Inhomogeneous Percolation on Ladder Graphs

Abstract

We define an inhomogeneous percolation model on “ladder graphs” obtained as direct products of an arbitrary graph $G = (V,E)$ and the set of integers ${\mathbb {Z}}$. (Vertices are thought of as having a “vertical” component indexed by an integer.) We make two natural choices for the set of edges, producing an unoriented graph ${\mathbb {G}}$ and an oriented graph $\vec {{\mathbb {G}}}$. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite “column” are open with probability q and all other edges are open with probability p. For all fixed q one can define the critical percolation threshold $p_\mathrm{c}(q)$. We show that this function is continuous in (0, 1).

1 Introduction

In this paper we examine how the critical parameter of percolation is affected by inhomogeneities. More specifically, we address the following problem. Suppose ${\mathbb {G}}$ is a graph with (oriented or unoriented) set of edges ${\mathbb {E}}$ and that ${\mathbb {E}}$ is split into two disjoint sets, ${\mathbb {E}} = {\mathbb {E}}' \cup {\mathbb {E}}''$. Consider the percolation model in which edges of ${\mathbb {E}}'$ are open with probability p and edges of ${\mathbb {E}}''$ are open with probability q. For $q \in [0,1]$, we can then define $p_\mathrm{c}(q)$ as the supremum of values of p for which percolation does not occur at p, q. What can be said about the function $q \mapsto p_\mathrm{c}(q)$?

This is the framework for the problem of interest of the recent reference [5]. In that paper, the authors consider an oriented tree whose vertex set is that of the d-regular, rooted tree, and containing “short edges” (with which each vertex points to its d children) and “long edges” (with which each vertex points to its $d^k$ descendants at distance k, for fixed $k \in {\mathbb {N}}$). Percolation is defined on this graph by letting short edges be open with probability p and long edges with probability q. It is proved that the curve $q \mapsto p_\mathrm{c}(q)$ is continuous and strictly decreasing in the region where it is positive.

In the present paper, we consider another natural setting for the problem described in the first paragraph, namely that of a “ladder graph” in the spirit of [6]. We start with an arbitrary (unoriented, connected) graph $G=(V,E)$ and construct ${\mathbb {G}}=({\mathbb {V}},{\mathbb {E}})$ by placing layers of G one on top of the other and adding extra edges to connect the consecutive layers. More precisely, ${\mathbb {V}} = V \times {\mathbb {Z}}$ and ${\mathbb {E}}$ consists of the edges that make each individual layer a copy of G, as well as edges linking each vertex to its copies in the layers above it and below it (see Fig. 1 for an example). With this choice (and other ones we will also consider), one would expect the aforementioned function $p_\mathrm{c}(q)$ to be constant in (0, 1). Our main result is that it is a continuous function. We also consider a similarly defined oriented model $\vec {{\mathbb {G}}}$ and obtain the same result. See Sect. 1.1 for a more formal description of the models we study and the results we obtain.

Fig. 1

The construction of ${\mathbb {G}}$ from G and a possible choice for the edge set ${\mathbb {E}}''$ (on which edges are open with probability q)

Our ladder graph percolation model is a generalization of the model of [12]. In that paper, Zhang considers an independent bond percolation model on ${\mathbb {Z}}^2$ in which edges belonging to the vertical line through the origin are open with probability q, while other edges are open with probability p. It then follows from standard results in percolation theory that $(0,1) \ni q \mapsto p_\mathrm{c}(q)$ is constant, equal to $\frac{1}{2}$, the critical value of (homogeneous) bond percolation on ${\mathbb {Z}}^2$. The main result of [12] is that, when p is set to this critical value and for any $q \in (0,1)$, there is almost surely no infinite percolation cluster. Since we are far from understanding the critical behaviour of homogeneous percolation on the more general graphs ${\mathbb {G}}$ and $\vec {{\mathbb {G}}}$ we consider here, analogous results to that of Zhang are beyond the scope of our work.

Let us briefly mention some other related works. Important references for percolation phase transition beyond ${\mathbb {Z}}^d$ are [3, 9]; see also [4] for a recent development. Concerning sensitivity of the percolation threshold to an extra parameter or inhomogeneity of the underlying model, see the theory of essential enhancements developed in [1, 2].

1.1 Formal Description of the Model and Results

Let $G=(V, E)$ be a connected graph with vertex set V and edge set E. Let ${\mathbb {V}}=V\times {\mathbb {Z}}$. We define the unoriented graph ${\mathbb {G}} = ({\mathbb {V}}, {\mathbb {E}})$ and the oriented graph $\vec {{\mathbb {G}}} = ({\mathbb {V}}, \vec {{\mathbb {E}}})$, where

$$\begin{aligned} {\mathbb {E}}=&\{\{(u,n), (v, n)\}: \{u, v\}\in E, n\in {\mathbb {Z}}\}\cup \{\{(u,n), (u, n+1)\}: u \in V, n\in {\mathbb {Z}}\},\\ \vec {{\mathbb {E}}}=&\{\langle (u,n), (v, n+1)\rangle : \{u, v\}\in E, n\in {\mathbb {Z}}\}; \end{aligned}$$

above, we denote unoriented edges by $\{\cdot , \cdot \}$ and oriented edges by $\langle \cdot , \cdot \rangle$. See Fig. 2 for an example. Note that $\vec {{\mathbb {G}}}$ is not necessarily connected.

Fig. 2

${\mathbb {G}}$ and $\vec {{\mathbb {G}}}$ for $G={\mathbb {Z}}$. Note that in this case, $\vec {{\mathbb {G}}}$ consists of two disjoint subgraphs; for clarity, we will only display one of these subgraphs further on

We consider percolation configurations in which each edge in ${\mathbb {E}}$ and $\vec {{\mathbb {E}}}$ can be open or closed. Let $\varOmega =\{0, 1\}^{{\mathbb {E}}}$ and $\vec {\varOmega }=\{0, 1\}^{\vec {{\mathbb {E}}}}$ be the sets of all possible configurations on ${\mathbb {G}}$ and $\vec {{\mathbb {G}}}$, respectively. Then for any $e \in {\mathbb {E}}$ or $\vec {{\mathbb {E}}}$, $\omega (e)=1$ corresponds to the edge being open and $\omega (e)=0$ closed.

An open path on ${\mathbb {G}}$ is a set of distinct vertices $(v_0, n_0), (v_1, n_1), \dots , (v_m, n_m)$ such that for every $i=0, \dots , m-1$, $\{(v_i, n_i),(v_{i+1}, n_{i+1})\}\in {\mathbb {E}}$ and is open. We say that (v, n) can be reached from $(v_0, n_0)$ either if they are equal or if there is an open path from $(v_0, n_0)$ to (v, n). Denote this event by $(v_0, n_0)\leftrightarrow (v, n)$. The set of vertices that can be reached from (v, n) is called the cluster of (v, n).

