Dimensional Crossover in Anisotropic Percolation on $Z^{d+s}$

TL;DR

This paper investigates bond percolation on a product lattice, deriving bounds for the critical curve near the threshold, and explores the transition from lower to higher-dimensional percolation behavior.

Contribution

It provides new bounds for the critical curve in anisotropic percolation on $ ext{Z}^{d+s}$, elucidating the dimensional crossover phenomenon.

Findings

Bounds for the critical curve in $(p, q)$ near $p_c( ext{Z}^d)$

Analysis of the dimensional crossover from $ ext{Z}^d$ to $ ext{Z}^{d+s}$

Insights into phase transition behavior in anisotropic percolation

Abstract

We consider bond percolation on $\Z^d\times \Z^s$ where edges of $\Z^d$ are open with probability $p<p_c(\Z^d)$ and edges of $\Z^s$ are open with probability $q$, independently of all others. We obtain bounds for the critical curve in $(p, q)$, with $p$ close to the critical threshold $p_c(\Z^d)$. The results are related to the so-called dimensional crossover from $\Z^d$ to $\Z^{d+s}$.

Equations

$$q_c(p)=\sup \{q:\theta (p,q)=0\}.$$

$${\chi }_d(p)\approx |p-p_c(d){|}^{-\gamma },$$

$$q_c(p)\approx |p-p_c(d){|}^{\psi }.$$

$$q<\frac{1}{2s{\chi }_d(p)},$$

$$\frac{1}{2s{\chi }_1(p)}\le q_c(p)\le \frac{\alpha }{{\chi }_1(p)},$$

$$q_c(p)\le \frac{\beta }{{\chi }_d(p)},$$

$$\chi (p,q)=\sum _{(\mathbf{u},\mathbf{t})\in {\mathbb{Z}}^{d+s}}{\mathbb{P}}_{p,q}(0_{d+s}\leftrightarrow (\mathbf{u},\mathbf{t})).$$

$$A_{i}=\{\{({\mathbf{u}}_{i-1},{\mathbf{t}}_{i})\leftrightarrow({\mathbf{u}}_{i},{\mathbf{t}}_{i})\mbox{ on ${\mathbb{Z}}^{d}\times\{{\mathbf{t}}_{i}\}$}\}\cap\{e_{({\mathbf{u}}_{i},{\mathbf{t}}_{i}),({\mathbf{u}}_{i},{\mathbf{t}}_{i+1})}\mbox{ is open}\}\},$$

$$A_{n}=\{\{({\mathbf{u}}_{n-1},{\mathbf{t}}_{n})\leftrightarrow({\mathbf{u}}_{n},{\mathbf{t}}_{n})\mbox{ on ${\mathbb{Z}}^{d}\times\{{\mathbf{t}}_{m}\}$}\}\},$$

$$\{0_{d+s}\leftrightarrow (\mathbf{u},\mathbf{t})\}=\bigcup _{n\ge 0}\bigcup _{\begin{array}{c}{\hat{\mathbf{u}}}_n:{\mathbf{u}}_n=\mathbf{u}\\ {\hat{\mathbf{t}}}_n:t_n=\mathbf{t}\end{array}}\{A_0\circ A_1\circ \cdots \circ A_n\},$$

$$\chi (p,q)\le \sum _{(\mathbf{u},\mathbf{t})\in {\mathbb{Z}}^{d+s}}\sum _{n\ge 0}\sum _{\begin{array}{c}{\hat{\mathbf{u}}}_n:{\mathbf{u}}_n=\mathbf{u}\\ {\hat{\mathbf{t}}}_n:t_n=\mathbf{t}\end{array}}\prod _{i=0}^n{\mathbb{P}}_{p,q}(A_i),$$

$$\chi (p,q)\le \sum _{n\ge 0}\sum _{{\hat{\mathbf{t}}}_n}\sum _{{\hat{\mathbf{u}}}_n}\prod _{i=0}^n{\mathbb{P}}_{p,q}(A_i),$$

$$\sum _{{\mathbf{u}}_n\in {\mathbb{Z}}^d}{\mathbb{P}}_{p,q}(A_n)\le {\chi }_d(p),$$

