1 Main results

Let $G=(V,E)$ be an infinite, connected, locally finite graph. A set of edges $F\subset E$ is called a cutset from a vertex v to $\infty $ if v belongs to a finite connected component of $(V,E\setminus F)$ . A cutset is called minimal if no proper subset of it is a cutset. Let $\mathcal {Q}_n(v)$ be the set of minimal cutsets from v to $\infty $ of cardinality n and consider the quantity

(1.1) $$ \begin{align} q_n:=\sup_{v\in V}|\mathcal{Q}_n(v)|. \end{align} $$

Here $|\emptyset |:=0$ . We emphasize that $q_n=\infty $ is possible, for example, for $G=\mathbb Z$ and $n=2$ . In this paper, we are interested in cases where the number of cutsets $q_n$ grows at most exponentially with n, and we define

(1.2) $$ \begin{align} \kappa(G):=\sup_{n\ge1} q_n^{1/n}. \end{align} $$

Let $\mathrm P_p$ denote (Bernoulli bond) percolation of parameter $p \in [0,1]$ on G, where each edge is open with probability p independently of the other edges. Consider the percolation probabilities $\theta _v(p):=\mathrm P_p(v\leftrightarrow \infty )$ , where $v\leftrightarrow \infty $ denotes the event that v belongs to an infinite open connected component. We define the critical parameter for uniform percolation as

(1.3) $$ \begin{align} p_c^*(G):=\inf\{p\in [0,1]: \theta^*(p)>0\}, \end{align} $$

where $\theta ^*(p):=\inf _{v\in V} \theta _v(p)$ .

By the classical Peierls argument [Reference Peierls19], if $\kappa (G)<\infty $ , then percolation on G has a uniformly percolating phase in the sense that $p_c^*(G)<1$ . Our first theorem establishes the converse.

Theorem 1.1. For every infinite, connected, locally finite graph G we have

(1.4) $$ \begin{align} p_c^*(G)<1 \iff \kappa(G)<\infty. \end{align} $$

Currently, the geometric condition $\kappa (G)<\infty $ is not well understood. Our second result gives a sufficient condition based on the simple random walk. Given a vertex v, let $\mathbb P_v$ be the law of a simple random walk $(X_t)_{t=0}^{\infty }$ on G starting at v. We say that G is uniformly transient if

(1.5) $$ \begin{align} \inf_{v\in V}\left[d_v \cdot \mathbb P_v(\forall t\ge 1: \ X_t\neq v)\right]>0, \end{align} $$

where $d_v$ denotes the degree of v.

2 Consequences and comments

In this section, all graphs are assumed to be infinite, connected, and locally finite. Given a set of vertices S in a graph $G=(V,E)$ , we define the boundary $\partial S$ to be the set of all edges $\{u,v\} \in E$ such that $u \in S$ but $v \not \in S$ , and we define the weight $|S|_G := \sum _{u \in S} d_u$ . The isoperimetric dimension of G is given by

$$\begin{align*}\operatorname{Dim}(G) := \sup\left\{ d \geq 1 : \inf_{ \substack{ S \subseteq V \\ 0 < |S|< \infty } } \frac{ |\partial S| } { |S|_G^{\frac{d-1}{d}} }> 0 \right\}. \end{align*}$$

1. 1. We remark that the uniform critical parameter $p^*_c(G)$ slightly differs from the most classical (nonuniform) one given by $p_c(G):=\inf \{p\in [0,1]: \theta (p)>0\}$ , where $\theta (p):=\sup _{v\in V} \theta _v(p)$ . However, these notions often coincide, such as for (quasi-)transitive graphs. See the introduction of [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9] for a survey of the rich history of the “ $p_c < 1$ ” question and its place in statistical mechanics. Let us just recall that all of the results about percolation here can be translated into analogous statements about many other models, most notably the Ising model.

2. 2. Duminil-Copin, Goswami, Raoufi, Severo, and Yadin proved that every quasitransitive graph of superlinear growth satisfies $p_c < 1$ [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9]. This had previously been a long-standing conjecture of Benjamini and Schramm [Reference Benjamini and Schramm5]. In fact, the authors of [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9] established that $p^*_c<1$ for every (not necessarily transitive) bounded degree graph G satisfying $\operatorname {Dim}(G)> 4$ , and this was known to imply the conjecture about transitive graphs by the classical works of Gromov [Reference Gromov11] and Trofimov [Reference Trofimov25].

3. 3. Theorem 1.2 establishes that $p^*_c < 1$ for every graph G satisfying $\operatorname {Dim}(G)> 2$ , since such graphs are uniformly transient (see, e.g., [Reference Lyons and Peres17, Theorem 6.41]). We therefore obtain stronger results than [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9], through a completely new proof. Theorem 1.2 fully realizes the idea at the heart of [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9] to exploit the transience of a simple random walk to prove $p_c < 1$ . In particular, we resolve [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9, Problem 1.4].

4. 4. Our proofs of Theorems 1.1 and 1.2 can also be run on finite graphs to establish the analogous results about giant clusters. (See [Reference Hutchcroft and Tointon15] for background.) In this setting, to define $q_n$ , one should instead count the number of minimal cutsets of cardinality n from a vertex v to another vertex u (and take the supremum over all choices for distinct u and v). The corresponding notion of uniform transience for a given family of finite graphs is that there exists a constant $C< \infty $ such that every graph $G=(V,E)$ in the family satisfies

   $$\begin{align*}\max_{u,v \in V} \mathcal R_G(u,v) \leq C, \end{align*}$$
   where $\mathcal R_G(u,v)$ denotes the effective resistance from u to v in the graph G.