An open path on $\vec {{\mathbb {G}}}$ can be defined similarly, but since edges are oriented upwards, (v, n) can only be reached from $(v_0, n_0)$ if $n\ge n_0$. Denote this event by $(v_0, n_0)\rightarrow (v, n)$. Hence, we will call the set of vertices that can be reached by an open path from (v, n) the forward cluster of (v, n). Denote by $C_{\infty }$ and $\vec {C}_{\infty }$ the events that there is an infinite cluster on ${\mathbb {G}}$ and an infinite forward cluster on $\vec {{\mathbb {G}}}$, respectively.

We examine the following inhomogeneous percolation setting. First, consider the unoriented graph ${\mathbb {G}}$. Fix finitely many edges and vertices

$$\begin{aligned} e_1=\{u_1, v_1\}, \dots , e_K=\{u_K, v_K\}\in E, \quad w_1, \dots w_L\in V, \end{aligned}$$

(1)

and let

$$\begin{aligned}&{\mathbb {E}}^i:=\{\{(u_i,n), (v_i, n)\}: n\in {\mathbb {Z}}\} \quad i=1, \dots , K; \end{aligned}$$

(2)

$$\begin{aligned}&{\mathbb {E}}^{K+j}:=\{\{(w_j,n), (w_j, n+1)\}: n\in {\mathbb {Z}}\} \quad j=1, \dots , L; \end{aligned}$$

(3)

that is, the set of “horizontal” edges on ${\mathbb {G}}$ between $u_i$ and $v_i$, and the set of “vertical” edges above and below vertex $w_j$, respectively (see Fig. 3 for an example). Further, let $\mathbf{q}=(q_1, \dots , q_{K+L})$ with $q_i\in (0,1)$ for all i and let $p\in [0, 1]$. Now let each edge of ${\mathbb {E}}^i$ be open with probability $q_i$, and each edge in ${\mathbb {E}}\setminus \cup _{i=1}^{K+L}{\mathbb {E}}^i$ be open with probability p. Denote the law of the open edges by ${\mathbb {P}}_{\mathbf{q}, p}$. Whether or not the event $C_{\infty }$ happens with positive probability depends on the parameters p and $\mathbf{q}$, so we can define the critical parameter as a function of $\mathbf{q}$:

$$\begin{aligned} p_\mathrm{c}(\mathbf{q}):=\sup \{p:{\mathbb {P}}_{\mathbf{q}, p}(C_{\infty })=0\}. \end{aligned}$$

We will show that this function is continuous:

Theorem 1

For fixed $K, L\in {\mathbb {N}}$, the function $\mathbf{q}\mapsto p_\mathrm{c}(\mathbf{q})$ is continuous in $(0,1)^{K+L}$.

Fig. 3

The edge sets ${\mathbb {E}}^1$ and ${\mathbb {E}}^2$ on ${\mathbb {G}}$ with $e_1=\{-1, 0\}$ and $w_1=1$, and on $\vec {{\mathbb {G}}}$ with $e_1=\{-1, 0\}$ and $e_2=\{1, 2\}$ (for $G={\mathbb {Z}}$)

We now turn to the oriented graph $\vec {{\mathbb {G}}}$. Fix finitely many edges

$$\begin{aligned} e_1=\{u_1, v_1\}, \dots , e_K=\{u_K, v_K\}\in E, \end{aligned}$$

(4)

and let

$$\begin{aligned} \vec {{\mathbb {E}}}^i:=\{\langle (u_i,n), (v_i, n+1)\rangle , \langle (v_i,n), (u_i, n+1)\rangle : n\in {\mathbb {Z}}\}; \end{aligned}$$

(5)

that is, the set of oriented edges on $\vec {{\mathbb {G}}}$ between $u_i$ and $v_i$ (see Fig. 3 for an example). Further, let $\mathbf{q}=(q_1, \dots , q_K)$ with $q_i\in (0,1)$ for all i and let $p\in [0, 1]$. Now let each oriented edge of $\vec {{\mathbb {E}}}^i$ be open with probability $q_i$, and each oriented edge in $\vec {{\mathbb {E}}}\setminus \cup _{i=1}^K\vec {{\mathbb {E}}}^i$ be open with probability p. Denote the law of the open edges by $\vec {{\mathbb {P}}}_{\mathbf{q}, p}$. Similarly as in the unoriented case we can define the critical parameter as a function of $\mathbf{q}$:

$$\begin{aligned} \vec {p_\mathrm{c}}(\mathbf{q}):=\sup \{p:\vec {{\mathbb {P}}}_{\mathbf{q}, p}(\vec {C}_{\infty })=0\}. \end{aligned}$$

We will show that this function is continuous:

Theorem 2

For fixed $K\in {\mathbb {N}}$, the function $\mathbf{q}\mapsto \vec {p_\mathrm{c}}(\mathbf{q})$ is continuous in $(0,1)^K$.

The proofs of both Theorems 1 and 2 rely on two coupling results which allow us to compare percolation configurations with different parameters $\mathbf{q}, p$. These coupling results are presented in Sect. 2. We prove Theorem 1 in Sect. 3 and Theorem 2 in Sect. 4.

1.2 Discussion on the Contact Process

Bond percolation on the oriented graph $\vec {{\mathbb {G}}}$ defined from $G = (V,E)$ is closely related to the contact process on G. The contact process is usually taken as a model of epidemics on a graph: vertices are individuals, which can be healthy or infected. In the continuous-time Markov dynamics infected individuals recover with rate 1 and transmit the infection to each neighbour with rate $\lambda > 0$ (“infection rate”). The “all healthy” configuration is a trap state for the dynamics; the probability that the contact process ever reaches this state is either equal to 1 or strictly less than 1 for any finite set of initially infected vertices. The process is said to die out in the first case and to survive in the latter. Whether it survives or dies out will depend on both the underlying graph G and $\lambda$, so one defines the critical rate $\lambda _\mathrm{c}$ as the supremum of the infection parameter values for which the contact process dies out on G. For a detailed introduction see [8].

The contact process admits a well-known graphical construction that is a “space-time picture” $G\times [0, \infty )$ of the process. We assign to each vertex $v\in V$ and ordered pair of vertices (u, v) satisfying $\{u, v\}\in E$ a Poisson point process $D_v$ with rate 1 and $D_{(u, v)}$ with rate $\lambda$, respectively. (All processes are independent.) For each event time t of $D_v$ we place a “recovery mark” at (v, t) and for each event time of $D_{(u, v)}$ an “infection arrow” from (u, t) to (v, t). An “infection path” is a connected path that moves along the timeline in the increasing time direction, without passing through a recovery mark and along infection arrows in the direction of the arrow. Starting from a set of initially infected vertices $A\subset V$, the set of infected vertices at time t is the set of vertices v such that (v, t) can be reached by an infection path from some (u, 0) with $u\in A$.

This representation can be thought of as a version of our oriented percolation model $\vec {{\mathbb {G}}}$ in which the “vertical”, one-dimensional component is taken as ${\mathbb {R}}$ rather than ${\mathbb {Z}}$. (Some other modifications have to be made to account for the “recovery marks” of the contact process, but this is unimportant for the present discussion.) In fact, one of the questions that originally motivated us was the following. Assume we take the contact process on an arbitrary graph G, and declare that the infection rate is equal to $\lambda > 0$ in every edge except for a distinguished edge $e^*$, in which the infection rate is $\sigma > 0$. Let $\lambda _\mathrm{c}(\sigma )$ be the supremum of values of $\lambda$ for which the process with parameters $\lambda , \sigma$ dies out (starting from finitely many infections). Is it true that $\lambda _\mathrm{c}(\sigma )$ is constant, or at least continuous, in $(0,\infty )$?