$$\sum _{\begin{array}{c}{\mathbf{u}}_k\in {\mathbb{Z}}^d\\ {\mathbf{t}}_k\in {\mathbb{Z}}^s\end{array}}{\mathbb{P}}_{p,q}(A_k)\le 2sq{\chi }_d(p),\forall k=0,\dots ,n-1.$$

$$\chi (p,q)\le \sum _{n=0}^{\infty }(2s{)}^n{\chi }_d(p{)}^{n+1}{q}^n.$$

$$A_{\mathbf{u}}(\omega )=\{\omega \in \Omega :\mathbf{u}\text{mbox}isgoodintheconfiguration\omega \},\mathbf{u}\in \{0\}\times {\mathbb{Z}}^s.$$

$${\mathbb{P}}_{p,q}(\mathbf{u}\text{mbox}isgood)={\mathbb{P}}_{p,q}\left({\mathbf{W}}_{\mathbf{u}}>\frac{1+p}{1-p}ϵ;F_{\mathbf{u}}\right).$$

$${\mathbb{P}}_{p,q}\left({\mathbf{W}}_{\mathbf{u}}>\frac{1+p}{1-p}ϵ\right)=p^{\frac{1+p}{1-p}ϵ}\ge 1-3ϵ.$$

$$F_{{\mathbf{u}},j}({\omega})=\{\mbox{ there exists at least one open edge with exactly one endvertex in ${\mathbf{W}}_{{\mathbf{u}}}$}$$

$$\mbox{ in direction $j$ in configuration ${\omega}$}\},\,\,\,j=1,\dots,s.$$

$$\mathbb{P}_{p,q}\left(F_{{\mathbf{u}}}|{\mathbf{W}}_{{\mathbf{u}}}>\frac{1+p}{1-p}\epsilon\right)=\mathbb{P}_{p,q}\left(\bigcap_{j=1}^{s}F_{{\mathbf{u}},j}\Big{|}{\mathbf{W}}_{{\mathbf{u}}}>\frac{1+p}{1-p}\epsilon\right)$$

$$\geq\left[\mathbb{P}_{p,q}\left(F_{{\mathbf{u}},1}\Big{|}{\mathbf{W}}_{{\mathbf{u}}}>\frac{1+p}{1-p}\epsilon\right)\right]^{s}\geq\left[1-(1-q)^{\frac{1+p}{1-p}\epsilon}\right]^{s}.$$

$$\mathbb{P}_{p,q}\left(F_{{\mathbf{u}}}\Big{|}{\mathbf{W}}_{{\mathbf{u}}}>\frac{1+p}{1-p}\epsilon\right)\geq\left[1-\left(1-\alpha\frac{1-p}{1+p}\right)^{\frac{1+p}{1-p}\epsilon}\right]^{s}\geq(1-e^{-\alpha\epsilon})^{s}.$$

$$\bar{p}:={\mathbb{P}}_{p,q}(\mathbf{u}\text{mbox}isgood)\ge (1-3ϵ)(1-e^{-\alpha ϵ}{)}^s.$$

$${\mathbb{P}}_{p,q}(\text{mbox}bondpercolationoccursin{\mathbb{Z}}^{1+s})\ge {\mathbb{P}}_{\bar{p}}(\text{mbox}sitepercolationoccursin{\mathbb{Z}}^s)>0.$$

Full text

Keywords: dimensional crossover; anisotropic percolation; critical threshold; phase diagram