5. 5. Babson and Benjamini conjectured that $\kappa < \infty $ for every transitiveFootnote ¹ graph of superlinear growth [Reference Babson and Benjamini1]. Notice that this purely geometric conjecture is a priori stronger than the above $p_c < 1$ conjecture of Benjamini and Schramm. Babson and Benjamini verified their conjecture in the special case of Cayley graphs of finitely presented groups by establishing that minimal cutsets in such graphs are coarsely connected. By [Reference Timár24, Reference Gromov11, Reference Trofimov25] (see also [Reference Contreras, Martineau and Tassion8, Lemma 2.1]), this extends to all transitive graphs satisfying $\operatorname {Dim}(G) < \infty $ . Given these results, it suffices to show that $\kappa < \infty $ for every transitive graph satisfying $\operatorname {Dim}(G) = \infty $ . Theorem 1.2 therefore resolves the $\kappa < \infty $ conjecture of Babson and Benjamini. (Alternatively, taking the results of [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9] for granted, this conjecture follows from Theorem 1.1).

6. 6. We establish the existence of a universal constant $\varepsilon> 0$ such that every transitive graph G satisfies $p_c = 1$ or $p_c \leq 1 - \varepsilon $ . When G is recurrent, this follows from the proof of [Reference Hutchcroft and Tointon15, Theorem 1.7], and when G is transient, this follows from our proof of Theorem 1.2 because there exists a universal constant $c> 0$ such that a simple random walk in any transient transitive graph has probability at least c never to return to where it started [Reference Tessera and Tointon22, Corollary 1.3]. Previous works had established this result if $\varepsilon $ is allowed to depend on the degree of vertices in G [Reference Hutchcroft and Tointon15, Theorem 1.7], or if we instead consider site percolation on a Cayley graph [Reference Panagiotis and Severo18, Reference Lyons, Mann, Tessera and Tointon16]. By the proof of Theorem 1.1, we also obtain a universal constant $K<\infty $ such that $\kappa <K$ for every transitive graph of superlinear growth.

7. 7. Much work has been motivated by a desire to find a sharp geometric criterion for a graph G to satisfy ${p_c}<1$ . Indeed, a well-known open conjecture of Benjamini and Schramm is that every (not necessarily transitive) graph G with $\operatorname {Dim}(G)> 1$ satisfies ${p_c}<1$ [Reference Benjamini and Schramm5]. We were very surprised to find that the geometric criterion $\kappa < \infty $ (which is arguably simpler and more natural than the isoperimetric criterion) is not just sharp but exact. Nevertheless, in light of Theorem 1.1 and this conjecture of Benjamini and Schramm, we encourage the reader to investigate the following:

   Conjecture 2.1. Every graph G with $\operatorname {Dim}(G)>1$ satisfies $\kappa < \infty $ .

The Peierls argument can be used to deduce results that are (a priori) much stronger than $p_c <1$ . To explore these, it helps to consider the isoperimetric profile $\psi $ of a graph $G=(V,E)$ , given by

$$\begin{align*}\psi(n) := \inf_{ \substack{ S \subseteq V \\ n \leq |S|_G < \infty } } |\partial S|. \end{align*}$$

1. 8. Every graph $G=(V,E)$ satisfying $\kappa < \infty $ admits a strongly percolating phase in the sense that for all $p \in (1-1/\kappa ,1]$ , there is a constant $c> 0$ such that

   (2.1) $$ \begin{align} \mathbb P_p(S \not\leftrightarrow \infty) \leq e^{-c \psi(|S|)} \qquad &\text{for every finite set } S \subseteq V; \nonumber\\ \mathbb P_p( n \leq |C_v| < \infty ) \leq e^{-c \psi(n) } \qquad &\text{for every } n \geq 1 \text{ and } v \in V. \end{align} $$
   Thus our work resolves [Reference Duminil-Copin, Goswami, Raoufi, Severo and Yadin9, Problem 1.6] and implies that percolation on every transitive graph of superlinear growth has a strongly percolating phase. It remains an important open problem to establish that on these graphs, such bounds hold for all $p \in (p_c,1]$ . Indeed, this is the “upper bound” half of [Reference Hermon and Hutchcroft13, Conjecture 5.1].

2. 9. Conversely, our proof of Theorem 1.1 (more precisely, Proposition 5.1) can be used to show that for every transitive graph $G=(V,E)$ and for every $p> p_c$ , there is a constant $c> 0$ such that

   $$\begin{align*}\mathbb P_p( n \leq |C_v| < \infty ) \geq e^{-c \psi(n) } \qquad \text{for every } n \geq 1 \text{ and 3} v \in V. \end{align*}$$
   This establishes the “lower bound” half of [Reference Hermon and Hutchcroft13, Conjecture 5.1].

3. 10. A major motivation for studying anchored isoperimetric inequalities for graphs and manifolds is the belief that – unlike (uniform) isoperimetric inequalities – anchored inequalities should typically be robust under small perturbations of the space [Reference Benjamini, Lyons and Schramm4, Section 6]. We obtain the following concrete statement to this effect by combining Theorem 1.1 with an argument of Pete [Reference Pete21, Theorem 4.1]: for every graph G satisfying $p^*_c(G)<1$ , there exists $\varepsilon> 0$ such that if G satisfies a d-dimensional anchored isoperimetric inequality for any $d \geq 1$ (or f-anchored isoperimetric inequality for any function f) then so does every infinite cluster formed by percolation of parameter $1-\varepsilon $ .