In case G is a vertex-transitive connected graph, one can show that $\lambda _\mathrm{c}(\sigma )$ is constant in $(0,\infty )$ by an argument similar to the one given in [7]. For general G, even continuity of $\lambda _\mathrm{c}(\sigma )$ is unproved, and the techniques we use here do not seem to be sufficient to handle that case (see Remark 2 below for an explanation of what goes wrong). This is surprising, since results for oriented percolation typically transfer automatically to the contact process (and vice versa). A recent result shows that the situation can be quite delicate: in [10], we exhibited a tree in which the contact process (with same rate $\lambda > 0$ everywhere) survives for any value of $\lambda$, but in which the removal of a single edge produces two subtrees in which the process dies out for small $\lambda$.

2 Coupling Lemmas

The proofs of both of our theorems are based on couplings which allow us to carefully compare percolation configurations sampled from measures with different parameter values. In the proof of Theorem 1 we use the following coupling lemma (Lemma 3.1 from [5]). The proof is omitted since it is quite simple and can be found in [5]; the idea of the coupling is reminiscent of Doeblin’s maximal coupling lemma (see [11] Chapter 1.4).

Lemma 1

Let ${\mathbb {P}}_{\theta }$ denote probability measures on a finite set S, parametrized by $\theta \in (0, 1)^N$, such that $\theta \mapsto {\mathbb {P}}_{\theta }(x)$ is continuous for every $x\in S$. Assume that for some $\theta _1$ and ${\bar{x}} \in S$ we have ${\mathbb {P}}_{\theta _1}({\bar{x}})>0$. Then, for any $\theta _2$ close enough to $\theta _1$, there exists a coupling of two random elements X and Y of S such that  $X\sim {\mathbb {P}}_{\theta _1}$, $Y\sim {\mathbb {P}}_{\theta _2}$ and

$$\begin{aligned} \mathbb {P}\left( \{X=Y\}\cup \{X={\bar{x}}\}\cup \{Y={\bar{x}}\}\right) =1. \end{aligned}$$

(6)

The following is a modified version of Lemma 1, to be used in the proof of Theorem 2.

Lemma 2

Let ${\mathbb {P}}_{\theta }$ denote probability measures on a finite set S, parametrized by $\theta \in (0, 1)^N$, such that $\theta \mapsto {\mathbb {P}}_{\theta }(x)$ is continuous for every $x\in S$. Let  be a non-trivial partition of S, and assume that for some $\theta _1$, ${\hat{x}}\in {\hat{S}}$ and  we have ${\mathbb {P}}_{\theta _1}({\hat{x}})>0$ and . Then, for any $\theta _2$ close enough to $\theta _1$, there exists a coupling of two random elements X and Y of S such that  $X\sim {\mathbb {P}}_{\theta _1}$, $Y\sim {\mathbb {P}}_{\theta _2}$ and

(7)

specifically

(8)

Proof

We write ${\hat{S}}=\{w_1, w_2, \dots , w_n, {\hat{x}}\}$ and , and for all $y\in S$ and  $k=1, 2$ let

Let U be a uniform random variable on [0, 1]. The values of X and Y will be given as functions of U. Clearly,

so we can cover the line segment [0, 1] with disjoint intervals with lengths equal to the summands of the left-hand side of the above equality with either $k=1$ or 2 (see Fig. 4). For any value of u we choose X and Y to be the element of S that corresponds to the interval u falls into in the first and second covers, respectively.

Fig. 4

The partitioning of the line segment [0, 1], and the sampling of (X, Y)

To guarantee that (7) is satisfied we arrange these intervals in a way that

• the interval corresponding to  in the first cover is entirely contained in the intervals corresponding to $\mathbb {P}_{\theta _2}({\hat{x}})$ and  in the second cover;

• the interval corresponding to  in the first cover is contained in the interval corresponding to  in the second cover;

• the interval corresponding to $p_{\theta _1}({\hat{S}})$ in the first cover is contained in the intervals corresponding to $\mathbb {P}_{\theta _2}({\hat{x}})$ and  in the second cover.

The above is possible since by continuity, as $\theta _2\rightarrow \theta _1: {\mathbb {P}}_{\theta _2}({\hat{x}})\rightarrow {\mathbb {P}}_{\theta _1}({\hat{x}})>0$, as well as . Therefore, if $\theta _2$ is sufficiently close to $\theta _1$, we have

$\square$

3 Proof of Theorem 1

We start showing that if the statement of Theorem 1 is proved for a given set of edges and vertices as in (1), then the same continuity statement automatically follows for smaller sets of edges and vertices. To prove this, let $e_1,\ldots , e_K,\;w_1,\ldots , w_L$ be edges and vertices as in (1), and let $w_{L+1}$ be an additional vertex. (We could alternatively take an additional edge with no change to the argument that follows.) We now compare two percolation models on ${\mathbb {G}}$: the first one with parameter values $\mathbf{q} = (q_1,\ldots , q_{K+L})$ for ${\mathbb {E}}^1,\ldots , {\mathbb {E}}^{K+L}$ and  p for all other edges, and the second one with parameter values $(\mathbf{q},q_{K+L+1})$ for ${\mathbb {E}}^1,\ldots , {\mathbb {E}}^{K+L+1}$ and p for all other edges.

Claim 1

If the function $(\mathbf{q}, q_{K+L+1})\mapsto {p_\mathrm{c}}(\mathbf{q}, q_{K+L+1})$ is continuous in $(0,1)^{K+L+1}$, then $\mathbf{q}\mapsto {p_\mathrm{c}}(\mathbf{q})$ is continuous in $(0,1)^{K+L}$.

Proof

Since $(0, 1)\ni q_{K+L+1}\mapsto {p_\mathrm{c}}(\mathbf{q}, q_{K+L+1})$ is non-increasing and by assumption continuous, there exists a unique $t^*\in (0, 1)$ such that $t^*= {p_\mathrm{c}}(\mathbf{q}, t^*)$. We claim that $t^*={p_\mathrm{c}}(\mathbf{q})$. Indeed, by the definition of ${p_\mathrm{c}}(\mathbf{q}, t^*)$,

$$\begin{aligned}&\forall t>t^*, \quad 0<{{\mathbb {P}}}_{(\mathbf{q}, t^*), t}(C_{\infty })\le {{\mathbb {P}}}_{(\mathbf{q}, t), t}(C_{\infty })={{\mathbb {P}}}_{\mathbf{q}, t}(C_{\infty }), \text { and } \\&\forall t<t^*, \quad 0={{\mathbb {P}}}_{(\mathbf{q}, t^*), t}(C_{\infty })\ge {{\mathbb {P}}}_{(\mathbf{q}, t), t}(C_{\infty })={{\mathbb {P}}}_{\mathbf{q}, t}(C_{\infty }), \end{aligned}$$

which implies ${p_\mathrm{c}}(\mathbf{q})= t^*$.