AMS 1991 subject classification: 60K35; 82B43; 82B26

1 Introduction and Results

1.1 Background

In this note we consider an anisotropic bond percolation process on the graph $({\mathbb{Z}}^{d+s},\mathcal{E}({\mathbb{Z}}^{d+s}))$, where $\mathcal{E}({\mathbb{Z}}^{d+s})$ is the set of edges between nearest neighbors of ${\mathbb{Z}}^{d+s}$. With some abuse of notation, we call this graph ${\mathbb{Z}}^{d+s}={\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}$. An edge of ${\mathbb{Z}}^{d+s}$ is called a ${\mathbb{Z}}^{d}$- edge (respectively a ${\mathbb{Z}}^{s}$- edge) if it joins two vertices which differ only in their ${\mathbb{Z}}^{d}$ (respectively ${\mathbb{Z}}^{s}$) component. Given two parameters $p,q\in[0,1]$, we declare each ${\mathbb{Z}}^{d}$- edge open with probability $p$ and each ${\mathbb{Z}}^{s}$- edge open with probability $q$, independently of all others. This model is described by the probability space $(\Omega,{\mathcal{F}},\mathbb{P}_{p,q})$ where $\Omega=\{0,1\}^{\mathcal{E}}$, ${\mathcal{F}}$ is the $\sigma$-algebra generated by the cylinder sets in $\Omega$ and $\mathbb{P}_{p,q}=\prod_{e\in{\mathbb{E}}}\mu(e)$, where $\mu(e)$ is Bernoulli measure with parameter $p$ or $q$ according to $e$ been a ${\mathbb{Z}}^{d}$- edge or a ${\mathbb{Z}}^{s}$- edge respectively.

Given two vertices $u$ and $v$, we say that $u$ and $v$ are connected in the configuration ${\omega}$ if there exists an open path starting in $u$ and ending in $v$. We denote the event where $v$ and $u$ are connected by \{{\omega}\in\Omega:v\leftrightarrow u\mbox{ in {\omega}}\} and write C({\omega})=\{u\in{\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}:u\leftrightarrow 0\mbox{ in {\omega}}\} for the open cluster containing the origin. We write $\theta(p,q)=\mathbb{P}_{p,q}({\omega}\in\Omega:|C({\omega})|=\infty)$ for the main macroscopic function in percolation theory and denote the mean size of the open cluster by $\chi(p,q)={\mathbb{E}}_{p,q}(|C({\omega}))|)$. Whenever necessary, we will write $\chi_{d}(p)$ and $p_{c}(d)$ in reference to the expected cluster size and critical threshold on ${\mathbb{Z}}^{d}$ with a single parameter $p\in(0,1)$. For a thorough background in independent percolation, we refer the reader to [9].

By a standard coupling argument we have that $\theta(p,q)$ is a monotone non decreasing function of the parameters $p$ and $q$. This allows us to define the function $q_{c}:[0,1]\rightarrow[0,1]$, where

[TABLE]

We observe that $q_{c}(0)=p_{c}(s)$ and $q_{c}(p)=0$ for $p\geq p_{c}(d)$. Moreover, $q_{c}(p)$ is a non increasing function of the parameter $p$. Our main interest is the behavior of $q_{c}(p)$ for values of $p$ less or equal than $p_{c}(d)$, in particular when $p\uparrow p_{c}(d)$.

1.2 Dimensional Crossover

Consider a bond percolation process on the graph ${\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}$, as defined in Section 1.1. The problem we address in this note is the following: what is the behavior of the critical curve $q_{c}(p)$ as $p\uparrow p_{c}(d)$?

A similar model was investigated in [3], where the authors study anisotropic percolation on the slab ${\mathbb{Z}}^{2}\times\{0,\dots,k\}$. In their model, vertical and horizontal edges are open with probability $p$ and $q$ respectively, independently of all others. They obtain bounds for the critical curves for these models and establish their continuity and strict monotonicity. Similar anisotropic ferromagnectic models have also been considered in the mathematical literature, see the works [5], [6] and [13].

A central open problem in percolation theory is the existence and determination of critical exponents. For instance, consider the isotropic percolation process with parameter $p$ on ${\mathbb{Z}}^{d}$. Quantities such as $\chi_{d}(p)$ are believed to diverge as $p\uparrow p_{c}(d)$ in the manner of a power law in $|p-p_{c}(d)|$, whose exponent is called a critical exponent (see Chapter 9 in [9] for details). That is, it is believed that there exists a $\gamma(d)>0$ such that

[TABLE]

when $p\uparrow p_{c}(d)$. Here the relation $a(p)\approx b(p)$ means log equivalence, i.e., $\frac{\log a(p)}{\log b(p)}\rightarrow 1$ when ${p\uparrow p_{c}(d)}$.

