Zd whose vertices are the d-tuplesof integers, and where two vertices are neighbors if, and only if, only one of theircoordinate disagree by one unit. The two main lattices considered here will be Z2
and Z3 called square lattice and cubic lattice, respectively. Denote by o the originof the lattice Zd, that is the d-tuple with zero on all entries.
The last definition in this section will be another frequent structure present inthe next chapters. Let a, b, c, d ∈ Z with a < b and c < d. Denote by
R = R([a, b)× [c, d)
)(1.1)
the subgraph of Z2 defined by
V (R) = [a, b]× [c, d] and
E(R) ={{(x, y), (x+ i, y + 1− i)}; (x, y) ∈ [a, b− 1]× [c, d− 1], i ∈ {0, 1}
},
where [a, b] denotes the set of all integers between a and b, including them both.R will be called a rectangle, and is the graph induced by [a, b]× [c, d] removed theedges in the right and top sides. See Figure 1.1.
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a b
c
d
Figure 1.1: Illustration of the rectangle R([a, b)×[c, d)
). Note that R
([a, b)×[c, d)
)is not equal to the graph induced by [a, b− 1]× [c, d− 1].
1.1.2 Percolation model
Consider G an infinite, locally finite, connected graph. A bond percolationconfiguration in G is an element
ω =(ω(e); e ∈ E(G)
)of the sample space {0, 1}E(G). A percolation event is an element of the σ-algebragenerated by the cylinder sets (events which only depend on the state of a finitenumber of edges). The edge e ∈ E(G) is said to be open if ω(e) = 1, otherwiseit is said to be closed. Note that ω can be seen as a subgraph of G that containsonly the edges which are open for ω, that is
V (ω) := V (G) and E(ω) := {e ∈ E(G); ω(e) = 1}.
Now let A,B, S ⊆ G, with A ∪ B ⊆ S. We denote {A ↔ B in S}, the eventthat there is a path (v0, v1, · · · , vn) with v0 ∈ A and vn ∈ B such that vi ∈ S forany 0 ≤ i ≤ n and that ω
({vi−1, vi}
)= 1 for any 1 ≤ i ≤ n. When S = G, we
drop it from notation. Also we define the cluster of A in ω, by
CA(ω) ={u ∈ V (G); there is v ∈ A such that ω ∈
{v ↔ u}
}.
For a vertex v ∈ G, let
{v ↔∞} ={ω ∈ {0, 1}E(G); |Cv(ω)| =∞
},
be the event that v is connected to infinity. A central question in percolation iswhether {v ↔∞} has positive probability, when ω is sampled at random.
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In the Bernoulli bond percolation model on G, the ω(e)’s are independentBernoulli random variables with mean pe ∈ [0, 1]. When pe = p for some p ∈ [0, 1]and for every edge e ∈ E(G), the model is said to be homogeneous. Otherwise itis said to be inhomogeneous. For the homogeneous percolation model, its law willbe denoted by Pp(·). The critical point of G is defined as
pc(G) = sup{p ∈ [0, 1]; Pp(v ↔∞) = 0
}.
Since G is connected, the FKG inequality implies that the critical point does notdepend on the choice of vertex v. When 0 < pc(G) < 1, we say that the modelundergoes a non-trivial phase transition, since the value of Pp(v ↔ ∞) changesfrom 0 when p < pc(G) to positive when p > pc(G).
Now consider the square lattice, Z2, and let R = R([a, b)× [c, d)
)be as in (1.1).
The horizontal and vertical crossing events in R are defined respectively as
Ch(R) ={{a} × [c, d]↔ {b} × [c, d] in R
}, (1.2)
Cv(R) ={
[a, b]× {c} ↔ [a, b]× {d} in R}. (1.3)
Similarly, in order to define the Bernoulli site percolation model on G, weconsider the percolation configuration
ω =(ω(v); v ∈ V (G)
)∈ {0, 1}V (G),
that can be seen as the subgraph of G induced by{v; ω(v) = 1
}. In the Bernoulli
site percolation model on G, the ω(v)’s are independent Bernoulli random vari-ables with mean pv ∈ [0, 1]. The connectivity and crossing events are definedanalogously. The horizontal and vertical crossing events in the rectangle R willbe denoted respectively by Cvh(R) and Cvv(R). For the homogeneous Bernoulli sitepercolation model, its law will be denoted by Psite
p (·). The critical point of G isdefined by
psitec (G) = sup
{p ∈ [0, 1]; Psite
p (v ↔∞) = 0}.
1.2 Motivation: a brief history of percolation
theory in random environments
Percolation theory first appeared in mathematical literature in 1957 [5] whenBroadbent and Hammersley modeled mathematically the flow of a fluid through aporous medium. They proved that for d ≥ 2 the critical point of the (homogeneous)Bernoulli percolation model is non-trivial, i.e., 0 < pc(Zd) < 1. In 1960, Harris[10] showed that for the square lattice P1/2(o ↔ ∞) = 0, and twenty years laterKesten [14] showed that pc(Z2) = 1/2. For a comprehensive list of references onpercolation theory see Grimmett’s book [9].
One of the main motivation for this work is the question:
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How the introduction of inhomogeneities modifies the existence of thephase transition or shifts the critical points in percolation models?
This type of question was posed in a number of different situations, as mentionedbelow.
In several percolation models, inhomogeneties arise by introduction of an en-vironment which specifies how to assign weights pe for each edge e of the graph.For instance, for the square lattice, Z2, one way to introduce inhomogeneities is byfixing columns (the environment) whose edges will have a probability p of beingopen, while the other edges will be open with probability q. Formally let Λ ⊆ Z,and set
Evert(Λ) :={{(x, y), (x, y + 1)}; x ∈ Λ, y ∈ Z
},
which corresponds to the edges belonging to the columns that project to Λ. LetPΛp,q(·) be the probability law in {0, 1}E(Z2) that governs the percolation model
whose weights of edges are given by
pe =
{p, if e ∈ Evert(Λ),q, if e 6∈ Evert(Λ).
In [25], Zhang nicely explores the ideas in [10] of constructing dual circuitsaround the origin, together with the Russo, Seymour and Welsh [21, 24] techniques,
in order to prove that P{0}p,q (o ↔ ∞) = 0 for any p ∈ [0, 1) and q ≤ pc(Z2) = 1/2.It follows from the results on percolation in half-spaces of Barsky, Grimmett andNewman [3] that P{0}p,q (o ↔ ∞) > 0 whenever q > pc(Z2). It is also known thatPZp,q(o↔∞) > 0 iff p+ q > 1 (see page 54 in [15] or Section 11.9 in [9]).
Suppose there is k such that for every l ∈ Z, Λ ∩ [l, l + k] 6= ∅. A classicalargument due to Aizenman and Grimmett [2] guarantees that for any ε > 0 thereis δ = δ(k, ε) > 0 such that PΛ
pc+ε,pc−δ(o↔∞) > 0, where pc = pc(Z2).We now consider models for which the environment is taken randomly. Let
υρ be the probability measure on Z under which {i ∈ Λ} are independent eventshaving probability ρ. Recently Duminil-Copin, Hilario, Kozma and Sidoravicius[6] showed that for any ε > 0 and ρ > 0 there is δ = δ(ρ, ε) > 0 such thatPΛpc+ε,pc−δ(o↔∞) > 0 for υρ-almost everywhere environment Λ.
Using the arguments in Bramson, Durrett and Schonmann [4] one can provethat for any ρ ∈ [0, 1), there is p < 1 large enough such that PΛ
0,p(o ↔ ∞) > 0for υρ-almost every environment Λ. The last result says that if one deletes alledges of columns according to Bernoulli trials with mean ρ, the phase transition ispresent. In [12], Hoffman study the case where both rows and columns are deletedindependently with the same probability ρ showing that the model still undergoesa non-trivial phase transition.
Another variation was studied by Kesten, Sidoravicius and Vares [16], whoconsidered a site percolation model on the oriented square lattice, with the edges
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oriented from left to right and from bottom to top. Let Λ be taken in the sameway as in [6]. The weights pv of the vertices will be chosen according to whetheror not they belong to diagonals given by Λ as follows
pv =
{p, if v = (x− y, y) for some x ∈ Λ and y ∈ Z,q, otherwise.
Let−→P Λp,q(·) the probability law that governs this model. The main result in [16]
shows that for any p > pvc(−→Z2) and q > 0, there is ρ < 1 such that
−→P Λp,q(o ↔ ∞) > 0 for υρ-almost everywhere environment Λ, where pc
(−→Z2)
de-
notes the critical point with respect to the measure−→P Λp,p(·).
Another related model is the Bernoulli line percolation introduced in the physicsliterature by Kantor [13] and studied mathematically by Hilario and Sidoravicius[11]. This site percolation model on Zd, d ≥ 3, consists of selecting lines parallel tothe coordinate axes at random, according to Bernoulli trials. A vertex is declaredclosed if it belongs to any of these lines, and open otherwise. In [11], the existenceof a phase transition, the number of infinite clusters and the connectivity decay arestudied. The presence of power-law decay contrasts sharply with the behavior ofmodels with finite-range dependencies where the decay is exponential, see [20, 1].This model was studied also from the numerical point of view in [23, 7].
1.3 Models and main results
In this section, we give a mathematical construction of the two models ofpercolation in random environments which will be studied in this thesis. We alsoprovide the statements of the main results achieved. The proofs will be given inChapter 2 and 3.
1.3.1 Percolation model with columnar disorder
In this section, we introduce a percolation model on the cubic lattice, Z3, featur-ing columnar disorder, consisting of a combination of the Bernoulli line percolationand the homogeneous Bernoulli bond percolation.
First consider the square lattice Z2 embedded in Z3 by identifying Z2 to the setof all the vertices of Z3 whose third coordinate vanishes, and call an environmentevery element of Λ ∈ {0, 1}Z2
. Slightly abusing notation, Λ can be seen as thesubset
{v ∈ Z2; Λ(v) = 1} ⊆ Z2.
Given an environment Λ and a vertex v ∈ Z2, the column {v} × Z ⊆ Z3 is said tobe present if Λ(v) = 1 and absent otherwise. Now define the set of excluded edges
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as
Eexc(Λ) ={e ∈ E(Z3); e has at least one of its endpoints in a absent column
}.
Consider now the percolation model on Z3, as defined in Section 1.1.2, whoseparameter pe = pe(Λ) of the edge e ∈ E(Z3) to be open is given by
pe =
{p, if e 6∈ Eexc(Λ)0, if e ∈ Eexc(Λ)
.
Denote by PΛp (·) the respective probability measure on {0, 1}E(Z3) under which
{ω(e)}e∈E(Z3) are independent Bernoulli random variables with mean pe. Themeasure PΛ
p is called the percolation law in Z3 with quenched columnar disorder Λ.Note that this model is equivalent to the homogeneous percolation model in
Λ× Z, viewed as a subgraph of Z3 induced by the present columns.The environments Λ will be taken at random. Denote by νρ the probability
measure on {0, 1}Z2under which {Λ(v)}v∈Z2 are independent Bernoulli random
variables with parameter ρ ∈ (0, 1]. Note that νρ is the law of the homogeneoussite percolation model on Z2, and regard an environment Λ as a percolation con-figuration in Z2. Set
ρc := psitec (Z2).