Assume that ${p_\mathrm{c}}(\mathbf{q}, t)= t$ for some $\mathbf{q}$ and t. By continuity, for all $\epsilon > 0$, if $\varvec{\delta } \in (0,1)^{K+L}$ is close enough to zero we have

$$\begin{aligned} {p_\mathrm{c}}(\mathbf{q}+\varvec{\delta }, t)\in (t-\epsilon , t+\epsilon ). \end{aligned}$$

As ${p_\mathrm{c}}$ is non-increasing in t, this yields

$$\begin{aligned} {p_\mathrm{c}}(\mathbf{q}+\varvec{\delta }, t-\epsilon )> t-\epsilon \text { and } {p_\mathrm{c}}(\mathbf{q}+\varvec{\delta }, t+\epsilon )< t+\epsilon . \end{aligned}$$

Hence, there exists $t'\in (t-\epsilon , t+\epsilon )$ such that ${p_\mathrm{c}}(\mathbf{q}+\varvec{\delta }, t')=t'$. This implies that $\mathbf{q}\mapsto {p_\mathrm{c}}(\mathbf{q})$ is continuous. $\square$

For our base graph $G=(V,E)$, $u, v \in V$ and $V' \subset V$, let $\text {dist}_G(u,v)$ be the graph distance between u and v, and let $\text {dist}_G(u,V')$ be the smallest graph distance between u and a point of $V'$. Fix $r \in {\mathbb {N}}$, $u_0 \in V$, and let

$$\begin{aligned} U:=B_r(u_0), \end{aligned}$$

(9)

that is the ball of radius r around $u_0$ with respect to the graph distance.

From now on, we will assume that the edges $e_1,\ldots , e_K$of (1) are all the edges with both endpoints belonging to U, and that the vertices $w_1,\ldots , w_L$of (1) are all the vertices of U. We are allowed to restrict ourselves to this case by Claim 1.

The proof of Theorem 1 will be a consequence of the following claim.

Claim 2

For all $p\in (0,1)$, $\mathbf{q}_\mathbf{0}\in (0,1)^{K+L}$ and $\epsilon \in (0, 1-p)$ there exists a $\delta >0$ such that for any $\mathbf{q}, \mathbf{q'}\in (0,1)^{K+L}$ satisfying $\Vert \mathbf{q}_\mathbf{0}-\mathbf{q}\Vert _{\infty }<\delta$ and $\Vert \mathbf{q}_\mathbf{0}-\mathbf{q'}\Vert _{\infty }<\delta$ we have

$$\begin{aligned} {\mathbb {P}}_{\mathbf{q},\;p}(C_{\infty })\le {\mathbb {P}}_{\mathbf{q'}, p+\epsilon }(C_{\infty }). \end{aligned}$$

Note that Claim 2 is trivial if $\mathbf{q'}-\mathbf{q}$ has non-negative coordinates.

Proof of Theorem 1

Fix $\mathbf{q}_\mathbf{0} \in (0,1)^{K+L}$ and $\epsilon > 0$. By Claim 2, if $\Vert \mathbf{q}_\mathbf{0}-\mathbf{q'}\Vert _{\infty }$ is close enough to zero, then

$$\begin{aligned}&{\mathbb {P}}_{\mathbf{q},\;p_\mathrm{c}(\mathbf{q}_\mathbf{0}) + \epsilon }(C_{\infty }) \ge {\mathbb {P}}_{\mathbf{q}_\mathbf{0} ,\;p_\mathrm{c}(\mathbf{q}_\mathbf{0}) + \frac{\epsilon }{2}}(C_{\infty }), \end{aligned}$$

(10)

$$\begin{aligned}&{\mathbb {P}}_{\mathbf{q},\;p_\mathrm{c}(\mathbf{q}_\mathbf{0}) - \epsilon }(C_{\infty }) \le {\mathbb {P}}_{\mathbf{q}_\mathbf{0} ,\;p_\mathrm{c}(\mathbf{q}_\mathbf{0}) - \frac{\epsilon }{2}}(C_{\infty }). \end{aligned}$$

(11)

By the definition of $p_\mathrm{c}(\mathbf{q}_\mathbf{0})$, the right-hand side of (10) is positive and the right-hand side of (11) is zero; hence, the two inequalities, respectively, yield

$$\begin{aligned} p_\mathrm{c}(\mathbf{q}) \le p_\mathrm{c}(\mathbf{q}_\mathbf{0}) + \epsilon \quad \text {and}\quad p_\mathrm{c}(\mathbf{q}) \ge p_\mathrm{c}(\mathbf{q}_\mathbf{0}) - \epsilon . \end{aligned}$$

This implies that $\mathbf{q} \mapsto p_\mathrm{c}(\mathbf{q})$ is continuous at $\mathbf{q}_\mathbf{0}$. $\square$

Proof of Claim 2

We start with several definitions. Recall the definition of U in (9), and for $n \in {\mathbb {Z}}$ let

$$\begin{aligned} {\mathbb {V}}_n = \{(v,m) \in {\mathbb {V}}:\; v\in ~B_{r+1}(u_0),\; (2L+2)n \le m \le (2L+2)(n+1)\} \end{aligned}$$

and

$$\begin{aligned} {\mathbb {E}}_n =\,&\{e \in {\mathbb {E}}:\; e \text { has both endpoints in } {\mathbb {V}}_n\}\\&\backslash \{e \in {\mathbb {E}}: e = \{(u, (2L+2)(n+1)),(v, (2L+2)(n+1))\} \text { for some } \{u,v\} \in E\}. \end{aligned}$$

We think of ${\mathbb {V}}_n$ as a “box” of vertices and of ${\mathbb {E}}_n$ as all the edges in the subgraph induced by this box, except for the “ceiling”. Note that the ${\mathbb {E}}_n$ are disjoint (though the ${\mathbb {V}}_n$ are not). Next, recall the definition of ${\mathbb {E}}^i$ for $1 \le i \le K+L$ from (2) and (3). Observe that $\cup _i{\mathbb {E}}^i\subsetneq \cup _n{\mathbb {E}}_n$, and define, for $n \in {\mathbb {Z}}$ and $1 \le i \le K+L$,

$$\begin{aligned} {\mathbb {E}}^i_n = {\mathbb {E}}_n \cap {\mathbb {E}}^i,\quad {\mathbb {E}}^\partial _n = {\mathbb {E}}_n \backslash \left( \cup _{i=1}^{K+L} {\mathbb {E}}^i_n\right) ,\quad {\mathbb {E}}_{\mathcal {O}} = {\mathbb {E}} \backslash \left( \cup _{n \in {\mathbb {Z}}} {\mathbb {E}}_n\right) . \end{aligned}$$