We introduce another critical exponent in the following way: consider the function defined in Equation (1). Is it true that there exists a constant $\psi>0$ (which depends on $d$) such that $q_{c}(p)\approx|p-p_{c}(d)|^{\psi}$ when $p\uparrow p_{c}(d)$? This problem arises in the physics literature as the dimensional crossover problem, and $\psi$ is called the crossover exponent. The term crossover is related to the study of percolative systems on $(d+s)$-dimensional lattices, where the $d$-dimensional parameter $p$ is close to $p_{c}(d)$ from below and the $s$-dimensional parameter $q$ is small.

It is commonly suggested in the physics literature that $\psi(d)$ and $\gamma(d)$ exist and are equal. Let us highlight a few papers that discuss this matter. In [8] the authors study anisotropic bond percolation on ${\mathbb{Z}}^{3}={\mathbb{Z}}^{2}\times{\mathbb{Z}}$ . Here ${\mathbb{Z}}^{2}$- edges are open with probability $p$ and ${\mathbb{Z}}$- edges are open with probability $q=Rp$, where $R$ is the anisotropy parameter. By means of a simulation, the authors estimate $\psi(2)$ by $2.3\pm 0.1$, which is compatible with the critical exponent $\gamma(2)$, whose estimate given in [16] is $\frac{43}{18}$. In [15], Redner and Stanley consider a percolation model on ${\mathbb{Z}}^{d}={\mathbb{Z}}^{d-1}\times{\mathbb{Z}}$ where ${\mathbb{Z}}ö^{d-1}$- edges are open with probability $p$ and ${\mathbb{Z}}$- edges parallel to $z$ are open with probability $q=Rp$. Through simulated results the authors show that in the limit $1/R\rightarrow 0$, the crossover exponent $\psi$ is equal to $1$ for all $d$. In the opposite limit $R\rightarrow 0$, their analysis suggests that $\psi(d-1)\neq\gamma(d-1)$. This result was later contradicted by Redner and Coniglio [14], where the authors argue the opposite relation, that is, $\psi(d-1)=\gamma(d-1)$. The interested reader can consult the works [2],[4],[7] and [12] for more results in this direction.

Based on the results mentioned above, we state the following:

Conjecture 1**.**

There exists a critical exponent $\psi=\psi(d)>0$ such that

[TABLE]

Besides, if $\gamma(d)$ exists, then $\psi(d)=\gamma(d)$.

In Section 1.3 we give a complete answer to this conjecture in the case $d=1$ and a partial answer for general $d$. We remark that the crossover critical exponent, if it exists, depends only on the dimension $d$ and not on $s$.

1.3 Results

We start with the following theorem, which gives a lower bound for the critical curve $q_{c}(p)$.

Theorem 1**.**

Consider a bond percolation process on ${\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}$ with parameters $(p,q)$, $p<p_{c}(d)$. If the pair $(p,q)$ satisfies

[TABLE]

then there is a.s. no infinite open cluster in ${\mathbb{Z}}^{d+s}$.

Unfortunately, we were not able to obtain an upper bound for $q_{c}(p)$ that would work for any $d$ and $s$. Nevertheless, in the particular case $d=1$, having full control of the one dimensional cluster enables us to build up the open cluster of the origin in ${\mathbb{Z}}^{1+s}$ by tuning the parameters appropriately. We adopt notation from Section 1.1 and say ${\mathbb{Z}}$- edges are open with probability $0<p<1$, independently of all other edges and ${\mathbb{Z}}^{s}$- edges are open with probability $0<q<1$, also independently of all other edges. The next theorem gives an upper bound for the critical curve $q_{c}(p)$.

Theorem 2**.**

Consider a bond percolation process on ${\mathbb{Z}}\times{\mathbb{Z}}^{s}$, $s>1$, with parameters $(p,q)$. There exists an $\alpha>0$ such that if $p$ is sufficiently close to $1$ and $q>\alpha\frac{1-p}{1+p}$, then there is a.s. an infinite open cluster in ${\mathbb{Z}}^{d+1}$.