4. 11. By combining the previous item with Theorem 1.2 and results of Thomassen [Reference Thomassen23] and Pemantle and Peres [Reference Pemantle and Peres20], we deduce that for every graph $G=(V,E)$ with $\operatorname {Dim}(G)> 2$ , and for every probability measure $\mu $ on $(0,\infty )$ , the random weighted network $(V,C)$ with $C = (C(e) : e \in E) \sim \mu ^{\otimes E}$ is almost surely transient. (This was previously known if $\operatorname {Dim}(G)> 4$ [Reference Hutchcroft14].)

5. 12. A standard analysis of Karger’s algorithm from computer science establishes that every finite graph $G=(V,E)$ with exactly n vertices contains at most $n \choose 2$ minimum cuts, that is, sets of edges F such that $(V,E\backslash F)$ is disconnected but there is no set of edges $F'$ with $|F'| < |F|$ such that $(V,E \backslash F')$ is also disconnected. In the same spirit, in the present paper, we design randomized algorithms to instead count minimal cutsets.

3 Background and notation

In this section, we fix $G=(V,E)$ a locally finite, connected graph.

Paths and connectivity. Let $S\subset V$ and $u,v\in S$ . A path from u to v in S is a finite sequence $\gamma =(\gamma _0,\gamma _1,\ldots ,\gamma _\ell )$ of distinct vertices of S such that $\gamma _0=u$ , $\gamma _\ell =v$ and $\{\gamma _{i-1},\gamma _{i}\}\in E$ for every $i\in \{1,\ldots ,\ell \}$ . When such a path exists, we say that u is connected to v in S. By extension, a set A is said to be connected to a set B in S if there exists a vertex of A that is connected to a vertex of B in S. A path from u to $\infty $ in S is an infinite sequence of distinct vertices $\gamma _0,\gamma _1,\ldots $ in S such that $\gamma _0=u$ and $\{\gamma _{i-1},\gamma _{i}\}\in E$ for every $i\in \{1,2,\ldots \}$ . When such a path exists, we say that u is connected to $\infty $ in S.

Exposed boundary. Let $S\subset V$ be a finite set. The exposed boundary of S is the set $\partial _\infty S$ of all the edges $\{u,v\}$ such that $u\in S$ and v is connected to $\infty $ in $V\setminus S$ . Notice that the exposed boundary is a subset of the standard boundary defined at the beginning of Section 2: for every finite set $S\subset V$ , we have $\partial _\infty S\subset \partial S$ .

Percolation configurations. An element $\omega \in \{0,1\}^E$ is called a percolation configuration. Given such a configuration, an edge $e\in E$ is said to be open if $\omega (e)=1$ and closed if $\omega (e)=0$ . By extension, a path is said to be open if all its edges are open. The cluster of a vertex $u\in V$ is the connected component of u in the graph $(V,\{e\in E: \omega (e)=1\})$ .

Percolation events. A measurable subset $A\subset \{0,1\}^E$ is called a percolation event. Given $S\subset V$ and $u,v\in S$ , we denote by the event that there exists an open path from u to v in S, and simply write when $S=V$ . Finally, denotes the event that there exists an open path from u to $\infty $ in V.

Percolation measures. A percolation measure on G is a probability measure on the product space $\{0,1\}^E$ . For $p\in [0,1]$ , we denote by $\mathrm P_p$ the standard Bernoulli percolation measure, under which each edge is open with probability p independently of the other edges.

Positive association. A percolation event $\mathcal E$ is called increasing if for all percolation configurations $\omega ,\xi $ satisfying $\omega \le \xi $ for the standard product (partial) ordering, we have $\omega \in \mathcal E\implies \xi \in \mathcal E$ . Typical examples of increasing events are the connection events (such as ) introduced above. A percolation measure $\mathrm P$ is said to be positively associated if

(3.1) $$ \begin{align} \mathrm P[\mathcal E\cap \mathcal F] \ge \mathrm P[\mathcal E] \mathrm P[\mathcal F] \end{align} $$

for all increasing events $\mathcal E,\mathcal F$ . This property is often referred to as the FKG inequality. We will use that Bernoulli percolation $\mathrm P_p$ is positively associated (for every fixed $p\in [0,1]$ ) as established by Harris [Reference Harris12].

4 Exposed boundaries and cutsets

In this section, we fix $G=(V,E)$ , an infinite, connected, locally finite graph. In our paper, we will use that minimal cutsets can be obtained by considering the exposed boundary of finite connected sets. In this section, we recall some well-known facts relating the two notions. The first elementary result is that the exposed boundary of a finite connected set is a minimal cutset.

The second elementary result identifies the exposed boundary under some simple conditions.

Lemma 4.2. Let $u\in V$ , let $\Pi $ be a minimal cutset from u to $\infty $ . Let A be the connected component of u in $(V,E\setminus \Pi )$ and $B=\{e\cap A,\, e\in \Pi \}$ be the set of inner vertices of $\Pi $ . For every set S of vertices, we have

(4.1) $$ \begin{align} (B \subset S \subset A) \implies (\partial_\infty S= \Pi). \end{align} $$

Proof. Since A is a maximal connected set in $(V,E\setminus \Pi )$ , all the edges at the boundary of A belong to $\Pi $ , and therefore $\partial _\infty A\subset \partial A\subset \Pi $ . By Lemma 4.1, $\partial _\infty A$ is a cutset from u to $\infty $ ; hence, by the minimality of $\Pi $ , the two inclusions above must be equalities:

(4.2) $$ \begin{align} \partial _\infty A =\partial A= \Pi. \end{align} $$

Now, let S be a set satisfying $B\subset S\subset A$ . Let $e\in \Pi $ . Since $e\in \partial _\infty A$ , one endpoint of e must belong to B and the other endpoint is connected to $\infty $ in $V\setminus A$ . Therefore, by hypothesis, one endpoint of e belongs to S and the other endpoint is connected to $\infty $ in $V\setminus S$ . This proves the inclusion

(4.3) $$ \begin{align} \Pi\subset \partial_\infty S. \end{align} $$

Let $e \in \partial _{\infty }S$ . Let u be the endpoint of e in S, and let v be the endpoint of e connected to $\infty $ in $V \backslash S$ . Then, by hypothesis, $u \in A$ and v is connected to $\infty $ in $V \backslash B$ . Since $\Pi = \partial A$ , every edge in $\Pi $ intersects A and hence intersects B. Therefore, there must exist an infinite path starting at v in the subgraph $(V,E \backslash \Pi )$ . In particular, $v \not \in A$ , and hence $e \in \partial A = \Pi $ . This proves that the inclusion above must be an equality.

5 Full connectivity via positive association

In this section, we consider the following problem: Let B be a finite set in a graph and $\mathrm P$ be a percolation measure. What is the probability that all the vertices of B are connected to each other? Or, in other words, what is the probability that all the vertices of B lie in the same cluster? We prove that this probability is at least exponential in the size of B when the measure is positively associated, and the probability for a point to be connected to B is uniformly lower bounded. This result, formally stated below, will allow us to construct random sets with a prescribed boundary.

Proposition 5.1. Let $G=(V,E)$ be a finite, connected graph. Let $\mathrm P$ be a positively associated percolation measure on G. Let $B \subset V$ , let $\theta ,p \in (0,1]$ and suppose that $\mathrm P(u \leftrightarrow B) \geq \theta $ for every $u \in V$ , and $\mathrm P(e \text { is open}) \geq p$ for every $e \in E$ . Then for every $o \in V$ ,

$$\begin{align*}\mathrm P\left( \bigcap_{b \in B} \{ o \leftrightarrow b \}\right) \geq c^{|B|}, \end{align*}$$

where $c := \left (\frac {p\theta }{2}\right )^{3/\theta }$ .

Proof. Say that a finite sequence of vertices $x_1,\dots ,x_k$ is chained if $x_1 = o$ and for all $i \in \{2,\dots ,k\}$ ,

(P1) $$ \begin{align} \frac{p\theta}{2} \leq \mathrm P\left( x_i \leftrightarrow \{x_1,\dots,x_{i-1}\} \right) \leq \frac{\theta}{2}. \end{align} $$

Since there exists at least one chained sequence (take $k = 1$ ) and V is finite, there must exist a chained sequence $x_1,\dots ,x_k$ that is maximal in the sense that for every vertex $x_{k+1}$ , the sequence $x_1,\dots ,x_{k+1}$ is not chained. Fix a maximal chained sequence $x_1,\dots ,x_{k}$ , and let $X := \{x_1,\dots ,x_{k}\}$ . We claim that, in addition to (P1), this sequence satisfies the following two properties, where $n:= |B|$ :

(P2)

(P3) $$ \begin{align} & k \leq \frac{2n}{\theta}. \end{align} $$

To prove (P2), consider the set of vertices $W\subset V$ that are connected to X with probability at least $\theta /2$ and suppose for contradiction that $W\neq V$ . Since W is non empty (because $X\subset V$ ) and G is connected, we can consider an edge $\{u,v\}$ such that $u\in W$ and $v\notin W$ . By positive association,

$$\begin{align*}\theta/2>\mathrm P( v \leftrightarrow X ) \geq \mathrm P(\{u,v\} \text{ is open}) \cdot \mathrm P(u \leftrightarrow X) \geq \frac{p \theta}{2}. \end{align*}$$

In particular, $x_1,\dots ,x_{k},v$ is a chained sequence, contradicting the maximality of $x_1,\dots ,x_k$ .

We now prove (P3). To this aim, for each $i \in \{1,\dots ,k\}$ , let $N_i$ denote the number of clusters that intersect both $\{x_1,\dots ,x_i\}$ and B. For every $i\in \{2,\dots ,k\}$ , the increment $N_{i}-N_{i-1}$ is equal to $1$ if $x_i$ is connected to B but not to the previous points $\{x_1,\ldots ,x_{i-1}\}$ , and it is equal to $0$ otherwise. Therefore, for every $i\in \{2,\ldots ,k\}$ , we have the deterministic inequality

(5.1)

Taking the expectation, using our hypothesis and (P1), for every $i\in \{2,\ldots , k\}$ , we get

(5.2)

Summing over $i\in \{2,\ldots , k\}$ and using , we get $\mathrm E[N_k] \geq \frac {\theta }{2}k$ . Since $N_k$ is deterministically bounded above by $|B| = n$ , this concludes the proof of (P3).

We now explain how the three properties above of the chained sequence imply the desired lower bound in the proposition. First, we estimate the event that all the vertices of X are connected to o: By (P1), (P3), and positive association, we have

(5.3)

Second, we estimate the event that all the vertices of B are connected to X: By (P2) and positive association, we have

$$\begin{align*}\mathrm P \left( \bigcap_{b\in B} \{ b \leftrightarrow X\} \right) \geq \left( \frac{\theta}{2} \right)^n. \end{align*}$$

If all the vertices of X are connected to o and all the vertices of B are connected to X, then all the vertices of B are connected to o. Hence, by the two displayed equations above and positive association, we obtain

where $c := \left (\frac {p\theta }{2}\right )^{3/\theta }$ .