The “edge boundary” ${\mathbb {E}}^\partial _n$ consists of edges of the form $\{(u,m),(u,m+1)\}$, with u such that $\text {dist}(u, u_0)=r+1$, and edges of the form $\{(u,m),(v,m)\}$, with $v \in U$ and $\text {dist}(u, u_0)=r+1$. Next, let

$$\begin{aligned} \varOmega _n^i = \{0,1\}^{{\mathbb {E}}_n^i},\quad \varOmega _n^\partial = \{0,1\}^{{\mathbb {E}}_n^\partial }, \quad \varOmega _n = \{0,1\}^{{\mathbb {E}}_n},\quad \varOmega _{\mathcal {O}} = \{0,1\}^{{\mathbb {E}}_{\mathcal {O}}}; \end{aligned}$$

note that

$$\begin{aligned} \varOmega = \varOmega _{\mathcal {O}} \times \prod _{n\in {\mathbb {Z}}} \varOmega _n = \varOmega _{\mathcal {O}} \times \prod _{n\in {\mathbb {Z}}}\left( \varOmega _n^\partial \times \prod _{i=1}^{K+L} \varOmega _n^i\right) . \end{aligned}$$

For each n, define the inner vertex boundary, consisting of the “floor”, “walls” and “ceiling” of the vertex box ${\mathbb {V}}_n$,

$$\begin{aligned} \partial {\mathbb {V}}_n =\,&\{(v,n) \in {\mathbb {V}}_n: \text {dist}(v, u_0)=r+1\} \\&\cup (U\times \{(2L+2)n\}) \cup (U \times \{(2L+2)(n+1)\}). \end{aligned}$$

Given any $\varnothing \ne A \subseteq \partial {\mathbb {V}}_n$ and $\omega _n \in \varOmega _n$, define

$$\begin{aligned} C_n(A,\omega _n) = \{(v,n) \in \partial {\mathbb {V}}_n: (v_0,n_0) {\mathop {\longleftrightarrow }\limits ^{\omega _n}} (v,n) \text { for some }(v_0,n_0) \in A\}, \end{aligned}$$

where the notation $(v_0,n_0) {\mathop {\longleftrightarrow }\limits ^{\omega _n}} (v,n)$ means that $(v_0,n_0)$ and (v, n) are connected by an $\omega _n$-open path of edges of ${\mathbb {E}}_n$. Note that $A \subseteq C_n(A,\omega _n)$.

Now fix p, $\mathbf{q}_\mathbf{0}$ and $\epsilon$, and for $\delta$ close enough to zero let $\mathbf{q}=(q_1, \dots , q_{K+L})$ and $\mathbf{q'}=(q'_1, \dots , q'_{K+L})$ be as in the statement of the claim. Note that $\Vert \mathbf{q}-\mathbf{q'}\Vert _{\infty }<2\delta$. We will define coupling measures $\mu _{\mathcal {O}}$ on $(\varOmega _{\mathcal {O}})^2$ and $\mu _n$ on $(\varOmega _n)^2$ satisfying the following properties. First,

$$\begin{aligned} \begin{aligned}&(\omega _{\mathcal {O}},\omega '_{\mathcal {O}}) \sim \mu _{\mathcal {O}} \Longrightarrow \omega _{\mathcal {O}} {\mathop {=}\limits ^{\text {(d)}}} {\mathbb {P}}_{\mathbf{q}, p}\vert _{{\mathbb {E}}_{\mathcal {O}}},\quad \omega '_{\mathcal {O}} {\mathop {=}\limits ^{\text {(d)}}} {\mathbb {P}}_{\mathbf{q'}, p+\epsilon }\vert _{{\mathbb {E}}_{\mathcal {O}}} \\&\quad \text { and } \omega _{\mathcal {O}} \le \omega '_{\mathcal {O}} \text { a.s.}\end{aligned} \end{aligned}$$

(12)

(We denote by ${\mathbb {P}}_{\mathbf{q}, p} \vert _{{\mathbb {E}}'}$ the projection of ${\mathbb {P}}_{\mathbf{q}, p}$ to ${\mathbb {E}}' \subset {\mathbb {E}}$.) Second,

$$\begin{aligned} \begin{aligned}&(\omega _n,\omega '_n) \sim \mu _n \Longrightarrow \omega _n {\mathop {=}\limits ^{\text {(d)}}} {\mathbb {P}}_{\mathbf{q}, p}\vert _{{\mathbb {E}}_n},\quad \omega '_n {\mathop {=}\limits ^{\text {(d)}}} {\mathbb {P}}_{\mathbf{q'}, p+\epsilon }\vert _{{\mathbb {E}}_n} \\&\quad \text { and } C_n(A,\omega _n) \subseteq C_n(A,\omega '_n) \text { for all } A \subset \partial {\mathbb {V}}_n \text { a.s.}\end{aligned} \end{aligned}$$

(13)

We then define the coupling measure $\mu$ on $\varOmega ^2$ by

$$\begin{aligned} \mu = \mu _{\mathcal {O}} \otimes \left( \otimes _{n \in {\mathbb {Z}}} \mu _n \right) . \end{aligned}$$

It is clear from (12) and (13) that, if $(\omega , \omega ') \sim \mu$, then $\omega \sim {\mathbb {P}}_{\mathbf{q}, p}$, $\omega ' \sim {\mathbb {P}}_{\mathbf{q'}, p+\epsilon }$, and almost surely if $C_{\infty }$ holds for $\omega$, then it holds for $\omega '$. Consequently,

$$\begin{aligned} {\mathbb {P}}_{\mathbf{q}, p}(C_{\infty })\le {\mathbb {P}}_{\mathbf{q'}, p+\epsilon }(C_{\infty }). \end{aligned}$$

Fig. 5

The deterministic configuration for $G={\mathbb {Z}}$, $U=\{-3, -2, -1, 0, 1, 2, 3\}$. In this case  $L=7$, $K=6$ and $w_1=-3, w_2=3, w_3=-2, w_4=2, w_5=-1, w_6=1, w_7=0$

The definition of $\mu _{\mathcal {O}}$ is standard. We take in some probability space a pair of random elements $Z=(Z_1, Z_2) \in \varOmega _{\mathcal {O}}^2$ such that $Z_1$ and $Z_2$ are independent on all edges of ${\mathbb {E}}_{\mathcal {O}}$ and they assign each edge to be open with probability p and $\frac{\epsilon }{1-p}$, respectively. We then let $\omega _{\mathcal {O}}=Z_1$ and $\omega '_O=Z_1\vee Z_2$, and $\mu _{\mathcal {O}}$ be the distribution of $(\omega _{\mathcal {O}},\omega '_{\mathcal {O}})$, so that (12) is clearly satisfied.