Since $\chi_{1}(p)=\frac{1+p}{1-p}\,,$ an immediate consequence of Theorems 1 and 2 is

Corollary 1**.**

Consider a bond percolation process on ${\mathbb{Z}}\times{\mathbb{Z}}^{s}$, $s>1$, with parameters $(p,q)$. Then

[TABLE]

for some $\alpha>0$ and $p$ sufficiently close to $1$.

Remark 1**.**

By Theorem 1, the lower bound for $q_{c}(p)$ in Corollary 1 remains valid for any $p\in[0,1)$.

Remark 2**.**

The case $s=1$ has been fully understood in [11]. It was shown that the critical curve is of the form $\psi(p,q)=p+q-1$, that is, if $\psi(p,q)<0$, then there is no infinite open cluster a.s., whereas if $\psi(p,q)>0$, then there exists an infinite open cluster a.s..

Another straightforward consequence of Theorems 1 and 2 is

Corollary 2**.**

We have the following relations for the critical exponents $\gamma$ and $\psi$:

$\psi(1)$* exists and is equal to $\gamma(1)=1$. Hence Conjecture 1 is true in the case $d=1$.* 2. 2.

If $\gamma(d)$ and $\psi(d)$ exist, then $\psi(d)\leq\gamma(d)$ for all $d$.

We end this section with the following conjecture, which together with Theorem 1 and the validity of Equation (2) implies that Conjecture 1 is true in general. In particular it establishes that, if the critical exponents $\psi(d)$ and $\gamma(d)$ exist, then they are equal.

Conjecture 2**.**

Consider a bond percolation model on ${\mathbb{Z}}^{d}\times{\mathbb{Z}}^{s}$, with parameters $(p,q)$, $p<p_{c}(d)$. Then

[TABLE]

for some $\beta>0$.

2 Proofs

2.1 Proof of Theorem 1

Let us first introduce some notation. For any $({\mathbf{u}},{\mathbf{t}})\in{\mathbb{Z}}^{d+s}$, ${\mathbf{u}}$ and ${\mathbf{t}}$ will always denote the ${\mathbb{Z}}^{d}$ and ${\mathbb{Z}}^{s}$ components of $({\mathbf{u}},{\mathbf{t}})$ respectively. We shall also consider sequences of points $\{{\mathbf{u}}_{n},\,n\in{\mathbb{Z}}^{+}\}$ and $\{{\mathbf{t}}_{n},\,n\in{\mathbb{Z}}^{+}\}$ in ${\mathbb{Z}}^{d}$ and ${\mathbb{Z}}^{s}$ respectively. We let $\mathbf{0}_{d}$ denote a $d$-dimensional vector of zeros. Finally, given ${\mathbf{u}}\in{\mathbb{Z}}^{d}$ and $\mathbf{s},{\mathbf{t}}\in{\mathbb{Z}}^{s}$ such that $\delta(\mathbf{s},{\mathbf{t}})=1$, write $e_{({\mathbf{u}},{\mathbf{t}}),({\mathbf{u}},\mathbf{s})}\in\mathcal{E}({\mathbb{Z}}^{s})$ for the edge with endpoints $({\mathbf{t}},{\mathbf{u}})$ and $({\mathbf{u}},\mathbf{s})$. Here, $\delta(\mathbf{x},\mathbf{y})=\sum_{i=1}^{s}(x_{i}-y_{i})$.

The following argument is based on techniques developed in [10]. We will bound $\chi(p,q)$, the expected size of the infinite open cluster in ${\mathbb{Z}}^{d+s}$. Clearly,

[TABLE]