We will also consider the annealed probability measure on {0, 1}E(Z3) given by
Pρp(·) :=
∫{0,1}Z2
PΛp (·)dνρ(Λ).
Note that, given a typical Λ, under the quenched measure PΛp , the state of
the non-excluded edges are independent of each other and the measure is invariantunder vertical shift. Under the annealed measure Pρp, the state of the edges presentin the same column are dependent, regardless of the distance between then, butthis measure is invariant under lattice shifts.
Given Λ, we can define pc(Λ) the critical percolation threshold for the quenchedlaw PΛ
p , that is, the value of p above which ω has an infinite cluster PΛp -almost surely
but below which such a component does not exist, PΛp -almost surely. Similarly,
define pc(ρ) the critical threshold for the annealed law Pρp. By standard ergodicityarguments one can show that pc(ρ) = pc(Λ) for νρ-almost any environment Λ.
This model was introduced in Grassberger [8] and was studied from both thenumerical and the mathematical point of view in Grassberger-Hilario-Sidoravicius[7] where the authors show rigorously that the connectivity decays with a power-law when ρ > ρc and p ∈
(pc(Z3), pc(ρ)
). Figure 1.2 shows the graph of ρ 7→ pc(ρ)
obtained numerically in [7].Before stating the result, some immediate properties fulfilled by ρ→ pc(ρ) will
be listed:
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Figure 1.2: The critical curve ρ 7→ pc(ρ). This graph was taken from [7].
1. pc(ρ) is non-increasing. Furthermore, pc(1) = pc(Z3).
2. For every ρ ≤ ρc, pc(ρ) = 1. In fact, νρ-almost all Λ consists of a count-able union of finite disjoint clusters, therefore the graph Λ × Z resembles acountable union of disjoint cigar-shaped graphs extending infinitely in onlyone direction.
3. For every ρ > ρc, pc(ρ) ≤ 12. In fact, νρ-almost all Λ contains an infinite
path. Conditional on the existence of such a path, the subset of Λ×Z whichprojects othogonally to this path resembles a cramped infinite sheet. It isindeed isomorphic to Z+ × Z whose critical point equals 1/2 (see [14]).
As noted in Figure 1.2, the bound pc(ρ) ≤ 1/2 for ρ > ρc should not be sharp:as one approaches ρc from the right the values of pc(ρ) fall strictly under 1/2. Arigorous proof of this fact was still missing and is one of the main contributions ofthis thesis.
Theorem 1.1. There exists δ > 0 such that for every ρ > ρc,
pc(ρ) ≤ 1
2− δ.
Roughly speaking, this result states that above ρc, the structure that remainsafter the columns are drilled are uniformly better to percolation than a simpletwo-dimensional sheet.
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1.3.2 Percolation model on the stretched square lattice
Let Λ = {x0, x1, · · · } ⊆ R be an increasing sequence which will be calledenvironment. Given an environment Λ, consider the lattice LΛ =
(V (LΛ
), E(LΛ))
defined as
V (LΛ) = Λ× Z+ ={
(x, y) ∈ R2;x ∈ Λ, y ∈ Z+)}
and
E(LΛ) ={{(xi, n), (xj,m)} ⊆ V (LΛ); |i− j|+ |n−m| = 1
}.
Roughly speaking, LΛ is a version of Z2+ horizontally stretched according to Λ.
Now let λ ∈ (0,∞) and consider the families{Pm}m∈Z+
and{Qm}m∈Z+
of
Poisson process with rate λ independent of each other. The percolation configura-tions ω ∈ {0, 1}E(Lλ) will be defined according to the realization of these Poissonprocesses as follows: for the edge e ∈ E(LΛ) set ω(e) = 1 iff
e ={
(xi, n), (xi+1, n)}
and Pn((xi, xi+1]
)= 0, or
e ={
(xi, n), (xi, n+ 1)}
and Qi((n, n+ 1]
)= 0.
Geometrically, points are marked on rows and columns of the lattice LΛ accordingto the above Poisson processes, and an edge is declared open iff it has not beenmarked. See Figure 1.3.
Λ : x0 x1 x2 x3 x4 x5
Figure 1.3: Illustration of the lattice LΛ. The dots represent the arrivals timesof the Poisson processes in the rows and columns of LΛ. The open edges arehighlighted by the thickness.
Denote by PΛλ (·) the probability law that governs this percolation model on
LΛ. Equivalently, PΛλ is the measure on {0, 1}E(LΛ) under which the variables ω(e)
10
(with e ∈ E(LΛ)) are independent Bernoulli random variables with mean
pe = exp(−λ|e|),
where |e| denotes the length of the edge e = {v1, v2} ∈ E(LΛ), defined by |e| =‖v1 − v2‖, being ‖·‖ the Euclidean norm in R2.
The goal is to study this model when the environment, Λ, is random anddistributed according to a renewal process as we describe now. Let ξ be a positiverandom variable, and let {ξi}i∈Z∗+ be copies of ξ independent of each other. Set
Λ =
{xk =
∑1≤i≤k
ξi; k ∈ Z+
}= {xi ∈ R;x0 = 0 and xk = xk−1 + ξk for k ∈ Z∗+}.
And denote υξ(·) the probability measure that governs the renewal process Λ withinterarrival time ξ. An overview of renewal processes will be provided in Section3.2.
This model can be seen as an inhomogeneous percolation model on Z2+, where
each edge e ∈ E(Z2+) is open with probability
pe =
{exp(−λ), if e = {(i, j), (i, j + 1)}exp(−λξi+1), if e = {(i, j), (i+ 1, j)} . (1.4)
The random variables ξi’s indicate how far apart the columns of the stretchedlattice lie from one another. Note that when ξ = 1 a.s. the model is homogeneous.
Our main results are
Theorem 1.2. Let ξ be a positive random variable. If E(ξη) <∞ for some η > 1,then for any λ > 0 sufficiently small, we have
PΛλ (o↔∞) > 0, for υξ-almost every environment Λ.
Theorem 1.3. Let ξ be a positive random variable. If E(ξη) =∞ for some η < 1,then for any λ ∈ (0,∞),
PΛλ (o↔∞) = 0, for υξ-almost every environment Λ.
Intuitively speaking, the absence of phase transition established in Theorem 1.3follows from the fact that the columns are typically very far apart from one another.The proof relies on a simple application of Borel-Cantelli Lemma. The proof ofTheorem 1.2 is more delicate and relies on the construction of a renormalizationscheme. Theorem 1.2 can be viewed as a generalization of a discrete version of theresults in [4]. Indeed the techniques there seem to apply only to the case whenthe ξ’s are geometric random variables. We are currently unable to tackle thecase when the lattice is stretched in both horizontal and vertical directions. Thisinteresting case would generalize the results in [12].
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Chapter 2
A non-trivial bound for thecritical threshold of a percolationmodel with columnar disorder
The chapter is devoted to the proof of Theorem 1.1. This proof has two maininputs: a classical block argument (Section 2.1) and an enhancing-type argument(Section 2.2). Below we give the summary of these strategies.
Fix ρ > ρc. The first input consists of constructing an infinite sequence ofcrossing events taking place inside overlapping boxes of Z2. The side length of theboxes are chosen, depending on ρ, in such a way as to guarantee that the crossingsoccur simultaneously for every box with positive probability. On the event thatthe crossings indeed occur, we pick one in each box carefully so that it splits thecorresponding box into two regions, being one of them unexplored.
The position of the boxes are chosen in such a way that, on the event that allthe crossings above occur, one can concatenate them and form an infinite openpath in Z2. The subgraph of Z3 that projects orthogonally to this infinite pathcan be regarded as a cramped sheet and is isomorphic to Z+ × Z. Therefore, thecritical point for Bernoulli bond percolation on it would equal 1/2. The idea nowis to attach to this cramped sheet some structure that will enhance the percolationprocess on it. Perhaps the most interesting point in our argument is how to havethis accomplished uniformly on ρ > ρc.
The key point is that, as one follows the infinite path, the environment on itsleft-hand side is fresh, that is, the state of every neighboring site on this side willbe unexplored. This allows us to attach to the path an infinite sequence of evenlyspaced strands of length one having one endpoint whose state is unexplored andthus dominate an i.i.d. sequence of Bernoulli random variables with mean ρc. Astheir state is revealed, the path will be decorated with a sequence of randomlyplaced strands whose endpoints are open. The edges that project to these strands
12
will serve to enhance the percolation on the cramped sheet.However, since their positions are random, the Aizenman and Grimmett enhan-
cement-type arguments do not apply directly. In order to show that the decoratedcramped sheet is strictly better for percolation than a simple cramped sheet, we usea modified version of the so-called Brochette Percolation ([6] and Section 2.2.1).
2.1 Block argument
The main goal of the present section is to provide the details involving theblock argument leading to the construction of an infinite path decorated withopen strands as explained at the beginning of this chapter. The procedure issummarized in the Lemma 2.1 below, but first we fix some notations.
Set[n] = {0, 1, · · · , n} and [n]∗ = {1, · · · , n}.
A subset A ⊆ Z is said to be 2-spaced if for every pair i 6= j in A, one has |i−j| ≥ 2.We denote by Γ the set of all the paths in Z2 and given an environment Λ ⊆ Z2
we say that a path γ is present if all of its vertices are present (that is, Λ(v) = 1for all v ∈ γ).
Let γ be a path in Z2. We say that a path γ′ is a subpath of γ if V (γ′) ⊆ V (γ)and both paths start and end on the same vertices. And we say that the path γis a minimal path when it is its only subpath.
Lemma 2.1. For any ρ > ρc and any k ∈ Z∗+ there exist L = L(ρ) ∈ Z+,
∆k = ∆k(ρ) ⊆ {0, 1}Z2measurable, and a function Φk : ∆k → Γ such that
(i) νρ(∩k∆k) ≥ 12.
(ii) For every Λ ∈ ∆k, Φk(Λ) is a Λ-present minimal path starting in BL ={0} × [0, L], with |Φk(Λ)| ≥ 2kL.
(iii) For every path γ = (v0, v1, . . . , vn) ∈ Φk(∆k), there is a 2-spaced Nγ ⊆ [n]and a set Sγ ⊆ ∂γ such that:
• for every i ∈ [n] \ Nγ, ∂vi ∩ Sγ 6= ∅,• and for every event A ∈ σ({Λ(v)}v∈Sγ ), νρ(A|Φk = γ) = νρ(A).