The measures $\mu _n$ will be defined as translations of each other, so we only define $\mu _0$. The construction relies on Lemma 1, with the finite set S of that lemma being here the set

$$\begin{aligned} \varOmega _0^1 \times \cdots \times \varOmega ^{K+L}_0 \times \varOmega ^\partial _0 \times \varOmega ^\partial _0. \end{aligned}$$

The two factors of $\varOmega ^\partial _0$ ensure the extra randomness needed for the coupling. We now define the deterministic element ${\bar{x}}$ of the above set that appears in the statement of Lemma 1. The definition is simple, but the notation is clumsy; a quick glimpse at Fig. 5 should clarify what is involved. We start assuming, without loss of generality, that the elements $w_1, \dots , w_L$ of U are enumerated so that

$$\begin{aligned} \text {dist}_G(w_j, V\setminus U)\le \text {dist}_G(w_{j+1}, V\setminus U) \quad \forall j=1, \dots , L-1. \end{aligned}$$

Let $\varGamma _j$ be the set of edges along a shortest path from $w_j$ to $U\setminus B_{r-1}(u_0)$. Further, for $m<m'$ let

$$\begin{aligned}{}[(w_i, m), (w_i, m')]:=\cup _{j=m}^{m'-1}\{(w_i, j), (w_i, j+1)\}. \end{aligned}$$

Now, ${\bar{x}}$ is defined in the following way:

• ${\bar{x}}= ({\bar{x}}^{U}, {\bar{x}}^{\partial , 1}, {\bar{x}}^{\partial , 2})$ with ${\bar{x}}^{U} \in \varOmega _0^1 \times \dots \times \varOmega _0^{K+L}$ and ${\bar{x}}^{\partial , 1}, {\bar{x}}^{\partial , 2} \in \varOmega _0^{\partial }$;

• ${\bar{x}}^{U}(e)=1$ if and only if for some $j=1, \dots L$,

  $$\begin{aligned}&e \in [(w_j, 0), (w_j, j)] \cup [(w_j, (2L+2)-j), (w_j, (2L+2))] \\&\quad \bigcup _{\{u, v\}\in \varGamma _j} \left( \{(u, j), (v, j)\} \cup \{(u, (2L+2)-j), (v, (2L+2)-j)\}\right) , \end{aligned}$$

  or

  $$\begin{aligned} e\in \bigcup _{u, v\in U} \{(u, L+1), (v, L+1)\}; \end{aligned}$$

• ${\bar{x}}^{\partial , 1}\equiv 0 \text { and } {\bar{x}}^{\partial , 2}\equiv 1$.

By Lemma 1, if $\delta$ is close enough to zero, then there exists a coupling of $(K+L+2)$-tuples of configurations

$$\begin{aligned} X=\,&(X^1, \dots , X^{K+L}, X^{\partial , 1}, X^{\partial , 2}),\; Y=(Y^1, \dots , Y^{K+L}, Y^{\partial , 1}, Y^{\partial , 2})\\&\in \varOmega ^1_0 \times \dots \times \varOmega ^{K+L}_0 \times \varOmega _0^{\partial } \times \varOmega _0^{\partial } \end{aligned}$$

such that

• the values of $X^1, \dots , X^{K+L}, X^{\partial , 1}, X^{\partial , 2}$ are independent on all edges;

• the values of $Y^1, \dots , Y^{K+L}, Y^{\partial , 1}, Y^{\partial , 2}$ are independent on all edges;

• $X^i$ assigns each edge to be open with probability $q_i$;

• $Y^i$ assigns each edge to be open with probability $q'_i$;

• $X^{\partial , 1}$ and $Y^{\partial , 1}$ assign each edge to be open with probability p;

• $X^{\partial , 2}$ and $Y^{\partial , 2}$ assign each edge to be open with probability $\frac{\epsilon }{1-p}$;

• (X, Y) satisfies

  $$\begin{aligned} \mathbb {P}\left( \{X=Y\}\cup \{X={\bar{x}}\}\cup \{Y= {\bar{x}}\} \right) =1. \end{aligned}$$

  (14)

Now let $\omega _0=(X^1, \dots , X^{K+L}, X^{\partial , 1})$ and $\omega _0'=(Y^1, \dots , Y^{K+L}, Y^{\partial , 1}\vee Y^{\partial , 2})$. Thus, $\omega _0'$ assigns each edge in ${\mathbb {E}}_0^{\partial }$ to be open with probability $p+\epsilon$. See Fig. 6 for $\omega _0$ and $\omega '_0$ if X or Y equals ${\bar{x}}$.

Fig. 6

$\omega _0$ and $\omega '_0$ on the fixed configurations for $G={\mathbb {Z}}$, $U=\{-3, -2, -1, 0, 1, 2, 3\}$

To check that the last property stated in (13) is satisfied, let us inspect $C_0(A,\omega _0)$ and $C_0(A,\omega '_0)$ in all possible cases listed inside the probability in (14):

• if $X=Y$, then $\omega _0(e)\le \omega _0'(e)$ for every $e\in {\mathbb {E}}_0$; thus, $C_0(A,\omega _0)\subseteq C_0(A,\omega '_0)$ for all A;

• if $X={\bar{x}}$, then $C_0(A,\omega _0)=A \subseteq C_0(A, \omega '_0)$ for all A;

• if $Y={\bar{x}}$, then $C_0(A,\omega '_0)=\partial {\mathbb {V}}_0 \supseteq C_0(A,\omega _0)$ for all A.

Hence, in all cases $C_0(A,\omega _0)\subseteq C_0(A,\omega '_0)$ for every $A\subseteq \partial {\mathbb {V}}_0$. We then let $\mu _0$ be the distribution of $(\omega _0,\omega '_0)$, completing the proof. $\square$

4 Proof of Theorem 2

We start with a similar reduction to a particular case as the one in the beginning of the previous section. As the proof of Claim 1 did not rely on any special properties of ${\mathbb {G}}$ (that $\vec {{\mathbb {G}}}$ does not have), we can repeat the same argument in the oriented case. We fix $r \in {\mathbb {N}}$, $u_0 \in V$, and let $U:=B_r(u_0)$ as in the unoriented case. From now on, we assume that the edges $e_1,\ldots , e_K$of (4) are all the edges with both endpoints belonging to U.

We again obtain the desired statement of Theorem 2 as a consequence of the following claim.

Claim 3

For all $p\in (0,1)$, $\mathbf{q}_\mathbf{0}\in (0,1)^{K}$ and $\epsilon \in (0, 1-p)$ there exists a $\delta >0$ such that for any $\mathbf{q}, \mathbf{q'}\in (0,1)^{K}$ satisfying $\Vert \mathbf{q}_\mathbf{0}-\mathbf{q}\Vert _{\infty }<\delta$ and $\Vert \mathbf{q}_\mathbf{0}-\mathbf{q'}\Vert _{\infty }<\delta$ we have

$$\begin{aligned} \vec {{\mathbb {P}}}_{\mathbf{q}, p}(\vec {C}_{\infty })\le \vec {{\mathbb {P}}}_{\mathbf{q'}, p+\epsilon }(\vec {C}_{\infty }). \end{aligned}$$

Theorem 2 follows from this claim by the same argument as in the unoriented case, so we omit the details.

Remark 1

The proof of Claim 3 is similar to that of Claim 2 but slightly more involved. In the proof of the unoriented case we used Lemma 1 with a single deterministic configuration ${\bar{x}}=({\bar{x}}^U, {\bar{x}}^{\partial , 1}, {\bar{x}}^{\partial , 2})$. This was possible because our choice of ${\bar{x}}$ was such that, for every $\omega _0\in \varOmega _0$ and $A\subseteq \partial {\mathbb {V}}_0$ we have

$$\begin{aligned}&C_0(A, ({\bar{x}}^U, {\bar{x}}^{\partial , 1}))=A \subseteq C_0(A, \omega _0),\\&C_0(A, ({\bar{x}}^U, {\bar{x}}^{\partial , 1} \vee {\bar{x}}^{\partial , 2}))=\partial {\mathbb {V}}_0 \supseteq C_0(A, \omega _0). \end{aligned}$$

However, we cannot find a configuration with similar properties in the oriented case (see Remark 3 at the end of the proof).