Now, the event $\{\mathbf{0}_{d+s}\leftrightarrow({\mathbf{u}},{\mathbf{t}})\}$ occurs if and only if there exist sequences $\hat{{\mathbf{u}}}_{n}=({\mathbf{u}}_{0},\dots\mathbf{u}_{n})$ and $\hat{{\mathbf{t}}}_{n}=({\mathbf{t}}_{0},\dots\mathbf{t}_{n})$ such that ${\mathbf{u}}_{j}\in{\mathbb{Z}}^{d}$, for all $0\leq j\leq n$, ${\mathbf{t}}_{i}\in{\mathbb{Z}}^{s}$ for all $1\leq i\leq n$ and $\delta({\mathbf{t}}_{i},{\mathbf{t}}_{i-1})=1$, with the following property: we start at $(\mathbf{0}_{d},{\mathbf{t}}_{0})=(\mathbf{0}_{d},\mathbf{0}_{s})$ and then we connect $(\mathbf{0}_{d},{\mathbf{t}}_{0})$ to $({\mathbf{u}}_{0},{\mathbf{t}}_{0})$ using only ${\mathbb{Z}}^{d}\times\{{\mathbf{t}}_{0}\}$- edges. Next, we move from $({\mathbf{u}}_{0},{\mathbf{t}}_{0})$ to $({\mathbf{u}}_{0},{\mathbf{t}}_{1})$ by a connection that uses a single open ${\mathbb{Z}}^{s}$- edge. We proceed looking for a connection from $({\mathbf{u}}_{0},{\mathbf{t}}_{1})$ to $({\mathbf{u}}_{1},{\mathbf{t}}_{1})$ that uses only ${\mathbb{Z}}^{d}\times\{{\mathbf{t}}_{1}\}$- edges that are not used to connect $(\mathbf{0}_{d},{\mathbf{t}}_{0})$ to $({\mathbf{u}}_{0},{\mathbf{t}}_{0})$. At the $m-th$ step we move from $({\mathbf{u}}_{m-1},{\mathbf{t}}_{m-1})$ to $({\mathbf{u}}_{m-1},{\mathbf{t}}_{m})$ by a connection that uses a single open ${\mathbb{Z}}^{s}$- edge which is not used in any of the previous steps of the construction. Next, we connect $({\mathbf{u}}_{m-1},{\mathbf{t}}_{m})$ to $({\mathbf{u}}_{m},{\mathbf{t}}_{m})$ using only ${\mathbb{Z}}^{d}\times\{{\mathbf{t}}_{m}\}$- edges, with the constraint that no ${\mathbb{Z}}^{d}$- edges used to connect $({\mathbf{u}}_{l-1},{\mathbf{t}}_{l})$ to $({\mathbf{u}}_{l},{\mathbf{t}}_{l})$, $1\leq l\leq m-1$, and $(\mathbf{0}_{d},{\mathbf{t}}_{0})$ to $({\mathbf{u}}_{0},{\mathbf{t}}_{0})$ are used to connect $({\mathbf{u}}_{m-1},{\mathbf{t}}_{m})$ to $({\mathbf{u}}_{m},{\mathbf{t}}_{m})$. We continue this procedure until it reaches $({\mathbf{u}}_{n},{\mathbf{t}}_{n})=({\mathbf{u}},{\mathbf{t}})$.

Given $\hat{{\mathbf{t}}}_{n}=({\mathbf{t}}_{0},\dots,{\mathbf{t}}_{n})$ and $\hat{{\mathbf{u}}}_{n}=({\mathbf{u}}_{0},\dots,{\mathbf{u}}_{n})$, consider the sequence of increasing events $\{A_{i}\}_{i=0}^{n}$, where, for $0\leq i\leq n-1$,

[TABLE]

and

[TABLE]

with the convention that ${\mathbf{u}}_{-1}=\mathbf{0}_{d}$. As a consequence of the above construction we have

[TABLE]

where, $A\circ B$ denotes disjoint occurrence of events $A$ and $B$. By the BK inequality (see [1]) we have

[TABLE]

or equivalently

[TABLE]

where the final two summations are over all appropriate sequences $\hat{{\mathbf{u}}}_{n}=({\mathbf{u}}_{0},\dots,{\mathbf{u}}_{n})$ and $\hat{{\mathbf{t}}}_{n}=({\mathbf{t}}_{0},\dots,{\mathbf{t}}_{n})$, where $({\mathbf{u}}_{0},{\mathbf{t}}_{0})=\mathbf{0}_{d+s}$. Clearly,

[TABLE]

and, since the events \{({\mathbf{u}}_{i-1},{\mathbf{t}}_{i})\leftrightarrow({\mathbf{u}}_{i},{\mathbf{t}}_{i})\mbox{ on {\mathbb{Z}}^{d}}\} and $\{e_{({\mathbf{u}}_{i},{\mathbf{t}}_{i}),({\mathbf{u}}_{i},{\mathbf{t}}_{i+1})}\mbox{ is open}\}$ are independent, we obtain

[TABLE]

Proceeding inductively we obtain

[TABLE]

If $p<p_{c}(d)$, then $\chi_{d}(p)<\infty$ a.s. and the result follows.