We now clarify the statements of Lemma 2.1. The events ∆k which will bedefined in Section 2.1.1 are given by the intersection of crossing events inside koverlapping boxes. The boxes in which the crossing take place will be arranged in aspiral-shaped region (see Figures 2.2 and 2.3). Imagine that the crossing event ∆k
occurs for some k. The functions Φk select carefully a minimal path of length at
13
least 2kL starting close to o and staying inside the spiral-shapped region mentionedabove. A vertex y ∈ Sγ will represent an endpoint of a strand that will be attachedto the site vi ∈ γ for which y ∈ ∂vi. Vertices of γ which are not attached to anystrand are indexed by the set Nγ. The fact that Nγ is 2-spaced implies that eachpair of neighboring vertices in γ contains at least one strand attached. Finally thestate of each strand attached to γ is independent of the event that γ was the pathselected.
The proof of Lemma 2.1 will be divided into two parts. In Section 2.1.1 weconstruct the relevant crossing events. Next, in Section 2.1.2 we show how toconstruct the decorated path. For the remainder of this section, fix ρ > ρc.
2.1.1 Proof of Lemma 2.1: (i)
Before we define the events ∆k, we set the lowest crossings in regions that arevirtually rectangles.
For a, b ∈ Z, consider the rectangle
R = R([0, ab)× [0, a)
),
and setr := {ab} × [0, a]
be its right side. For a fixed path γ in Z2 such that ∂γ ∩ R 6= ∅ and ∂γ ∩ r = ∅,write lγ := ∂γ ∩R. Let Γ(R, γ) stand for the set of all minimal paths γ′ in R that
γ′
D(R, γ, γ′)
lγ
γ
r
R
Figure 2.1: Representation of a rectangle R and a path γ. ∂γ intersects R but notits right side r. The path γ′ is a crossing from lγ to r. The dashed area correspondsto D(R, γ, γ′).
14
start in lγ and end in r. Define {lγR↔ r} the event that lγ and r are connected in
R, i.e. the event that Γ(R, γ) contains at least one present path.
Given an environment Λ ∈ {lγR↔ r}, denote by LR,γ = LR,γ(Λ) the lowest
present path of Γ(R, γ) on Λ. For γ′ ∈ Γ(R, γ), denote by D(R, γ, γ′) the regionof R below γ′ (including the vertices of γ’) and to the right of lγ (see Figure 2.1).Note that, for all γ′ ∈ Γ(R, γ), the event
{LR,γ(Λ) = γ′} = {Λ ∈ {0, 1}Z2
; LR,γ(Λ) = γ′}
is measurable with respect to the state of the vertices in D(R, γ, γ′).Now for any given L > 1 consider the sequence of rectangles Ri defined recur-
sively as follows (see Figure 2.2):
• w1 := (0, 0) and R1 := R([0, 4L)× [0, L)
);
• Let, e1 = (1, 0) ∈ R2 and θπ/2 be the rotation by π/2 about the origin. Definerecursively,
wn+1 = wn + 2n+1Lθn−1π/2 (e1), Rn+1 = wn+1 + 2nθnπ/2(R1).
The union of the overlapping rectangles Ri form a spiral region where we aregoing to find an infinite decorated path. Note that the rectangles Ri have dimen-sions 2i−1L by 2i+1L.
Suppose γ0 a path such that ∂γ0 intersects the rectangle R1, but does notintersects its right side. Let LR1,γ0 be the lowest present path connecting γ0 to theright side of R1, if there is such path. We will give the analogous definitions for R2,
R1
R2
R3
R4
(0, 0) (4L, 0)(2L, 0)
(4L, 8L)
(0, L)
Figure 2.2: Illustration of R1, R2, · · · creating the so-called spiral of rectangles.
15
R3 and R4 where we replace right side and lowest by top side and rightmost, leftside and highest, bottom side and leftmost respectively. We repeat the procedurefor all indexes i > 4. Now let us use the definitions and notations above on thelowest minimal paths for the rectangles Ri with i ≡ 1 (mod 4). In an analogousway, we set the rightmost (resp. the highest and leftmost) minimal paths for therectangles Ri with i ≡ 2 (resp. 3, 4) (mod 4). See Figure 2.3.
Considerγ0 = {−1} × [0, L].
Let {γi}i∈Z∗+ be a sequence of paths, where γi ∈ Γ(Ri, γi−1) for all i ∈ Z∗+. Foreach k set
∆γ1,...,γk :=k⋂i=1
{LRi,γi−1= γi},
the set of environments in which the paths γi’s are the “lowest” (according to thediscussion of the previous paragraph) in their respective regions. Notice that
∆γ1,...,γk ∩∆γ′1,··· ,γ′k 6= ∅ iff γi = γ′i for all i ∈ [k].
Now set the decreasing nested sequence of events
∆k =⋃γ1∈
Γ(R1,γ0)
⋃γ2∈
Γ(R2,γ1)
· · ·⋃γk∈
Γ(Rk,γk−1)
∆γ1,γ2,...,γk .
The following lemma gives us the existence of L(ρ) which guarantees thatνρ(∩k∆k) ≥ 1
2.
Lemma 2.2. For any ρ > ρc, there is L = L(ρ) such that
νρ
( ⋂k∈Z∗+
∆k
)≥ νρ
( ⋂k∈Z∗+
{Rk is crossed in the hard direction
})≥ 1
2.
Proof. Let (Z2)∗ be the so-called matching graph of Z2 i.e. the square lattice withthe diagonal edges added,
V((Z2)∗
)= Z2 and
E((Z2)∗
)=
{{(x1, y1), (x2, y2)
}∈ (Z2)2; max{|x1 − x2|, |y1 − y2|} = 1
}}.
Given an environment Λ ∈ {0, 1}Z2, we define the environment Λ∗ ∈ {0, 1}(Z2)∗ by
Λ∗(v) = 1− Λ(v) for all v ∈ V (Z2) = V((Z2)∗
).
16
And let ν∗1−ρ(·) be the site percolation measure on (Z2)∗. Since ρ > ρc, it fol-lows from the arguments due to Menshikov [20] and Russo [22] that there isα = α(ρ) > 0 such that
ν∗1−ρ(v and w are connected) ≤ exp(−α‖v − w‖∞), for any v, w ∈ Z2,
where ‖·‖∞ denotes the maximum norm on Z2. Once we know the length of thesides of Rk, it follows from the exponential decay that
ν∗1−ρ(Rk is crossed in the easy direction) ≤ 2k+1L exp(−α2k−1L
)(2.1)
≤ exp((k + 1) ln(2) + ln(L)− α2k−1L
).
Since Rk is crossed in the hard direction with respect to Λ iff Rk is not crossedin the easy direction with respect to Λ∗, we have
νρ
( ⋂k∈Z∗+
∆k
)≥ νρ
( ⋂k∈Z∗+
{Rk is crossed in the hard direction
})= ν∗1−ρ
( ⋂k∈Z∗+
{Rk is not crossed in the easy direction
})≥
∏k∈Z∗+
ν∗1−ρ
({Rk is not crossed in the easy direction
})≥ 1−
∑k∈Z∗+
ν∗1−ρ
({Rk is crossed in the easy direction
}),
where the second inequality follows from FKG inequality. By (2.1) the sum aboveconverges, and we just choose L = L(α) = L(ρ) large enough so that this sum isless than 1/2.
For the rest of the text, we will fix L = L(ρ) as given by Lemma 2.2.
2.1.2 Proof of Lemma 2.1: (ii) and (iii)
Fix k ∈ Z∗+, and consider
γ0 = {−1} × [0, L].
Now fix Λ ∈ ∆k, and let
γi = γi(Λ) = L(Ri, γi−1)
defined in a recursive way for all i ∈ Ik. Note that for all i ∈ I∗k ,
|γi ∩ ∂γi−1| = 1
17
Dk
Φk
w1
w4w3
w2
Figure 2.3: Representation of the decorated path Φk(Λ), and its outermost regionthe spiral of rectangles.
and let wi be such a vertex. Now we set the decorated path Φk(Λ) to be the pathstarting at w1, concatenated by γ1 until the first vertex of γ1 that is a neighbor ofw2. Φk(Λ) then concatenates with γ2 through w2 and follow along γ2 until its firstvertex that is a neighbor of w3. The procedure is iterated until the concatenationwith γk through wk and follows to the endpoint of γk.
Two observations are straitforward:
1. |Φk(Λ)| ≥ 2L+∑k−1
i=2 (2i − 2i−2)L = (2k + 2k−1 − 1)L ≥ 2kL,
2. Φk(Λ) is a present minimal path of Λ.
Now let γ ∈ Φk(∆k) and note that γ divides the “spiral” ∪ki=1Ri into tworegions (an innermost and an outermost region the spiral of rectangles). Let usdenote by
Dk = Dk(γ) =k⋃i=1
D(Ri, γi(Λ), γi−1(Λ))
the outermost region. Suppose that γ = (v0, v1, . . . , vn). We say that the vertex viis not attached, if ∂vi ⊆ Dk. Note that if vi and vi+1 are not attached, we wouldhave vi−1 ∼ vi+2 contradicting that γ is a minimal path, therefore the set
Nγ = {i ∈ Z+; vi is not attached}
is 2-spaced.To define the strands of γ ∈ Φk(∆k), choose an arbitrary, but fixed ordering of
Z2. For all i ∈ [n] \Nγ choose yi ∈ ∂vi \Dk following this ordering, and call yi the
18
strand of vi. SetSγ =
⋃i∈[n]\Nγ
{yi}.
Note that the event ∆γ = Φ−1k (γ) is measurable with respect to the states of
vertices of Dk(γ), thus the states of the vertices in Sγ are independent of ∆γ andindependent of each other. It follows that for every event A measurable in Sγ,
νρ(A|∆γ) = νρ(A).
This finishes the proof of Lemma 2.1.
2.2 Enhancements
This section is divided into two parts. In Section 2.2.1 we define a percolationmodel featuring enhancements along vertical columns resembling the percolationmodel studied by Duminil-Copin, Hilario, Kozma and Sidoravicius in [6]. Westate a theorem which is a modification of the one considered there. The proofof Theorem 1.1 will be based on a coupling of the decorated path including itsstrands obtained in Lemma 2.1 and the percolation model with enhancementsalong vertical columns. This coupling is made in Section 2.2.2.
2.2.1 Enhancements along vertical columns
Consider the half-lattice Z+ × Z and let N ⊆ Z+ be a 2-spaced subset. Thepercolation model with enhancements along vertical columns is obtained by firstdeclaring the columns generated by N weak and randomly choosing independentlyof each other vertical columns of Z+ × Z will be declared strong. On this randomenvironment we define an independent percolation process declaring each edgeto be open acording whether it belongs to a weak or a strong column. In theremainder of this section we will give a precise construction of this model andstate the new form of the result of [6].
Let Λ ∈ {0, 1}Z+ , where Λ(j) = 0 for any j ∈ N . We call the column {i} × Zof strong column if Λ(i) = 1 and we call weak column otherwise. Now we declarean edge e to be open with probability pe, where
pe =
{p, if e belongs to a strong column,q, otherwise.
Denote by PΛ
p,q the corresponding probability measure on Z+ × Z under Λ.