6 Proof of Theorem 1.1

Let $G=(V,E)$ be an infinite, connected, locally finite graph. In this section, we prove Theorem 1.1, in the following form.

(6.1) $$ \begin{align} \big (\exists p<1 \ \exists \theta>0 \ \forall u \in V \quad\mathrm P_p(u\leftrightarrow \infty)\ge \theta \big )\iff \big (\exists K<\infty \ \forall u\in V\ \forall n\ge1\quad |\mathcal Q_n(u)|\le K^n\big). \end{align} $$

The implication $\Leftarrow $ is well known and follows from the Peierls argument [Reference Benjamini2, Theorem 4.11], which we now recall for completeness. Let $u\in V$ . If the cluster of u is finite, then by Lemma 4.1, its exposed boundary is a finite minimal cutset from u to $\infty $ , and all its edges are closed. Hence, by the union bound, for every $p\in [0,1]$ we have

(6.2) $$ \begin{align} \mathrm P_p(|C_u|<\infty)\le \sum_{n\ge 1} q_n (1-p)^n. \end{align} $$

If $q_n\le K^n$ for some constant $K<\infty $ , then the right-hand side above converges to $0$ as p tends to $1$ . Since the bound is uniform in u, there exists $p<1$ such that

(6.3) $$ \begin{align} \forall u\in V \quad \mathrm P_p(u\leftrightarrow \infty)\ge 1/2. \end{align} $$

We now prove the implication $\Rightarrow $ . Fix $\theta ,p\in (0,1)$ such that $\mathrm P_{p}(u \leftrightarrow \infty ) \geq \theta $ for every $u\in V$ . Fix $o\in V$ and $n\ge 1$ . Writing C for the cluster of o, we show that for every minimal cutset $\Pi $ from o to $\infty $ with $|\Pi | = n$ ,

(6.4) $$ \begin{align} \mathrm P_p (\partial_{\infty} C = \Pi ) \geq 1/K^n, \end{align} $$

where $K=K(p,\theta )\in (0,\infty )$ is a finite constant depending on p and $\theta $ only (in particular, it does not depend on the chosen vertex o). This concludes the proof since

(6.5) $$ \begin{align} 1\ge \sum_{ \Pi \in \mathcal Q_n(o) } \mathrm P_p(\partial_{\infty} C = \Pi ) \overset{~(6.4)}\ge |\mathcal{Q}_n(o)| /K^n. \end{align} $$

Let us now prove the lower bound (6.4). As in Lemma 4.2, let A be the connected component of o in $(V,E\setminus \Pi )$ and B the set of inner vertices of $\Pi $ . Since any infinite open path from a vertex $u\in A$ must intersect B before exiting A, the hypothesis $\mathrm P_p(u\leftrightarrow \infty )\ge \theta $ implies

(6.6)

Let $\mathcal E$ be the event that every vertex in B is connected to o by an open path in A. By Proposition 5.1 applied to the finite subgraph of G induced by A, we have $\mathrm P_{p}(\mathcal E) \geq c^n$ , where $c=(p\theta /2)^{3/\theta }>0$ . Let $\mathcal F$ be the event that all the edges of $\Pi $ are closed. By independence, we have

(6.7) $$ \begin{align} \mathrm P_p(\mathcal E\cap \mathcal F)= \mathrm P_p(\mathcal E)\mathrm P_p(\mathcal F)\ge c^n (1-p)^n. \end{align} $$

If the event $\mathcal E\cap \mathcal F$ occurs, then the cluster C of o satisfies $B\subset C\subset A$ . Hence, by Lemma 4.2 we must have $\partial _\infty C=\Pi $ . This concludes that

(6.8) $$ \begin{align} \mathrm P_p(\partial_{\infty} C=\Pi)\ge \mathrm P_p(\mathcal E\cap \mathcal F)\ge c^n(1-p)^n, \end{align} $$

which establishes the desired lower bound (6.4) with $K=\frac 1{c(1-p)}=\frac 1{(p\theta /2)^{3/\theta }(1-p)}$ .

7 A covering lemma for Markov chains

In this section, we give conditions under which a killed Markov chain survives long enough to visit every state and then return to its initial state.Footnote ² We will apply this in the next section to prove Theorem 1.2. Here $[n]$ denotes the set $\{1,\dots ,n\}$ .

Lemma 7.1. Let $n \geq 1$ . Let $P=(p_{i,j})_{i,j \in [n]}$ be a symmetric matrix of nonnegative entries such thatFootnote ³ $\sum _{j \in [n]} p(i,j) \leq 1$ for all $i \in [n]$ . Let $\Gamma $ be the set of all sequences $\gamma =(\gamma _0,\gamma _1,\ldots ,\gamma _k)$ in $[n]$ (for any $k \geq 1$ ) with $\gamma _0=1$ such that the unique element $i\in [k]$ satisfying both $\gamma _i=1$ and $\{\gamma _0,\gamma _1,\ldots ,\gamma _i\} = [n]$ is $i=k$ . For every such sequence $\gamma $ , define

$$\begin{align*}p(\gamma) := \prod_{i=1}^k p\left(\gamma_{i-1},\gamma_i\right). \end{align*}$$

For each $\varepsilon> 0$ , if every nonempty proper subset I of $[n]$ satisfies

(7.1) $$ \begin{align} \sum_{i \in I} \sum_{j \in [n] \backslash I} p(i,j) \geq \varepsilon, \end{align} $$

then $\delta :=\frac {\varepsilon ^2}{16e^2}$ satisfies

$$\begin{align*}\sum_{\gamma \in \Gamma} p(\gamma) \geq \delta^n. \end{align*}$$

Proof. Let $e_1,\dots ,e_{2n-2} \in [n]^2 \sqcup \{\emptyset \}$ be an independent and identically distributed (IID) sequence of random variables such that for all $u,v \in [n]$ ,

$$\begin{align*}\mathbb P\left( e_1 = (u,v) \right) =\frac{p(u,v)}{n}. \end{align*}$$

Such random variables exist because these probabilities sum to at most 1. Let H be the undirected multigraph with vertex set $[n]$ and edges $e_1,\dots ,e_{2n-2}$ . Even though $[n]^2$ consists of ordered pairs, we think of each $e_i \in [n]^2$ as encoding an undirected edge, loops allowed. (When $e_i = \emptyset $ , we simply do not include an edge.)