2.2 Proof of Theorem 2

We shall construct an independent site percolation process in ${\mathbb{Z}}^{s}$ which is induced by the bond percolation process in ${\mathbb{Z}}^{1+s}$. We show that, under the appropriate hypothesis, site percolation occurs in ${\mathbb{Z}}^{s}$, and therefore, by stochastic dominance, bond percolation will occur in ${\mathbb{Z}}^{1+s}$.

Let ${\mathbf{u}}=(u,u_{1},\dots,u_{s}),{\mathbf{z}}=(z,z_{1},\dots,z_{s})\in{\mathbb{Z}}^{1+s}$. Given a configuration $\omega\in\Omega$, we say that ${\mathbf{u}}$ is updownwards connected to ${\mathbf{z}}$ in the configuration $\omega$ if $u_{i}=z_{i}$ for all $i=1,\dots,s$, and there exists a open path using only ${\mathbb{Z}}\times\{u_{1},\dots,u_{s}\}$- edges starting at ${\mathbf{u}}$ and ending at ${\mathbf{z}}$. Denote this event by $\{\omega\in\Omega:{\mathbf{u}}\updownarrow{\mathbf{z}}\mbox{ in }\omega\}$.

Now, for ${\mathbf{u}}\in\{0\}\times{\mathbb{Z}}^{s}$, let ${\mathbf{W}}_{{\mathbf{u}}}(\omega)=\{{\mathbf{z}}\in{\mathbb{Z}}^{1+s}:{\mathbf{z}}\updownarrow{\mathbf{u}}\mbox{ in }\omega\}.$ We proceed to construct an auxiliary site percolation process of good vertices on $\{0\}\times{\mathbb{Z}}^{s}$. Given $\epsilon>0$ and a configuration $\omega$ we declare each vertex ${\mathbf{u}}\in\{0\}\times{\mathbb{Z}}^{s}$ as good if

${\mathbf{W}}_{{\mathbf{u}}}(\omega)>\frac{1+p}{1-p}\epsilon$ 2. 2.

there exists at least one open edge with exactly one endvertex in ${\mathbf{W}}_{{\mathbf{u}}}$ in each of the $s$ possible directions of increasing coordinate-value in configuration $\omega$.

Consider the sequence of independent events $\{A_{{\mathbf{u}}}(\omega)\}_{{\mathbf{u}}\in\{0\}\times{\mathbb{Z}}^{s}}$, where

[TABLE]

We aim to obtain an estimate for the probability of $A_{{\mathbf{u}}}$. If we denote the event in condition 2. by $F_{{\mathbf{u}}}$, then for any ${\mathbf{u}}\in\{0\}\times{\mathbb{Z}}^{s}$, we have

[TABLE]

Now, for $\epsilon>0$ sufficiently small and $p$ sufficiently close to 1,

[TABLE]

Define

[TABLE]

[TABLE]

Note that $\{F_{{\mathbf{u}},j}\}_{j}$, $j=1,\dots,s$, is a collection of independent events with equal probabilities. Then

[TABLE]

[TABLE]

If for some $\alpha>0$ we choose $q>\alpha\frac{1-p}{1+p}$, then

[TABLE]

Plugging Equations (4) and (5) into Equation (3) we obtain

[TABLE]

Taking $\alpha=\alpha(\epsilon)$ sufficiently large, this can be made strictly larger than the critical threshold of site percolation on ${\mathbb{Z}}^{s}$. Since the anisotropic bond process with parameters $(p,q)$ stochastically dominates the isotropic site process with parameter $\bar{p}$, we obtain

[TABLE]

This completes the proof.

Acknowledgements

Rémy Sanchis was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by Fundacão de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG), grant PPM 00600/16. Roger W. C. Silva was partially supported by FAPEMIG, grant APQ-02743-14.