19
Let also νNρ be the probability measure on Z+ under which {Λ(i)}i∈Z+\N areindependent Bernoulli random variables with parameter ρ, and let the annealedpercolation law
Pρ,(N )
p,q (·) =
∫{0,1}Z+\N
PΛ
p,q(·)dνNρ (Λ).
The theorem below is a simple adaptation of the main result of [6], which willbe discussed below.
Theorem 2.1. For every ε ∈ (0, 1/2) and ρ > 0, there exists δ = δ(ε, ρ) > 0 andα = α(ε, ρ, δ) > 0, such that for all N ⊆ Z+ 2-spaced subset we have
Pρ,(N )12
+ε, 12−δ(0↔∞) > α.
The idea due to [6] consists in reducing the problem to the result in [16] (quotedin Section 1.2) using a one-step renormalization procedure. The authors divideZ into blocks of size n (overlapping) that will play the role of the columns inrenormalization. They show that these columns are good (in the sense that theblocks have the original strong columns next to each other) with high probability,and that the crossing events in boxes inside the good columns also have highprobability. This renormalized model dominates stochastically the model of [16],and hence percolates. The choice of the size of the blocks requires knowledges ofnear-critical percolation.
The proof of Theorem 2.1 is virtually identical to the main result in [6], with asingle modification of its Lemma 8. This Lemma shows that columns are declaredgood (by the above renormalization) with high probability. The other variationsare minimal. In the statement of the Theorem 2.1 we use the annealed measureinstead of the quenched measure that appears in the original version of [6]. Thelast modification is the exchange of lattice from Z2 to Z+ × Z which is irrelevantto the paper [6].
It is therefore sufficient to show that the following lemma (adaptation of Lemma8 from [6] to Theorem 2.1) is satisfied.
Lemma 2.3 (Lemma 8 for Theorem 2.1). Let ρ > 0, and N be a 2-spaced set.Denote the interval [2n(i− 1), 2n(i+ 1)] by c(i). The interval c(i) is called good ifΛ intersects every sub-interval of c(i) with length d4
ρlog(2n)e. For every i ≥ 1,
limn→∞
νNρ (c(i) is good) = 1.
Proof. For any x ∈ Z and Λ ∈ {0, 1}Z let
l(x) = l(x,Λ) := inf{y − x; y ∈ Λ, y > x}.
20
note thatνNρ(l(x) > k
)≤ (1− ρ)b
k2c ≤ e
−ρk2
and hence
νNρ
(l(x) > k for some x in
[2i(n− 1), 2i(n+ 1)
])≤ 4ne
−ρk2 .
For k = d4ρ
log(2n)e, the right-hand side is at most equal to 1n
while the left-hand
side contains the event that c(i) is bad. Concluding the proof of lemma.
2.2.2 The coupling scheme
We will construct the coupling between the percolation model with colum-nar disorder (on Z3) and the percolation model with enhancements along verticalcolumns (on Z+ × Z). To do this, fix ρ > ρc, k ∈ Z∗+ and a decorated pathγ = (v0, v1, . . . , vn) ∈ Φk(∆k).
First we construct a coupling between the environments of the two models. LetΛ ∈ ∆γ be an environment on Z2 that establishes the columnar disorder. Recallthe sets Nγ and Sγ defined in Section 2.1.2 and set the environment Λ ∈ {0, 1}Z+
as follows:
(i) if i ∈ [n] \ Nγ, set Λ(i) = Λ(yi), where yi is the strand of vi,
(ii) otherwise, set Λ(i) = 0.
Note that by Lemma 2.1, we have that{
Λ(i)}i∈[n]\Nγ
is a family of Bernoulli
random variables with parameter ρ, with respect to measure νρ(·|∆γ). But thesevariables are not independent, because the same vertex y ∈ Sγ can be strand ofup to 4 vertices of γ. Now replace each strand y ∈ Sγ by 4 vertices (y1, y2, y3 andy4) so that no pair of vertices of γ share the same strand. Declare now each oneof these new strands as present independently with probability ρ = ρ(ρ) where
ρ = 1− (1− ρ)4.
It means that each original strand is present if and only if at least one of the newstrands is present. Let {Λ(i)}i∈[n]\Nγ a family of i.i.d. Bernoulli random variables
with parameter ρ and set Λ(i) = 0 for any i ∈ Nγ. Let νρ be the correspondingproduct measure. Note that {Λ(i)}i∈[n]\Nγ stochastically dominates {Λ(i)}i∈[n]\Nγ ,i.e., for all increasing event A ∈ {0, 1}[n]\Nγ ,
νρ(A|∆γ) = νρ(A) ≥ νρ(A).
With respect to Λ, we will declare when an edge of Z+×Z is open or not, in orderto we establish a parallel with the percolation model with enhancements along
21
vertical columns defined in Section 2.2.1. Now we say that the column {i}×Z, fori ∈ Z+, is strong if Λ(i) = 1, and it is weak otherwise. This nomenclature comesfrom the fact that if Λ(i) = 1, then, in the model with columnar disorder, thecolumn {yi} × Z of strand of vi “strengthens” the column {vi} × Z when we seethe decorated path as a cramped sheet. This strengthening is gained by doing atrick of handles on Ci = {vi, yi}×Z as we describe below ( this trick is illustratedin Figure 2.4).
Figure 2.4: Representation on the left of the decorated path and its present strands.On the right, the division of the horizontal edge of a strand into two edges, showingthe strengthening of the column corresponding of the cramped sheet.
Replace each horizontal edge fk ={
(vi, k), (yi, k)}
of Ci into 2 edges (a multi-edge f 1
k , f 2k , one for the bottom handle and the other for the top handle) and declare
independently each of these 2 edges open with probability p′ = p′(p), where
p = 1− (1− p′)2.
This means that each horizontal edge fk is closed if, and only if, the edges f 1k and
f 2k are both closed. Then divide each vertical edge ek =
{(yi, k), (yi, k + 1)
}of
Ci into 4 edges (a multi-edge e1k, e
2k, e
3k and e4
k, one for each of the 4 divisions ofthe strand yi). Declare independently each of these 4 edges open with probabilityp′′ = p′′(p), where
p = 1− (1− p′′)4.
This means that each vertical edge is closed if and only if the corresponding 4multiple edges are closed. Now declare the edge
{(i, k), (i, k+ 1)
}of Z+ × Z to be
open with probabilityp = p(p) = p+ (1− p)p′2p′′,
which corresponds to either{
(vi, k), (vi, k + 1)}
being open or f 2k , ejk and f 1
k+1
being all open. Lastly, we declare each edge of Z+ × Z which does not belong toa strong column to be open with probability p.
22
Since all columns {i} × Z, for i > n, are declared weak, Pρ,(Nγ)
p,p is not the lawof the coupled model on Z+ × Z. Even so, the coupling scheme described abovegives us that
Pρp({v0} × {0} ↔ {vn} × Z|∆γ
)≥ Pρ,(Nγ)
p,p
(o↔ {n} × Z
). (2.2)
2.3 Proof of Theorem 1.1
Proof of Theorem 1.1. We will use the notations present in Section 2.2.2. We firstnote that
p(1
2
)=
1
2+ ε, for some ε > 0.
We then choose δ′ > 0 such that
p(p) >1
2+ε
2, for all p ∈
[1
2− δ′, 1
2
].
From Theorem 2.1 we get
δ = δ( ε
2, ρ)> 0 and α = α
( ε2, ρ, δ
)> 0,
and let
δ = min{δ′, δ
}and p =
1
2− δ.
For each k ∈ Z+, let γ = (v0, v1, · · · , vn) ∈ Φk(∆k) be a decorated path andset ∆γ = Φ−1
k (γ). Denote by C the cluster of the segment {0}× [0, L]×{0} in themodel with columnar disorder and note that v0 ∈ {0}× [0, L]×{0}. Then we have
Pρp(|C| > n) ≥ 1
2Pρp(|C| > n|∆k)
=1
2
∑γ∈Φk(∆k)
Pρp(|C| > n|∆γ)νρ(∆γ|∆k)
≥ 1
2
∑γ∈Φk(∆k)
Pρp({v0} × {0} ↔ {vn} × Z|∆γ)νρ(∆γ|∆k)
(2.2)
≥ 1
2
∑γ∈Φk(∆k)
Pρ,(Nγ)
p,p
(o↔ {n} × Z
)νρ(∆γ|∆k).
By Lemma 2.1 we have n > 2k, and it follows from Theorem 2.1 that
Pρp(|C| ≥ 2k) ≥ α
2.
23
Since the above inequality holds for every k ∈ Z∗+, we have that
Pρp(|C| =∞) ≥ α
2,
thereforePρp(o↔∞) > 0.
Concluding that pc(ρ) ≤ 12− δ.
24
Chapter 3
Percolation on a horizontallystretched square lattice
In this chapter we present the results on independent percolation on the latticewhich is obtained from the square lattice by stretching by the same random amountall the edges that connect sites in two consecutive vertical columns. More precisely,every edge of the square lattice that links sites on the i-th and (i+ 1)-th columnsis replaced by a segment of random length ξi. The ξi’s are assumed to be i.i.d..The edges of the resulting lattice are declared open independently with probabilitype = p|e| where p ∈ [0, 1] and |e| is the length of edge e. The construction of thismodel and the notation can be found in Section 1.3.2. We relate the occurrenceof nontrivial phase transition for this model to moment properties of ξ1. Moreprecisely, we prove that the model undergoes a nontrivial phase transition whenE(ξη1 ) < ∞, for some η > 1 whereas, when E(ξη1 ) = ∞ for some η < 1, nophase transition occurs. This results were obtained in collaboration with AugustoTeixeira, Associate Professor of IMPA.
This chapter is organized as follows. In Section 3.1 we will prove Theorem 1.3which rules out a non-trivial phase transition when the ξi’s have sufficiently heavytails. The rest of the chapter will be dedicated to the proof of Theorem 1.2which establishes the existence of a non-trivial phase transition when the tails aresufficiently light. Section 3.2 contains a brief review on renewal processes and someresults that will be useful for proving a decoupling inequality crucial in the proofof Theorem 1.2. Those who are familiar with the theory of renewal processes canperhaps skip reading this section.
In Section 3.3, we develop the multiscale scheme used to prove Theorem 1.2.First, in Section 3.3.1 we define a fast-growing sequence of numbers which corre-spond to the scales in which we analyze the model. Then we partition Z+ into theso-called k-blocks which are intervals whose length are related to the k-th scale.The k-blocks will be declared either bad or good hierarchically in such a way that
25
being bad indicates that the renewal process within the block has arrivals that areclose to each other. We will show that bad k-blocks are extremely rare under theappropriate moment condition. Section 3.3.2 will be dedicated to the constructionof crossing events in rectangles whose bases are k-blocks and with large heights.Such crossings will have a very high probability on good k-blocks. In Section 3.4,we finish the proof of Theorem 1.2, using the results established throughout thechapter.