Consider the IID spanning subgraphs $H_1$ and $H_2$ of H that contain only the edges $e_1,\dots ,e_{n-1}$ and $e_{n},\dots ,e_{2n-2}$ , respectively. We will lower bound the probability that each of these graphs is connected. Consider any $k \in [n-1]$ . Suppose that we are given all of the connected components $C_1,\dots ,C_r$ of the spanning subgraph of H that contains only the edges $e_1,\dots ,e_{k-1}$ . If $r \geq 2$ , then the conditional probability that $e_{k}$ connects two of these components is

$$\begin{align*}\sum_{z=1}^r \sum_{ i\in C_z} \sum_{ j \in [n] \backslash C_z} \frac{p(i,j)}{n} \overset{(7.1)}\geq \frac{r \varepsilon}{n}. \end{align*}$$

Therefore by induction on k, and by using the elementary bound $\frac {n^n}{n!} \leq e^n$ in the third inequality,

(7.2) $$ \begin{align} \mathbb P\left( H_1 \text{ is connected} \right) \geq \prod_{r=2}^{n} \frac{r \varepsilon}{n} \geq \frac{n! \cdot \varepsilon^n}{n^{n} } \geq \frac{\varepsilon^n}{e^{n}}. \end{align} $$

Let $\gamma =(\gamma _0,\gamma _1,\ldots ,\gamma _k)$ be a sequence in $\Gamma $ . Say that $\gamma $ is present if there exists an injection $\sigma : [k] \to [2n-2]$ such that for every $i \in [k]$ , we have $e_{\sigma (i)} = (\gamma _{i-1},\gamma _i)$ or $(\gamma _i,\gamma _{i-1})$ . Assume that $k \leq 2n-2$ , and note that $\gamma $ cannot be present otherwise. There are at most $(2n-2)^k$ choices of $\sigma $ , and given $\sigma $ , for each i, the probability that $e_{\sigma (i)} = (\gamma _{i-1},\gamma _i)$ is the same as the probability that $e_{\sigma (i)}=(\gamma _i,\gamma _{i-1})$ , both given by $\frac {1}{n}p( \gamma _{i-1},\gamma _i ) = \frac {1}{n}p( \gamma _i,\gamma _{i-1} )$ . So by a union bound,

(7.3) $$ \begin{align} \mathbb P\left( \gamma \text{ is present} \right) \leq (2n-2)^k \prod_{i=1}^k \frac{2 }{n} p\left( \gamma_{i-1},\gamma_i \right) \leq 4^k p(\gamma) \leq 4^{2n}p(\gamma). \end{align} $$

On the other hand, when $H_1$ is connected and $H_2$ is connected, then some $\gamma \in \Gamma $ must be present in H because every multigraph that contains two edge-disjoint spanning trees must also contain a spanning subgraph that is connected and Eulerian [Reference Catlin7, Corollary 2.3A].Footnote ⁴ Thanks to (7.2), this occurs with probability at least $\varepsilon ^{2n}/e^{2n}$ . So by a union bound,

(7.4) $$ \begin{align} \frac{\varepsilon^{2n}}{e^{2n}} \leq \sum_{\gamma \in \Gamma} \mathbb P(\gamma \text{ is present}) \leq 4^{2n} \sum_{\gamma \in \Gamma} p(\gamma). \end{align} $$

The conclusion follows by rearranging.

8 Proof of Theorem 1.2

Let $G=(V,E)$ be an infinite, connected, locally finite graph such that for some constant $\varepsilon> 0$ , for every vertex $v \in V$ , the simple random walk $(X_t)_{t=0}^{\infty }$ on G started at v satisfies

$$\begin{align*}d_v \mathbb P_v(\forall t\ge 1 : \ X_t\neq v) \geq \varepsilon. \end{align*}$$

Let $G'=(V',E')$ be the graphFootnote ⁵ obtained from G by replacing each edge by a path of length 2. View V as a subset of $V'$ , and let $m: E \to V'$ map each edge to its midpoint. Let $\mathbb P_u'$ be the law of simple random walk in $G'$ started from a given vertex u, and let $\tau := \sup \{ t \geq 0 : X_t = X_0 \}$ . We claim that for all $z \in V'$ ,

(8.1) $$ \begin{align} d_z \mathbb P_z'(\tau =0 ) \geq \varepsilon_1:=\frac{2\varepsilon}{4+\varepsilon}. \end{align} $$