Proof of Claim 3

Let

$$\begin{aligned} {\mathbb {V}}_n=\{(v, m)\in {\mathbb {V}}: v\in B_{r+1}(u_0), (2K+2)n\le m \le (2K+2)(n+1)\} \end{aligned}$$

and

$$\begin{aligned} \vec {{\mathbb {E}}}_n=\{ e\in \vec {{\mathbb {E}}}: e \text { has both endpoints in }{\mathbb {V}}_n\}. \end{aligned}$$

Note that $\vec {{\mathbb {E}}}_n$ are disjoint. Next, recall the definition of $\vec {{\mathbb {E}}}^{i}$ from (5) and define, for $n\in {\mathbb {Z}}$ and $1\le i \le K$,

$$\begin{aligned} \vec {{\mathbb {E}}}^i_n = \vec {{\mathbb {E}}}_n \cap \vec {{\mathbb {E}}}^i,\quad \vec {{\mathbb {E}}}^\partial _n = \vec {{\mathbb {E}}}_n \backslash \left( \cup _{i=1}^{K} \vec {{\mathbb {E}}}^i_n\right) ,\quad \vec {{\mathbb {E}}}_{\mathcal {O}} = \vec {{\mathbb {E}}} \backslash \left( \cup _{n \in {\mathbb {Z}}} \vec {{\mathbb {E}}}_n\right) . \end{aligned}$$

The “edge boundary” $\vec {{\mathbb {E}}}^\partial _n$ consists of edges of the form $\langle (u,m),(v,m+1)\rangle$, with $u, v \in {\mathbb {V}}_n$ and at least one of u and v at distance $r+1$ from $u_0$. Define corresponding sets of configurations $\vec {\varOmega }_n^i$, $\vec {\varOmega }_n^\partial$ and $\vec {\varOmega }_{\mathcal {O}}$.

For each n, define the boundary sets

$$\begin{aligned}&{\underline{\partial }} {\mathbb {V}}_n = \{(v,n) \in {\mathbb {V}}_n: \text {dist}(v, u_0)=r+1\} \cup ({\mathbb {V}}_n \cap ({\mathbb {V}} \times \{(2K+2)n\})),\\&{\overline{\partial }}{\mathbb {V}}_n = \{(v,n) \in {\mathbb {V}}_n: \text {dist}(v, u_0)=r+1\} \cup ({\mathbb {V}}_n \cap ({\mathbb {V}} \times \{(2K+2)(n+1)\})), \end{aligned}$$

so that ${\underline{\partial }} {\mathbb {V}}_n$ consists of “walls and floor” and ${\overline{\partial }}{\mathbb {V}}_n$ consists of “walls and ceiling” of the box ${\mathbb {V}}_n$. Given any $\varnothing \ne A\subseteq {\underline{\partial }}{\mathbb {V}}_n$ and $\omega _n \in \vec {\varOmega }_n$, define

$$\begin{aligned} \vec {C}_n(A,\omega _n) = \{(v,n) \in {\overline{\partial }} {\mathbb {V}}_n: (v_0,n_0) {\mathop {\longrightarrow }\limits ^{\omega _n}} (v,n) \text { for some }(v_0,n_0) \in A\}, \end{aligned}$$

where the notation $(v_0,n_0) {\mathop {\longrightarrow }\limits ^{\omega _n}} (v,n)$ means that $(v_0,n_0)$ and (v, n) are connected by an $\omega _n$-open path of edges of $\vec {{\mathbb {E}}}_n$.

Fix p, $\mathbf{q}_\mathbf{0}$ and $\epsilon$, and for $\delta$ close enough to zero let $\mathbf{q}=(q_1, \dots , q_{K})$ and $\mathbf{q'}=(q'_1, \dots , q'_{K})$ be as in the statement of the claim. We will define coupling measures $\vec {\mu }_{\mathcal {O}}$ on $(\vec {\varOmega }_{\mathcal {O}})^2$ and $\vec {\mu }_n$ on $(\vec {\varOmega }_n)^2$ that satisfy similar properties as in the unoriented case. First,

$$\begin{aligned} \begin{aligned} (\omega _{\mathcal {O}},\omega '_{\mathcal {O}}) \sim \vec {\mu }_{\mathcal {O}} \Longrightarrow \,&\omega _{\mathcal {O}} {\mathop {=}\limits ^{\text {(d)}}} \vec {{\mathbb {P}}}_{\mathbf{q}, p}\vert _{\vec {{\mathbb {E}}}_{\mathcal {O}}},\quad \omega '_{\mathcal {O}} {\mathop {=}\limits ^{\text {(d)}}} \vec {{\mathbb {P}}}_{\mathbf{q'}, p+\epsilon }\vert _{\vec {{\mathbb {E}}}_{\mathcal {O}}} \\&\text { and } \omega _{\mathcal {O}} \le \omega '_{\mathcal {O}} \text { a.s.} \end{aligned} \end{aligned}$$

(15)

Second,

$$\begin{aligned} \begin{aligned} (\omega _n,\omega '_n) \sim \vec {\mu }_n \Longrightarrow \,&\omega _n {\mathop {=}\limits ^{\text {(d)}}} \vec {{\mathbb {P}}}_{\mathbf{q}, p}\vert _{\vec {{\mathbb {E}}}_n},\quad \omega '_n {\mathop {=}\limits ^{\text {(d)}}} \vec {{\mathbb {P}}}_{\mathbf{q'}, p+\epsilon }\vert _{\vec {{\mathbb {E}}}_n} \\&\text { and } \vec {C}_n(A,\omega _n) \subseteq \vec {C}_n(A,\omega '_n) \quad \text { for all } A \subset {\underline{\partial }} {\mathbb {V}}_n \text { a.s.}\end{aligned} \end{aligned}$$

(16)

We then define the coupling measure $\vec {\mu }$ on $\vec {\varOmega }^2$ by

$$\begin{aligned} \vec {\mu } = \vec {\mu }_{\mathcal {O}} \otimes \left( \otimes _{n \in {\mathbb {Z}}} \vec {\mu }_n \right) . \end{aligned}$$

It is clear from (15) and (16) that, if $(\omega , \omega ') \sim \vec {\mu }$, then $\omega \sim \vec {{\mathbb {P}}}_{\mathbf{q}, p}$, $\omega ' \sim \vec {{\mathbb {P}}}_{\mathbf{q'}, p+\epsilon }$, and almost surely if $\vec {C}_{\infty }$ holds for $\omega$, then it holds for $\omega '$. Consequently,

$$\begin{aligned} \vec {{\mathbb {P}}}_{\mathbf{q}, p}(\vec {C}_{\infty })\le \vec {{\mathbb {P}}}_{\mathbf{q'}, p+\epsilon }(\vec {C}_{\infty }). \end{aligned}$$