3.1 Proof of Theorem 1.3
Proof of Theorem 1.3. Let η < 1 be such that E(ξη) =∞. Then∑n=0
P(ξη > n) =∞. (3.1)
Setting ε > 0 such that η−1 = 1 + 2ε, (3.1) reads∑n=0
P(ξ > n1+2ε) =∞.
Now consider the events Fi = {ξi ≥ i1+2ε}, with i ∈ Z∗+ and recall that {ξi}i∈Z∗+are independent copies of ξ. Since the Fi are independent and
∑i P(Fi) =∞, we
have that υξ(Fi i.o.) = 1.Let us now fix Λ ∈ {Fi i.o.} together with an increasing subsequence ik = ik(Λ),
k ∈ Z+ such that Fik occurs for every k. Roughly speaking, the columns thatproject to xik−1
and xik are too distant from each other to allow paths to connectbetween them.
We now fix a λ ∈ (0,∞) and show that PΛλ (o↔∞) = 0. Recall the definition
of the model on Z2+ whose inhomogeneities are given by pe in (1.4) and also recall
the crossing events introduced in (1.1), (1.2) and (1.3). For each k ∈ Z+, let
Rk = R([
0, ik)×[0, dexp
(i1+εk
)e)),
and note that
PΛλ (o↔∞) ≤ PΛ
λ
(Ch(Rk)
)+ PΛ
λ
(Cv(Rk)
). (3.2)
The probability of Ch(Rk) is bounded above by the probability that there is anopen edge between the columns {ik− 1}×Z+ and {ik}×Z+. Since these columnsare very far apart, the height of Rk is not large enough to ensure the existence ofan open edge with good probability. In fact, let
Jk = {0, 1, · · · , dexp(i1+εk
)e − 1}
26
and note that
PΛλ
(Ch(Rk)
)≤ PΛ
λ
( ⋃j∈Jk
{{(ik − 1, j), (ik, j)} is open
})≤ dexp
(i1+εk
)e exp(−λξik)
≤ dexp(i1+εk
)e exp
(−λi1+2ε
k
) k→∞−→ 0, (3.3)
where we used the definition of ik(Λ) in the last inequality.In order to estimate the probability of Cv(Rk), we note that, on this event there
must be at least one vertical edge connecting the j-th and (j + 1)-th layer in Rk,for j ∈ Jk. The height of Rk is large enough to guarantee that this event hasvanishing probability as k grows. Let
λ = ln(1− exp(−λ)
)< 0 (3.4)
and note that
PΛλ
(Cv(Rk)
)≤ PΛ
λ
( ⋂j∈Jk
ik−1⋃l=0
{{(l, j), (l, j + 1)} is open
})=
(1−
(1− exp(−λ)
)ik)|Jk|(3.4)=
(1− exp
(λik))|Jk|
≤ exp(− exp
(λik)|Jk|)
≤ exp(− exp
(λik + i1+ε
k
)) k→∞−→ 0. (3.5)
In the second inequality sign above we used 1− x ≤ exp(−x).Combining (3.2), (3.3) and (3.5),
PΛλ (o↔∞)
k→∞−→ 0,
which concludes the proof.
3.2 Renewal process
The purpose of this section is to prove a decoupling inequality (see Lemma 3.4),which will be used as a fundamental tool on our multiscale analysis in Section3.3.1. For this a brief outline of some results on renewal processes, it taken from[19], will be presented.
27
3.2.1 Definition and notation
Let ξ be a positive integer-valued random variable, called interarrival time,and χ be a non-negative integer-valued random variable, called delay. As before,let {ξi}i∈Z∗+ be i.i.d. copies of ξ which are also independent of χ. We define therenewal process
X = X(ξ, χ) = {Xi}i∈Z+
recursively as:
X0 = χ, and Xi = Xi−1 + ξi for i ∈ Z∗+.
We say that the i-th renewal takes place at time t if Xi−1 = t.It is convenient to define two other processes
Y = Y (ξ, χ) = {Yn}n∈Z+ and Z = Z(ξ, χ) = {Zn}n∈Z+ ,
as
Yn =
{1, if a renewal of X occurs at time n,0, otherwise.
(3.6)
and the residual life
Zn = min{Xi − n; i ∈ Z+ and Xi − n ≥ 0}. (3.7)
The processes Y and Z will also be called renewal process with interarrival timeξ and delay χ, since each one of X, Y and Z fully determines the two others(see Figure 3.1). The probability law that governs these renewal processes willbe denoted by υχξ (·). It is worth noting that Z is a Markov chain with transitionkernel given by
P(Zn = i|Zn−1 = j) =
P(ξ = i+ 1), if j = 0
1, if i+ 1 = j > 00, otherwise
, (3.8)
for all n ∈ Z∗+, and i, j ∈ Z+.For m ∈ Z+ consider θm : Z∞ 7→ Z∞, the shift operator given by
θm(x0, x1, · · · ) = (xm, xm+1, · · · ).
It is desirable that the renewal process Z be invariant under shifts, i.e.
θmZd= Z for any m ∈ Z∗+, (3.9)
28
X0 X1 X2 X3 X4
1 1 1 1 10 0 0 0 0 0 0 0 0
1 0 1 0 4 3 2 1 0 2 1 0 0 · · ·
ξ1 ξ2 ξ3 ξ4 ξ5χ
0X :
Yn :
Zn :
Figure 3.1: Illustration of the processes X, Y, Z. In this realization, χ = 1, ξ1 = 2,ξ2 = 5, ξ3 = 3 and ξ4 = 1.
whered= means equality in distribution. If E(ξ) < ∞, we can define a random
variable ρ = ρ(ξ) with distribution
ρk = P(ρ = k) :=1
E(ξ)
∑i=k+1
P(ξ = i), for any k ∈ Z+. (3.10)
Using ρ as the delay, implies that the resulting renewal process Z(ξ, ρ) satisfies(3.9). In fact, using (3.8) we have
P(Z1 = k) = P(Z1 = k|Z0 = k + 1)P(Z0 = k + 1) + P(Z1 = k|Z0 = 0)P(Z0 = 0)
= 1 · ρk+1 + P(ξ = k + 1)ρ0
=1
E(ξ)
∑i=k+2
P(ξ = i) +P(ξ = k + 1)
E(ξ)
=1
E(ξ)
∑i=k+1
P(ξ = i) = ρk.
ThereforeZ1
d= Z0
d= ρ.
Using the Markov property and induction, it follows that θmZd= Z. Consequently
it also follows that θmYd= Y for Y = Y (ξ, ρ). For a fixed ξ, ρ is called stationary
delay.From same properties of expectation, one obtain
if E(ξ1+ε) <∞ then E(ρε) <∞. (3.11)
3.2.2 Loss of memory
The purpose of this section is to show how the renewal process Z = {Zn}n∈Z+
forgets its initial state when n→∞. This fact will be important in what follows.
29
We say that a random variable ξ is aperiodic if
gcd{k ∈ Z∗+; P(ξ = k) > 0
}= 1.
For the rest of this section we assume ξ is aperiodic and also E(ξ) <∞.Let Y = Y (ξ, χ) and Y ′ = Y (ξ, χ′) be two independent renewal processes with
interarrival time ξ and delay χ and χ′. Set
T := min{k ∈ Z∗+; Yk = Y ′k = 1},
the coupling time of the renewal processes X and X ′. And denote υχ,χ′
ξ (·) the
product measure υχξ ⊗ υχ′
ξ .The next lemma will ensure that the processes X and X ′ meet a.s..
Lemma 3.1. If ξ aperiodic and E(ξ) <∞, then
υχ,χ′
ξ (T <∞) = 1. (3.12)
Proof. Set Y = Y (ξ, χ, χ′) ={Yn}n∈Z+
as
Yn := Yn · Y ′n, for all n ∈ Z+.
Notice that Y is also a renewal process, since ξ is aperiodic. Denote ξ its interarrivaltime. It is worth noting that ξ is nondefective, i.e.∑
i∈Z∗+
P(ξ = i
)= 1.
Otherwise we would have P(ξ =∞) > 0 which would imply
limn→∞
υρ,ρξ(Yn = 1
)= 0,
where ρ is the stationary delay with respect to ξ. But this cannot happen, since
υρ,ρξ(Yn = 1
)= υρξ (Yn = 1)υρξ (Y
′n = 1)
= υρξ (Zn = 0)υρξ (Z′n = 0) = ρ2
0 =
(1
E(ξ)
)2
> 0,
where we used the stationarity of ρ for equality between the lines. Hence
υδ0,δ0ξ
(Yn = 1 i.o.
)= 1,
where δx denotes the delta distribution concentrated in x ∈ Z.
30
Now since ξ is aperiodic, it follows that there is n0 ∈ Z+ such that
υδ0ξ (Yn = 1) > 0 for all n ≥ n0.
(This result can be seen in [17], Lemma 1.30) Therefore, for m ∈ Z,
1 = υδ0,δ0ξ
(Y = 1 i.o.
)(3.13)
= υδ0,δ0ξ
(Y = 1 i.o.|(Yn0 , Y
′n0+m) = (1, 1)
)υδ0,δ0ξ
((Yn0 , Y
′n0+m) = (1, 1)
)+
+υδ0,δ0ξ
(Y = 1 i.o.|(Yn0 , Y
′n0+m) 6= (1, 1)
)υδ0,δ0ξ
((Yn0 , Y
′n0+m) 6= (1, 1)
).
Note that if
xz + y(1− z) = 1 for x, y ∈ [0, 1] and z ∈ (0, 1),
then x = 1. Therefore, it follows from (3.13) that
υδ0,δmξ
(Y = 1 i.o.
)= υδ0,δ0ξ
(Y = 1 i.o.|(Yn0 , Y
′n0+m) = (1, 1)
)= 1.
Therefore for any i, j ∈ Z+,
υδi,δjξ
(Y = 1 i.o.
)= υ
δ0,δ|i−j|ξ
(Y = 1 i.o.
)= 1.
The result can finally be established
υχ,χ′
ξ (T <∞) ≥ υχ,χ′
ξ
(Y = 1 i.o.
)=
∑i∈Z+
∑j∈Z+
υδi,δjξ
(Y = 1 i.o.
)P(χ = i)P(χ′ = j) = 1,
where the above equality follows the conditioning in the first renewal (delay).
The lemma below shows that renewal processes forget the delay.
Lemma 3.2 (Renewal theorem). If ξ aperiodic and E(ξ) <∞, then for all eventA,
limn→∞
∣∣∣υχξ (θnY ∈ A)− υχ′
ξ (θnY′ ∈ A)
∣∣∣ = 0 (3.14)
for any delays χ and χ′.Moreover,
limn→∞
υχξ (Zn = k) = ρk (3.15)
for all k ∈ Z+ and any delay χ.
31
Proof. For every event A, we have∣∣∣υχξ (θnY ∈ A)− υχ′
ξ (θnY′ ∈ A)
∣∣∣ ≤ υχ,χ′
ξ (T > n). (3.16)
Taking the limit as n→∞ on both sides of (3.16) and using (3.12) we have
limn→∞
υχ,χ′
ξ (T > n) = υχ,χ′
ξ (T =∞) = 0.