This is trivial when $z \in V$ , even with $\varepsilon _1 = \varepsilon /2$ , because simple random walk on $G'$ induces lazy simple random walk on G. Otherwise, when $z = m(\{u,v\})$ for some $\{u,v\} \in E$ , this follows from the corresponding bounds for u and v by rearranging the following elementary calculation, where $\ell _x := |\{t \geq 0 : X_t = x\}|$ :

$$ \begin{align*} \frac{1}{\mathbb P_z^{\prime}(\tau = 0)} = \sum_{n \geq 0}\mathbb P_z^{\prime}(\tau> 0)^n = \mathbb E_z^{\prime}[\ell_z] &= \sum_{t \geq 0} \mathbb P_z^{\prime}(X_t = z)\\ &= 1 + \sum_{t \geq 1}\left[ \mathbb P_z^{\prime}(X_{t-1} = u) \cdot \frac{1}{d_u} + \mathbb P_z^{\prime}(X_{t-1} = v) \cdot \frac{1}{d_v}\right]\\ &= 1 + \frac{\mathbb E_z^{\prime}[\ell_u]}{d_u} + \frac{\mathbb E_z^{\prime}[\ell_v]}{d_v} \\&\leq 1 + \frac{\mathbb E_u^{\prime}[\ell_u]}{d_u} + \frac{\mathbb E_v^{\prime}[\ell_v]}{d_v} = 1 + \frac{1}{d_u \mathbb P_u^{\prime}(\tau = 0)} + \frac{1}{d_v \mathbb P_v^{\prime}(\tau = 0)}. \end{align*} $$

Let $C := \{X_t : 0 \leq t \leq \tau \}$ and $\partial := \{ e \cap C : e \in \partial _\infty C \}$ . Fix $o \in V$ , and pick a neighbor $o' \in m(E)$ of o in $G'$ . Fix a finite minimal cutset $\Pi $ from o to $\infty $ in G, and set $n := |\Pi |$ . We will show that for some finite constant $K=K(\varepsilon ) \in (0,\infty )$ depending only on $\varepsilon $ ,

(8.2) $$ \begin{align} \mathbb P_{o^{\prime}}^{\prime} \left( \partial = m(\Pi) \right) \geq 1/K^n. \end{align} $$

This implies that $\kappa (G) < \infty $ because for all $o \in V$ and $n \geq 1$ ,

$$\begin{align*}1 \geq \sum_{\Pi \in \mathcal Q_n(o)} \mathbb P_{o^{\prime}}^{\prime} \left( \partial = m(\Pi) \right) \overset{(8.2)}\geq |\mathcal Q_n(o)|/K^n. \end{align*}$$

Let A be the connected component of o in $(V,E \backslash \Pi )$ , let $U := m(\Pi ) \cup \{o^{\prime }\}$ , and let $I := A \cup m( \{e \in E : e \subset A\} )$ . For all $u,v \in U \cup I$ , let

$$\begin{align*}p(u,v) := \mathbb P_{u}^{\prime}\left( \exists t \geq 1 : X_1,\dots,X_{t-1} \in I \backslash \{u\} \text{ and } X_t = v \right). \end{align*}$$

Extend this to sets of vertices by $p(L,R) := \sum _{u \in L; v \in R} p(u,v)$ , and similarly, $p(u,L) := p(\{u\},L)$ and $p(L,u) := p(L,\{u\})$ . We would like to apply Lemma 7.1 to the matrix $P:=(p(u,v))_{u,v \in U}$ . By time-reversing trajectories, we have $p(u,v) = p(v,u)$ whenever $d_u = d_v$ , which is, for example, the case when $u,v \in U$ . So P is symmetric, and clearly the entries of P are nonnegative and sum to at most 1 along each row. We claim that for every nontrivial partition $U = L \sqcup R$ ,

(8.3) $$ \begin{align} p(L,R) \geq \varepsilon_2 := \varepsilon_1^2/64. \end{align} $$

Indeed, for each $x \in U \cup I$ , consider the function (the unit voltage)

$$\begin{align*}F(x) := \mathbb P_x^{\prime}\left( \exists t \geq 0 : X_0,\dots,X_{t-1} \not\in L \text{ and }X_t \in R \right). \end{align*}$$

Given $u \in L$ , there existsFootnote ⁶ $x \in A$ such that $\{u,x\} \in E'$ , and if $F(x) \geq 1/2$ , then we are done because

$$\begin{align*}p(u,R) \geq \mathbb P_u'(X_1 = x) \cdot F(x) \geq 1/2 \cdot 1/2 \geq \varepsilon_2. \end{align*}$$

In particular, we may assume that there exists $x \in A$ with $F(x) < 1/2$ . By a similar argument, we may assume that there exists $y \in A$ with $F(y)> 1/2$ . Since A is connected in G, we can therefore find $\{x',y'\} \in E$ satisfying $F(x') \leq 1/2 \leq F(y')$ . Let $z := m(\{x',y'\})$ , which has degree 2. Note that

$$\begin{align*}F(z) \geq \mathbb P_z'(X_1 = y') \cdot F(y') \geq 1/2 \cdot 1/2, \end{align*}$$

and by a union bound,

$$\begin{align*}F(z) \leq \sum_{n = 0}^{\infty} \mathbb P_z'(\tau> 0)^n p(z,R) = \frac{p(z,R)}{\mathbb P_z'(\tau = 0)} \overset{(8.1)}\leq \frac{p(z,R)}{\varepsilon_1/2}. \end{align*}$$

So by rearranging, $p(z,R) \geq \varepsilon _1/8$ . By a similar argument (i.e.,by replacing F by $1-F$ , which switches the roles of L and R, and by recalling that $p(L,z)=p(z,L)$ ), we deduce that $p(L,z) \geq \varepsilon _1/8$ . Now (8.3) follows because $p(L,R) \geq p(L,z) p(z,R)$ .