The measure $\vec {\mu }_{\mathcal {O}}$ is defined using the same standard coupling as the corresponding measure in the proof of Claim 2. The measures $\vec {\mu }_n$ will again be taken as translations of each other, so we only define $\vec {\mu }_0$. The construction relies on Lemma 2. The finite set S and the decomposition  of the statement of that lemma are given by

where $\vec {\varLambda }_0^i$ is the set of configurations in $\vec {\varOmega }_0^i$ in which edges from height K to height $K+1$ are closed. The definition of ${\hat{x}}$ and  is as follows (see Fig. 7 for a specific example):

• ${\hat{x}}= ({\hat{x}}^1,\ldots , {\hat{x}}^K, {\hat{x}}^{\partial , 1}, {\hat{x}}^{\partial , 2})$ with ${\hat{x}}^i \in \vec {\varLambda }_{0}^i$ and ${\hat{x}}^{\partial , 1}, {\hat{x}}^{\partial , 2} \in \vec {\varOmega }_0^{\partial }$;

• with  and ;

• ${\hat{x}}^{\partial , 1}\equiv 0$, ${\hat{x}}^{\partial , 2}\equiv 1$ and for each i,  ${\hat{x}}^{i}(e)=0$ if and only if e goes from height K to $K+1$,;

• ,   and for each i, .

Fig. 7

The deterministic configurations for $G={\mathbb {Z}}$ and $U=\{-1, 0, 1, 2\}$. In this case ${K=3}$. Note that only one of the two disjoint subgraphs of $\vec {{\mathbb {G}}}$ is displayed

By Lemma 2, if $\delta$ is close enough to zero, there exists a coupling of $(K+2)$-tuples of configurations

$$\begin{aligned} X=\,&(X^1, \dots , X^{K}, X^{\partial , 1}, X^{\partial , 2}), \;Y=(Y^1, \dots , Y^{K}, Y^{\partial , 1}, Y^{\partial , 2}) \\&\in \vec {\varOmega }^1_0 \times \dots \times \vec {\varOmega }^K_0 \times \vec {\varOmega }_0^{\partial } \times \vec {\varOmega }_0^{\partial } \end{aligned}$$

such that

• the values of $X^1, \dots , X^K, X^{\partial , 1}, X^{\partial , 2}$ are independent on all edges;

• the values of $Y^1, \dots , Y^K, Y^{\partial , 1}, Y^{\partial , 2}$ are independent on all edges;

• $X^i$ assigns each edge to be open with probability $q_i$;

• $Y^i$ assigns each edge to be open with probability $q'_i$;

• $X^{\partial , 1}$ and $Y^{\partial , 1}$ assign each edge to be open with probability p;

• $X^{\partial , 2}$ and $Y^{\partial , 2}$ assign each edge to be open with probability $\frac{\epsilon }{1-p}$;

• (X, Y) satisfies

  

  (17)

Now let $\omega _0=(X^1, \dots , X^K, X^{\partial , 1})$ and $\omega _0'=(Y^1, \dots , Y^K, Y^{\partial , 1}\vee Y^{\partial , 2})$. Thus, $\omega _0'$ assigns each edge in $\vec {{\mathbb {E}}}_0^{\partial }$ to be open with probability $p+\epsilon$. See Fig. 8 for $\omega _0$ and $\omega '_0$ if X or Y equals ${\hat{x}}$ or .

Fig. 8

$\omega _0$ and $\omega '_0$ on the fixed configurations for $G={\mathbb {Z}}$, $U=\{-1, 0, 1, 2\}$

To check that the last property in (16) is satisfied, we need to show that in any of the situations listed inside the probability in (17), we have $\vec {C}_0(A, \omega _0) \subseteq \vec {C}_0(A, \omega '_0)$ for any $\varnothing \ne A \subseteq {\underline{\partial }}{\mathbb {V}}_n$. $\{X = {\hat{x}}\}$ entails $\vec {C}_0(A, \omega _0)=A\cap {\overline{\partial }}{\mathbb {V}}_0$ and $\{X = Y\}$, $\{X \in {\hat{S}},\; Y = {\hat{x}}\}$ as well as  lead to $\omega _0(e)\le \omega _0'(e)$ for every $e\in \vec {{\mathbb {E}}}_0$. The remaining case is when  and $Y = {\hat{x}}$. In this case, $(v_0, n_0)\xrightarrow []{\omega _0}(v_1, n_1)$ can only happen if $v_0, v_1 \in U, n_0=0$ and $n_1=(2K+2)$. But then we also have $(v_0, n_0)\xrightarrow []{\omega '_0}(v_1, n_1)$.

Finally, we let $\vec {\mu }_0$ be the distribution of $(\omega _0, \omega '_0)$, completing the proof. $\square$

Remark 2

As mentioned in Sect. 1.2, the approach we used to prove Theorem 2 is not readily applicable when the oriented model is replaced by a “continuous-time” version such as the contact process. The essential difficulty is that our approach involves finding a configuration that is better than any other in connecting points of any possible boundary set A to other boundary points. In a continuous-time setting, the set of configurations inside a finite box is infinite, so such an optimal configuration cannot exist. (In a standard construction involving Poisson processes, one can always introduce extra arrivals between those of a fixed configuration.) As a potential strategy, one could attempt to sophisticate our method by partitioning the configuration space not in two, but in infinitely many parts, proving a corresponding version of Lemma 2, and finding a sequence of finer and finer configurations which could produce an effective coupling.

Fig. 9

Examples of why we need two configurations in the oriented case. • denotes the vertices of $C_0(\circ , \cdot ) \backslash \{\circ \}$ in each configuration ($G={\mathbb {Z}}$, $U=\{-1, 0, 1, 2\}$)

Remark 3

In the oriented case we cannot find a configuration with similar properties as the one in Remark 1. If ${\hat{x}}=({\hat{x}}^U, {\hat{x}}^{\partial , 1}, {\hat{x}}^{\partial , 2})$ is such that ${\hat{x}}^U$ contains at least one closed edge, depending on the topology of $G\vert _{U}$, the induced subgraph of G on U, we can find a configuration $\omega _0\in \vec {\varOmega }_0$ and a set $A\subseteq {\underline{\partial }} {\mathbb {V}}_0$ such that

$$\begin{aligned} \vec {C}_0(A, ({\hat{x}}^U, {\hat{x}}^{\partial , 1} \vee {\hat{x}}^{\partial , 2})) \nsupseteq \vec {C}_0(A, \omega _0). \end{aligned}$$

In case  is such that every edge in  is open, then we can always find a configuration $\omega '_0\in \vec {\varOmega }_0$ and a set $B\subseteq {\underline{\partial }} {\mathbb {V}}_0$ such that

(see Fig. 9 for examples). This is the reason why we needed to apply Lemma 2, involving two deterministic configurations, to make the coupling work. The trick was to choose ${\hat{x}}$ and  in a way that for every $A\subseteq {\underline{\partial }} {\mathbb {V}}_0$,