This establishes (3.14). In order to prove (3.15), plug χ′ = ρ in (3.14) and notethat {Zn = k} = {Yn+k = 1},
limn→∞
υχξ (Zn = k) = limn→∞
υρξ (Zn = k) = υρξ (Z0 = k) = ρk.
3.2.3 Decoupling
The goal of this section is to prove a decoupling for stationary renewals. Toprove decoupling inequality we will use (3.16). We will also need to bound the
value of υχ,χ′
ξ (T > n). The upper bound for υχ,χ′
ξ (T > n) will be obtained by theMarkov inequality applied to the random variable T ε, where ε > 0. The lemmabelow guarantees that T ε has finite expected value and will be crucial to provethe decoupling inequality. Its proof was taken from [19] (Theorem 4.2, page 27)and/or [18] (Proposition 1).
Lemma 3.3. Let ξ be an aperiodic positive integer-valued random variable. Sup-pose that for some ε ∈ (0, 1), E(ξ1+ε) < ∞, and that χ, χ′ are non-negativeinteger-valued random variables with E(χε) and E(χ′ε) finite. Then E(T ε) <∞.
Proof. Denote by Eχ,χ′
ξ (·) the expected value with respect to measure υχ,χ′
ξ , andnote that
Eχ′,χ′
ξ (T ε) = Eχ,χ′
ξ
((min(χ, χ′) + T −min(χ, χ′)
)ε)≤ Eχ,χ
′
ξ
(min(χ, χ′)ε
)+ Eχ,χ
′
ξ
((T −min(χ, χ′)
)ε)≤ Eχ,χ
′
ξ
(min(χ, χ′)ε
)+ Eδ0,ϕξ (T ε),
where ϕ = max(χ, χ′) − min(χ, χ′). By hypothesis Eχ,χ′
ξ
(min(χ, χ′)ε
)< ∞ and
Eχ,χ′
ξ (ϕε) < ∞. Therefore without loss of generality, we will assume that χ = δ0.
To simplify the notation we will omit the variables ξ, χ and χ′ in Eχ,χ′
ξ when thereis no risk of ambiguity.
32
LetY = Y (ξ, δ0) and Y ′ = Y (ξ, χ′)
mutually independent (as well as X and X ′). The idea behind the proof is to makea series of attempts for processes Y and Y ′ find each other, where each attempt willhave positive probability of occurring. The uniform lower bound for the successof each attempt is obtained by taking n0 ∈ Z+ and σ > 0 such that
υδ0ξ (Zn = 0) ≥ σ for all n ≥ n0.
This follows from the Lemma 3.2, because
limn→∞
υδ0ξ (Zn = 0) =1
E(ξ).
Define the random variables Fn, Hn and vn for n ∈ Z∗+ inductively by F0 = 0and,
H2n = min{X ′i − F2n; X ′i − F2n ≥ 0} = X ′v2n− F2n,
F2n+1 = X ′v2n+n0,
H2n+1 = min{Xi − F2n+1; Xi − F2n+1 ≥ 0} = Xv2n+1 − F2n+1,
F2n+2 = = Xv2n+1+n0 .
Note that Hn = 0 corresponds to a success. See Figure 3.2.The reason for addingn0 steps to the variables Xv1 , X
′v2, · · · is that the probability of success is at least
equal to σ. Now define
τ = min{k ∈ Z+; Hk = 0}
andU1 = Xv1 −X ′v0
, U2 = X ′v2−Xv1 , · · · .
Consider also the filtration {Ni}i∈Z+ , where Ni is generated by the variables
Xj, X′k, where j ≤ vi and k ≤ vi−1 + n0, when i is odd, or
Xj, X′k, where k ≤ vi and j ≤ vi−1 + n0, when i is even.
Note that,
T ≤ X ′0 +τ∑i=1
Ui = χ′ +∑i=1
UiI{τ≥i}, (3.17)
where IS denotes the indicator function of the set S. Since(∑i=1
xi
)ε≤∑i=1
xεi (3.18)
33
F0 F1 F2
X
X ′X′v0 X′v2
Xv1
H0
H1
Figure 3.2: In this example we have n0 = 3, H0 > 0, H1 > 0 and H2 = 0. Henceτ = 2 and we have a successful coupling.
when xi ≥ 0 and ε ∈ (0, 1), the inequality (3.17) implies that
Eδ0,χ′(T ε)≤ E(χ′ε) +
∑i=1
E(U εi I(τ≥i)). (3.19)
For each term of the above sum,
E(U εi I(τ≥i)) = E
(E(U ε
i I(τ≥i)|Ni−1))
≤ E(E(U ε
i |Ni−1)I(τ≥i)
). (3.20)
To conclude the proof it suffices to show that E(U εi |Ni−1) < K, for some
constant K, because combining (3.19), (3.20), E(χ′ε) <∞ and the choice of n0
Eδ0,χ′(T ε)≤ E(χ′ε) +
∑i=1
KP(τ ≥ i)
≤ E(χ′ε) +∑i=1
K(1− σ)i <∞.
Applying one more time (3.18)
E(U εi |Ni−1) = E
((Xvi−1+n0 −Xvi−1
+Hi)ε|Ni−1
)= n0E(ξε) + E(Hε
i |Hi−1). (3.21)
Now think that Fi−1 = 0, and note that Hi = Z∗k , where Z∗ = Z(δ0, ξ) as defined
34
in (3.7) and k = Hi−1 +Xvi−1+n0 −Xvi−1. Note that
E((Z∗k)ε
)=
∑i=1
iεP(Z∗k = i)
=∑i=1
iεk−1∑m=0
P(Z∗m = 0)P(ξ = k + i−m)
≤∑i=1
iε∑m=i
P(ξ = m)
=∑i=1
(1ε + 2ε + · · ·+ iε)P(ξ = i)
≤∑i=1
(i+ 1)1+εP(ξ = i) <∞, (3.22)
where the last conclusion follows from E(ξ1+ε) <∞. The lemma follows combining(3.21) and (3.22).
We can now prove the decoupling.
Lemma 3.4 (Decoupling). Let ξ be a positive integer-valued, aperiodic randomvariable with E(ξ1+ε) < ∞, for some ε > 0, and consider the renewal processY = Y
(ξ, ρ(ξ)
)defined in (3.6).
Then there is c1 = c1(ξ, ε) ∈ (0,∞) such that for all n,m ∈ Z+ and for allevents A and B, where
A ∈ σ(Yi; 0 ≤ i ≤ m) and B ∈ σ(Yi, i ≥ m+ n)
we haveυρξ (A ∩B) ≤ υρξ (A)υρξ (B) + c1n
−ε.
Proof. For simplicity the indices ξ will be omitted in this proof. If υρ(A) = 0 thereis nothing to be proved. Suppose then that υρ(A) > 0. Recall the definition of Zin (3.7), and note that
υρ(A ∩B) = υρ(A ∩B ∩ {Zm > n/2}) + υρ(A ∩B ∩ {Zm ≤ n/2})≤ υρ(Zm > n/2) + υρ(A)υρ(B ∩ {Zm ≤ n/2}|A)
≤ υρ(Zm > n/2) + υρ(A)∑
0≤i≤bn/2cυρ(Zm=i|A)>0
υρ(B|A,Zm = i)υρ(Zm = i|A)
≤ υρ(Zm > n/2) + υρ(A) max0≤j≤bn/2c
υδm+j(B)∑
0≤i≤bn/2c
υρ(Zm = i|A)
≤ υρ(Zm > n/2) + υρ(A) max0≤j≤bn/2c
υδm+j(B). (3.23)
35
Now we compare υδm+j(B) with υρ(B), when 0 ≤ j ≤ bn/2c. Using thatυδm+j(B) = υδ0
(θm+j(B)
)and by the stationarity of ρ∣∣υδm+j(B)− υδρ(B)
∣∣ =∣∣υδ0(θm+j(B)
)− υρ
(θm+j(B)
)∣∣(3.16)
≤ υδ0,ρξ (T > n− j)≤ υδ0,ρξ (T > n/2). (3.24)
By (3.23), (3.24) and by the fact Zmd= Z0
d= ρ
υρ(A ∩B) ≤ υρ(A)υρ(B) + υρ(ρ > n/2) + υδ0,ρξ (T > n/2)
≤ υρ(A)υρ(B) + 2εE(ρε)n−ε + 2εEδ0,ρξ (T ε)n−ε,
where the last inequality follows from the Markov inequality for ρε and T ε. Finallytake c1 = 2εE(ρε) + 2εEδ0,ρξ (T ε) which is finite by (3.11) and Lemma 3.3.
3.3 The multiscale scheme
Throughout this section we fix ξ positive, integer-valued and aperiodic. Wealso assume E(ξ1+ε) < ∞ for a some ε > 0 and denote ρ = ρ(ξ) the respectivestationary delay given in (3.10). These conditions on ξ allow us to redefine thepercolation model on horizontally stretched square lattice to obtain an equivalentmodel on Z2
+ as follows:Consider the environment Λ ⊆ Z+ distributed as υξ and set
Evert(Λc) :=
{{(x, y), (x, y + 1)} ∈ E(Z2
+);x 6∈ Λ, y ∈ Z+
}.
Let each edge e ∈ E(Z2+) be open independently with probability
pe =
{0, if e ∈ Evert(Λ
c),p, if e 6∈ Evert(Λ
c).
Edges which are not open are called closed.Geometrically, this formulation consists in preserving the columns of the Z2
+
lattice that project to Λ while deleting the ones that project to Λc. The resultinggraph is similar to the stretched lattice LΛ defined in Section 1.3.2, however, theedges are now split into unit length segments. Each one of these edges is openindependently with probability p.
Note that, we can recover the original formulation on LΛ by declaring an edgeopen if all the corresponding unitary edges in Z2
+ are open in the new formulation.Therefore, these two formulations are equivalent and we slightly abusing notation,denoting PΛ
p (·) the law of this new model.
36
Since the model is now defined on Z2+, one can define the rectangle R =
R([a, b) × [c, d)
)for any a, b, c, d ∈ Z+ and the corresponding (horizontal and
vertical) crossing events as in (1.1), (1.2) and (1.3). In the remainder of thissection only this new definition of the model will be adopted.
3.3.1 Environments
Let us fix constants
α ∈(
0,ε
2
]and γ ∈
(1, 1 +
α
α + 2
). (3.25)
which will appear as exponents in several expressions below. The exponent γ willgive the rate of growth for the scales in which we study the environment while αwill give the rate of decay of probability that bad events occur in each scale (see(3.26) and (3.30)).
Let us also fix L0 = L0(ξ, ε, α, γ) ∈ Z+ sufficiently large so that
(i) Lγ−10 ≥ 3,
(ii) Lε−α0 ≥ E(ρε) and
(iii) Lc20 ≥ c1 + 1, where c1 is given by the Lemma 3.4 and
c2 = 2 + 2α− γα− 2γ.