Therefore by Lemma 7.1, the event $\mathcal E$ that the random walk visits every vertex in U then returns to $o'$ before exiting $U \cup I$ satisfies

(8.4) $$ \begin{align} \mathbb P_{o^{\prime}}^{\prime}\left( \mathcal E \right) \geq \varepsilon_3^{|U|} \geq \varepsilon_3^{n+1} \end{align} $$

for some constant $\varepsilon _3> 0$ depending only on $\varepsilon _2$ . So by Lemma 4.2 and the strong Markov property,

(8.5) $$ \begin{align} \mathbb P_{o^{\prime}}^{\prime}\left( \partial = m\left(\Pi \right) \right) \geq \mathbb P_{o^{\prime}}^{\prime}\left( \mathcal E \right) \cdot \mathbb P_{o^{\prime}}^{\prime}\left( \tau = 0 \right) \overset{{(8.1)},{(8.4)}}\geq \varepsilon_3^{n+1} \cdot \varepsilon_1/2. \end{align} $$

By expanding the definitions of $\varepsilon _1,\varepsilon _2,\varepsilon _3$ we deduce that (8.2) holds with $K := 2^{20}/\varepsilon ^5$ .

9 Alternative proof of Theorem 1.2 using the Gaussian free field

Here we sketch an alternative, slightly less elementary proof of Theorem 1.2 along the lines of the proof of Theorem 1.1. Let G be an infinite, connected, locally finite graph that is uniformly transient. Consider the graph $\tilde {G}=(\tilde {V},{\tilde {E}})$ obtained by replacing each edge by a path of length $3$ . Similarly to the proof in Section 8, one can prove that $\tilde {G}$ is also uniformly transient. Let $\varphi \in \mathbb {R}^{\tilde {V}}$ with law $\mathbb P$ be the (centered) Gaussian free field (GFF) on $\tilde G$ – see, for example, [Reference Berestycki and Powell6, Section 1.1] for the required background and definitions. Uniform transience implies that there exists $\varepsilon> 0$ such that $\mathrm {Var}(\varphi (x))\leq 1/\varepsilon $ for every $x\in \tilde {V}$ .

Fix $o\in V$ and let $\tilde {C}$ be the cluster of o in the percolation model induced by the excursion set $\{\varphi \geq 0\}:=\{x\in \tilde {V}:~\varphi (x)\geq 0\}$ . Given every edge e of G, we associate the corresponding mid-edge $\tilde {e}$ in $\tilde {G}$ , with both endpoints of degree $2$ . For a subset $\Pi $ of edges in G, we denote by $\tilde {\Pi }$ the associated set of mid-edges in $\tilde {G}$ . We claim that there exists $c=c(\varepsilon )>0$ , depending only on $\varepsilon $ , such that for every $\Pi \in \mathcal {Q}_n(o)$ ,

(9.1) $$ \begin{align} \mathbb{P} ( \partial_\infty \tilde{C}=\tilde{\Pi}) \geq c^n. \end{align} $$

Similarly to the previous sections, Theorem 1.2 follows readily from (9.1).

We now proceed to prove (9.1). Enumerate $\tilde {\Pi }$ by $\tilde {e}_i=\{x_i,y_i\}$ , $1\leq i\leq n$ , where $x_i$ and $y_i$ are the inner and outer endpoints, respectively. We first observe that, for some constant $c_1=c_1(\varepsilon )>0$ ,

(9.2) $$ \begin{align} \mathbb{P}(\varphi(y_i)\in [-2,-1] \text{ and } \varphi(x_i)\in [1,2]~~\forall~ 0\leq i\leq n)\geq c_1^n. \end{align} $$

Indeed, this follows by successively demanding the desired event at each vertex. Here we use the Markov property of the GFF (see [Reference Berestycki and Powell6, Theorem 1.10]) and the fact that the conditional variance of the next vertex given the previous ones is between $1/2$ (since they have degree $2$ ) and $1/\varepsilon $ , while the conditional mean remains bounded between $-2$ and $2$ .

Let $\mathcal {F}$ be the event in (9.2) and A be the component of o in $(\tilde {V},\tilde {E}\setminus \tilde {\Pi })$ . Notice that

By the Markov property, conditionally on $\mathcal {F}$ , the process $\{\varphi \geq 0\}\cap A$ stochastically dominates $\{\varphi _A\geq -1\}$ , where $\varphi _A$ is the centered GFF on A (i.e., associated to the random walk on A killed when reaching $\partial A=\{x_1,\dots ,x_n\}$ ). Therefore, it is enough to prove that, for some constant $c_2=c_2(\varepsilon )>0$ ,

(9.3)

Indeed, since the GFF is positively associated (see [Reference Berestycki and Powell6, Theorem 3.38]), the desired inequality (9.3) follows readily from Proposition 5.1 and the following inequality

(9.4)

for some constant $c_3=c_3(\varepsilon )>0$ . The latter follows easily from the Markov property of the GFF. Indeed, let $\mathcal {S}$ be the union of all clusters of $\{\varphi _A\geq -1\}$ intersecting $\partial A$ and note that its closure $\overline {\mathcal {S}}$ (i.e., the union of S with its neighbors) is a stopping set. Clearly, one has $\text {sgn}(\varphi _A(u)+1)=1$ almost surely on the event . On the complementary event $\mathcal {G}^c$ and conditionally on the field on $\overline {\mathcal {S}}$ , the Markov property implies that we have a GFF on $A\setminus \mathcal {S}$ with boundary conditions $<-1$ . In particular, $\text {sgn}(\varphi _A(u)+1)$ has a negative conditional expectation on $\mathcal {G}^c$ . These observations readily imply the first inequality of (9.4). The second inequality follows from the fact that the variance of $\varphi _A(u)$ is at most $1/\varepsilon $ .