Note that c2 > 0 by the choice of γ in (3.25).
Once L0 is fixed, we can define recursively the sequence of scales (Lk)k∈Z+ byputting
Lk = Lk−1bLγ−1k−1c, for any k ≥ 1. (3.26)
Item (i) in the definition of L0 together with (3.25) and (3.26) implies that thescales grow superexponentially. In fact,(
2
3
)kLγ
k
0 ≤ · · · ≤2
3Lγk−1 ≤ Lk ≤ Lγk−1 ≤ · · · ≤ Lγ
k
0 . (3.27)
The items (ii) and (iii) are necessary to prove the Lemma 3.5 below.For k ∈ Z+, consider the partition of R+ into the intervals
Ikj =[jLk, (j + 1)Lk
), with j ∈ Z+.
37
The interval Ikj is said to be the j-th k-block. Note that the k-blocks have size Lkand are formed by bLγ−1
k−1c (k − 1)-blocks. More precisely, for k ≥ 1
Ikj =⋃i
Ik−1i , with i ∈
{jbLγ−1
k−1c, · · · , (j + 1)bLγ−1k−1c − 1
}. (3.28)
Now fix an environment Λ ⊆ Z+. Blocks will be labeled good or bad dependingon Λ in a recursive way. For k = 0 declare the j-th 0-block, I0
j , good if Λ∩ I0j 6= ∅,
and bad otherwise. Once labeled the (k−1)-blocks, declare a k-block bad, if thereare at least two non-consecutive bad (k − 1)-blocks contained in it. Otherwise,we declare the k-block good. More precisely, for j, k ∈ Z+ consider the events Akjdefined recursively by
A0j = {Λ ⊆ Z+; Λ ∩ I0
j = ∅} = {I0j is bad}, and
Akj =⋃
j−k ≤i1,i2≤j+k
|i1−i2|≥2
(Ak−1i1∩ Ak−1
i2
)= {Ikj is bad}, for k ≥ 1, (3.29)
where j−k and j+k correspond respectively to the smallest and greatest among the
indices of the (k − 1)-blocks that form Ikj . By (3.28) we know that
j−k = jbLγ−1k−1c and j+
k = (j + 1)bLγ−1k−1c − 1.
It follows from (i) that every block is formed by at least three blocks from theprevious scale, which results that the events Akj (for k ≥ 1) are non trivialitydefined.
We now definepk := υρ(Ak0) = υρ(Akj )
where the equality follows from the stationarity of ρ.The next lemma establishes an upper bound for the pk’s, which is a power law
in Lk with exponent α.
Lemma 3.5. For every k ∈ Z+ we have
pk ≤ L−αk . (3.30)
Proof. We proceed by induction on k. First observe that
p0 = υρ(A00)
(3.7)= υρ(Z0 > L0)
(3.10)= P(ρ > L0) ≤ E(ρε)
Lε0,
where the last inequality follows from the Markov’s inequality for ρε. Therefore(ii) implies p0 ≤ L−α0 .
38
Now suppose that for some k ∈ Z+, pk ≤ L−αk . Set
I :={
(i1, i2) ∈ Z2+; i1, i2 ≤ bLγ−1
k c − 1 and |i1 − i2| ≥ 2},
and note that
pk+1 = υρ(Ak+10 )
(3.29)
≤∑
(i1,i2)∈I
υρ(Aki1 ∩ Aki2
)
≤ bLγ−1k c
2(p2k + c1L
−εk )
≤ L2γ−2k (L−2α
k + c1L−εk )
≤ (1 + c1)L2γ−2−2αk ,
where the second inequality follows from the Lemma 3.4 and the third inequalityfollows from the induction hypothesis.
Therefore
pk+1
L−αk+1
≤ (1 + c1)L2γ−2−2αk Lαk+1 ≤ (1 + c1)L2γ−2−2α+γα
k
(iii)
≤ 1.
Finishing the proof.
3.3.2 Crossings
In this section we will define crossing events in certain rectangles of Z2+. The
base of such an rectangle will be a k-block, for some k. Also, the rectangles willbe very elongated on the vertical direction, meaning that the height is a stretchedexponential function of the length of the base. First fix
µ ∈(
1
γ, 1
)(3.31)
and define recursively the sequence of heights (Hk)k∈Z+ by
H0 = 100 and Hk = 2dexp(Lµk)eHk−1, for k ≥ 1.
The choice H0 = 100 is arbitrary and we could have used any other positive integer.Recall of the crossing events defined in (1.2) and (1.3). For i, j, k ∈ Z+ consider
the events (see Figure 3.3).
Cki,j = Ch
((Iki ∪ Iki+1
)×[jHk, (j + 1)Hk
))and (3.32)
39
Iki Iki+1
jHk
(j + 1)Hk
(j + 2)Hk
Figure 3.3: Illustration of the occurrences of events Cki,j and Dk
i,j.
Dki,j = Cv
(Iki ×
[jHk, (j + 2)Hk
)). (3.33)
Now set
qk(i, j) = max
{max
Λ; Iki and
Iki+1are good
PΛλ ({Ck
i,j}c), maxΛ; Iki is
good
PΛλ ({Dk
i,j}c)
}.
And note that for any k ∈ Z+,
qk := qk(0, 0) = qk(i, j), for any i, j ∈ Z+. (3.34)
The next lemma will guarantee that the crossing events above have high prob-ability, since the bases of these rectangles are good blocks. Using this lemma, wewill construct an infinite cluster with positive probability, by taking intersectionsof these rectangles. But before stating it, note that by (3.31), we have γµ− γ < 0and so we can choose
β ∈ (γµ− γ + 1, 1). (3.35)
Lemma 3.6. There are c3 = c3(γ, L0, µ, β) ∈ Z+ sufficiently large and λ =λ(γ, L0, µ, β, c3) > 0 sufficiently small such that
qk ≤ exp(−Lβk
), for any k ≥ c3.
To prove the previous lemma it suffices to prove the following auxiliary lemmas:
40
Lemma 3.7. Let λ < 1, then there is c4 = c4(γ, L0, µ, β) ∈ Z+, such that for allk ≥ c4 we have the implication
qk ≤ exp(−Lβk
)⇒ PΛ
λ (Ck+10,0 ) ≤ exp
(−Lβk+1
),
for all environment Λ satisfying that the blocks Ik+10 and Ik+1
1 are good.
Lemma 3.8. There is c5 = c5(γ, L0, µ, β) ∈ Z+, such that for all k ≥ c5 we havethe implication
qk ≤ exp(−Lβk
)⇒ PΛ
λ (Dk+10,0 ) ≤ exp
(−Lβk+1
),
for all environment Λ satisfying that the block Ik+10 is good.
Now we give the proofs of the above lemmas.
Proof of Lemma 3.6. Lemmas 3.7 and 3.8 imply that it is enough to setc3 = max{c4, c5} and choose λ = λ(γ, L0, µ, β, c3) > 0 small enough so thatqc3 ≤ exp
(−Lβc3
).
Proof of Lemma 3.7. Fix an environment Λ such that Ik+10 and Ik+1
1 are goodblocks. Both (k + 1)-blocks can be formed by up to two (consecutive) bad k-blocks. Although the probability of crossing a bad k-block is small, this will becompensated for giving a lot of opportunities. For this, divide the height Hk+1
into bands of size 2Hk, and see if there are crossings in these bands (see Figure3.4). Recall the notations (1.1), (1.2) and consider the events
Gj = Ch
(R([0, 2Lk+1)× [jHk, (j + 2)Hk)
))and note that {Ck+1
0,0 }c ⊆ ∩jGc2j, where 0 ≤ j ≤ dexp
(Lµk+1
)e − 1. This along with
the independence and invariance of events G’s implies that
PΛλ
({Ck+1
0,0 }c)≤ PΛ
λ (Gc0)dexp(Lµk+1)e ≤ PΛ
λ (Gc0)exp(Lµk+1). (3.36)
To obtain the crossing G0 we will intercept the crossing events C’s and D’swhose bases are good k-blocks that form Ik+1
0 and Ik+11 , plus the crossing of the
bad k-blocks, which we shall denote by B0 and B1 (with respect to Ik+10 and Ik+1
1
respectively), see Figure 3.5.Now we formally define the events Bl for l ∈ {0, 1}. If all the k-blocks that
form Ik+1l are good, define Bl = ∅. Otherwise denote by jl the lowest index of a
bad k-block that forms Ik+1l and set the interval
I∗l = (Ikjl−1 ∪ Ikjl ∪ Ikjl+1 ∪ Ikjl+2) ∩ (Ik+1
0 ∪ Ik+11 ) ⊆ Z+.
41
Ik+10 Ik+1
1
0
2Hk
4Hk
6Hk
...
Hk+1
G2
......
Figure 3.4: Illustration of the event G2, implying in the occurrence of {Ck+10,0 }c.
Note that the interval I∗l contains all the bad k-blocks that form Ik+1l , plus the
previous k-blocks and the posterior one (as long as they are contained in Ik+10 ∪
Ik+11 ). And set
Bl ={
all edges of the form{
(m, 0), (m+ 1, 0)}
with m ∈ I∗l are open}.
Since λ < 1, and by the definition of B0 and B1, we have
PΛλ (B0 ∩B1) ≥ exp(−8λLk) ≥ exp(−8Lk). (3.37)
0
Hk
2Hk
Ik0 Ik1 · · · · · ·Ikj0 Ikj1 Ikj1+1
I∗0 I∗1
Ik+10 Ik+1
1
Figure 3.5: Illustration of the occurrence of the events C’s, D’s, B0 and B1 thatimply in the occurrence of G0. Ikj0 , Ikj1 and Ikj1+1 correspond to bad k-blocks.
42
Note that ( ⋂i; Iki ,I
ki+1
are good
Cki,0
)∩( ⋂j; Ikj is
good
Dkj,0
)∩B0 ∩B1 ⊆ G0 (3.38)
where 0 ≤ i, j ≤ 2bLγ−1k c − 1. Since the events B’s, C’s and D’s are increasing
events, it follows from the FKG inequality, (3.34), (3.37) and (3.38) that
PΛλ (G0) ≥ (1− qk)4bLγ−1
k c exp(−8Lk)
≥ (1− 4Lγ−1k qk) exp(−8Lk). (3.39)
Now suppose qk ≤ exp(−Lβk) and let c6 = c6(L0, γ, β) ∈ Z+ sufficiently largesuch that for k ≥ c6 we have
1− 4Lγ−1k qk ≥ 1− 4Lγ−1
k exp(−Lβk
)≥ 1/2. (3.40)
By (3.36), (3.39) and (3.40)
PΛλ
({Ck+1
0,0 }c)
exp(−Lβk+1
) ≤ exp(Lβk+1
)(1− exp(−8Lk)
2
)exp(Lµk+1)
≤ exp(Lβk+1 − exp
(−8Lk + Lµk+1 − log 2
))(3.27)
≤ exp(Lγβk − exp(−8Lk + 0.66µLγµk − log 2)
),
where in the second inequality we use that 1− x ≤ exp(−x).Since γµ > 1, by (3.31), we can take c4 = c4(γ, L0, µ, β, c6) ≥ c6 sufficiently
large such that for any k ≥ c4 the above inequality becomes at most 1. Thusproving the lemma.
Proof of Lemma 3.8. Fix an environment Λ such that Ik+10 is good. We will es-
timate PΛp (Dk+1
0,0 ) using a Peierls-type argument in a renormalized lattice. Each
rectangle Iki ×[jHk, (j + 1)Hk
)will correspond to a vertex (i, j) in this renormal-
ized lattice. This renormalized lattice is then just the Z2+ lattice and the vertex
(i, j) ∈ Z2+ is declared open if the event Ck
i,j ∩ Dki,j occurs, see Figure 3.6. This
gives rise to a dependent percolation process in the the renormalized lattice.Since Ik+1
0 is good, either Iki is good for every i ∈{
0, 1, · · · ,⌊
12bLγ−1
k c⌋− 1}
or
Iki is good for every i ∈{⌊
12bLγ−1
k c⌋
+ 1, · · · , bLγ−1k c− 1
}. Assume without loss of
generality that the former holds and define
L :=⌊
12bLγ−1
k c⌋− 1. (3.41)
43
Consider the rectangle
R = R([
0, L)×[0, 4dexp
(Lµk+1
)e)),
and the event Cv(R) that this rectangle is crossed vertically. Note that
PΛp (Dk+1
0,0 ) ≥ P(Cv(R)
).
Now we use the Peierls argument: suppose that the event Cv(R) does not occur.Then there exists a sequence of distinct vertices (i0, j0), (i1, j1), · · · , (in, jn) in Rsuch that
1. max{|il − il−1|, |jl − jl−1|
}= 1,
2. (i0, j0) ∈ {0}×[0, 4dexp
(Lµk+1
)e]
and (in, jn) ∈ {L}×[0, 4dexp
(Lµk+1
)e]
and
3. (ik, jk) is closed for every k = 0, · · · , n.
Note that there are at most 4dexp(Lµk+1
)e8n sequences with n+ 1 vertices that
satisfy 1. and 2. Also, the probability that a vertex of R be open is at least
1− 2qk(p) ≥ 1− 2 exp(−Lβk
).
By the geometry of the crossing events in the original lattice, for any (i, j) ∈ Z2+,
the event {(i, j) is open} in the renormalized lattice depends on {(i′, j′) is open}for, at most 7 distinct vertices (i′, j′) (see Figure 3.6). Therefore, for every setcontaining n+1 vertices, there are at least bn/7c vertices whose states are mutuallyindependent.
Therefore,
P(Cv(R)c
)≤∑n
P(there is a sequence of n+ 1 vertices satisfying 1., 2. and 3.)
≤∑n≥L
4dexp(Lµk+1
)e8n(
2 exp(−Lβk
))bn/7c≤ 4dexp
(Lµk+1
)e∑n≥L
exp(n ln 8 + bn/7c ln 2− bn/7cLβk
)(3.41)
≤ c7 exp(Lµk+1 − c8 · Lβ+γ−1
k
),
for some c7 = c7(γ, L0, β) > 0 and c8 = c8(γ, L0, β) > 0 sufficiently large.Therefore,
PΛp ({Dk+1
0,0 }c)
exp(−Lβk+1
) ≤ c7 exp(Lγµk + Lγβk − c8L
β+γ−1k
)(3.42)
44
Iki Iki+1· · · · · ·
jHk
(j + 1)Hk
(j + 2)Hk
i
j
Figure 3.6: On the left, we illustrate the occurrence of the event Cki,j ∩ Dk
i,j onthe original lattice. On the right, we depict the renormalized square lattice wherethe circles represent the sites. The occurrence of Ck
i,j ∩Dki,j in the original lattice
implies that the site (i, j) (represented as a black circle) is open in the renormalizedlattice. The state of the site (i, j) only depends on the state of the other six sitesrepresented as circles with a dot inside.
It follows from the choice of β in (3.35), that
β + γ − 1 > max{γβ, γµ}.
The proof now follows by choosing c5 = c5(γ, L0, µ, β) sufficiently large so that theright-hand side of the (3.42) is less than 1 whenever k ≥ c5.
3.4 Proof of Theorem 1.2
Proof of Theorem 1.2. In the hypotheses of Theorem 1.2, the random variable ξassumes values in R. It turns out that, it is enough to consider the case whereξ is a positive integer-valued, aperiodic random variable. In fact, suppose thatTheorem 1.2 holds whenever ξ has support on Z and be aperiodic. Now let ξ beany positive random variable such that E(ξη) <∞ for a given η > 1. Let
m = gcd{k ∈ Z∗+; P(dξe = k) 6= 0
}45
and consider the positive integer-valued, aperiodic random variable ξ′ = dξe/m.Note that E
((ξ′)η
)<∞ so that for every λ sufficiently large, PΛ′
λ (o↔∞) > 0 forυξ′- almost every environment Λ′. Now if Λ is distributed according to υξ, PΛ
λ/m
dominates stochastically PΛ′
λ as it can be seen by a simple coupling argument.Therefore, PΛ
λ/m(o ↔ ∞) is also strictly positive. In view of this fact we assumefrom now on that ξ is positive integer-valued and aperiodic. So we can use all theresults developed in Section 3.3.
By Lemma 3.5
υρξ
( ⋃0≤i≤bLγ−1
k c−1
{Iki is bad})≤ Lγ−1
k L−αk(3.31)
≤ L−α
2k .
Since∑
k L−α/2k < ∞, it follows from Borel-Cantelli Lemma that for (υρξ )-almost
every environment Λ there is c9 = c9(Λ, c3) > c3 such that, for every k ≥ c9, all
Lk 2Lk 3LkIik−1Ijk−20
0
Hk
2Hk
Hk−1
2Hk−1
2Hk−2
Figure 3.7: Representation of the intersections of events C’s e D’s, showing theexistence of the infinite cluster.
46
the k-blocks that form the first (k + 1)-block are good. Now fix Λ one of theseenvironments.
Recall the definitions (3.32) and (3.33) and note that⋂k≥c10
(⋂i
(Cki,0 ∩Dk
i,0)
)⊆ {there is an infinite cluster},
where i ∈{
0, 1, · · · , bLγ−1k c − 2
}and c10 is any positive integer (see Figure 3.7).
It follows from FKG inequality and from (3.34) that
PΛλ (there is an infinite cluster) ≥
∏k≥c10
(1− 2qk)bLγ−1k c−1
≥ 1−∑k≥c10
2Lγ−1k qk
≥ 1−∑k≥c10
2Lγ−1k exp
(−Lβk
), (3.43)
where the last inequality follows from Lemma 3.6.The theorem is proved by choosing c10 = c10(γ, L0, β, c9) > c9 sufficiently large
so that the summation in (3.43) is less that 1.
47
Bibliography
[1] Aizenman, M., Barsky, D. J. (1987). Sharpness of the phase transition inpercolation models. Communications in Mathematical Physics, 108(3), 489-526.
[2] Aizenman, M., Grimmett, G. (1991). Strict monotonicity for critical points inpercolation and ferromagnetic models. Journal of Statistical Physics, 63(5-6),817-835.
[3] Barsky, D. J., Grimmett, G. R., Newman, C. M. (1991). Percolation in half-spaces: equality of critical densities and continuity of the percolation proba-bility. Probability Theory and Related Fields, 90(1), 111-148.
[4] Bramson, M., Durrett, R., Schonmann, R. H. (1991). The contact processesin a random environment. The Annals of Probability, 960-983.
[5] Broadbent, S. R., Hammersley, J. M. (1957). Percolation processes: I. Crystalsand mazes. Mathematical Proceedings of the Cambridge Philosophical Society,53(3), 629-641.
[6] Duminil-Copin, H., Hilario, M. R., Kozma, G., Sidoravicius, V. (2018). Bro-chette percolation. Israel Journal of Mathematics, 225(1), 479-501.
[7] Grassberger, P. (2017). Universality and asymptotic scaling in drilling perco-lation. Physical Review E, 95(1), 010103.
[8] Grassberger, P., Hilario, M. R., Sidoravicius, V. (2017). Percolation in Mediawith Columnar Disorder. Journal of Statistical Physics, 168(4), 731-745.
[9] Grimmett, G. (1999). Percolation, Grundlehren der Mathematischen Wis-senschaften.Vol.321 Springer-Verlag, Berlin, second edition.
[10] Harris, T. E. (1960). A lower bound for the critical probability in a certainpercolation process. Mathematical Proceedings of the Cambridge PhilosophicalSociety, 56(1), 13-20.
48
[11] Hilario, M. R., Sidoravicius, V. (2019). Bernoulli line percolation. StochasticProcesses and their Applications.
[12] Hoffman, C. (2005). Phase transition in dependent percolation. Communica-tions in mathematical physics, 254(1), 1-22.
[13] Kantor, Y. (1986). Three-dimensional percolation with removed lines of sites.Physical Review B, 33(5), 3522.
[14] Kesten, H. (1980). The critical probability of bond percolation on the squarelattice equals 1/2. Communications in mathematical physics, 74(1), 41-59.
[15] Kesten, H. (1982). Percolation theory for mathematicians vol. 423, Boston:Birkhauser.
[16] Kesten, H., Sidoravicius, V., Vares, M. E. (2012). Oriented percolation in arandom environment. arXiv preprint arXiv:1207.3168.
[17] Levin, D. A., Peres, Y. (2017). Markov chains and mixing times. AmericanMathematical Soc. Vol. 107.
[18] Lindvall, T. (1979). On coupling of discrete renewal processes. ProbabilityTheory and Related Fields, 48(1), 57-70.
[19] Lindvall, T. (2002). Lectures on the coupling method. Dover.
[20] Menshikov, M. V. (1986). Coincidence of critical points in percolation prob-lems. Soviet Mathematics Doklady 33, 856-859.
[21] Russo, L. (1978). A note on percolation. Zeitschrift fur Wahrscheinlichkeits-theorie und verwandte Gebiete, 43(1), 39-48.
[22] Russo, L. (1981). On the critical percolation probabilities. Zeitschrift furWahrscheinlichkeitstheorie und verwandte Gebiete, 56(2), 229-237.
[23] Schrenk, K. J., Hilario, M. R., Sidoravicius, V., Araujo, N. A. M., Herrmann,H. J., Thielmann, M., & Teixeira, A. (2016). Critical fragmentation propertiesof random drilling: How many holes need to be drilled to collapse a woodencube?. Physical review letters, 116(5), 055701.
[24] Seymour, P. D., Welsh, D. J. (1978). Percolation probabilities on the squarelattice. Annals of Discrete Mathematics, Elsevier 3, 227-245.
[25] Zhang, Y. (1994). A note on inhomogeneous percolation. The Annals of Prob-ability, 22(2), 803-819.
49
