On percolation critical probabilities and unimodular random graphs

We investigate generalisations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p_c}$ defined by Duminil-Copin and Tassion (2015) to bounded degree unimodular random graphs. We further examine Schramm's conjecture in the case of unimodular random graphs: does $p_c(G_n)$ converge to $p_c(G)$ if $G_n\to G$ in the local weak sense? Among our results are the following: 1. $p_c=\tilde{p_c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and $p_T<p_c$; i.e., the classical sharpness of phase transition does not hold. 2. We give conditions which imply $\lim p_c(G_n) = p_c(\lim G_n)$. 3. There are sequences of unimodular graphs such that $G_n\to G$ but $p_c(G)>\lim p_c(G_n)$ or $p_c(G)<\lim p_c(G_n)<1$. As a corollary to our positive results, we show that for any transitive graph with sub-exponential volume growth there is a sequence $T_n$ of large girth bi-Lipschitz invariant subgraphs such that $p_c(T_n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.


Motivation and results
There are several definitions of the critical probability for percolation on the lattices Z d , which have turned out to be equivalent not only on Z d , but also in the more general context of arbitrary transitive graphs [26,1,16,4,11]. One of our goals is to investigate the relationship between these different definitions when the graph G is an ergodic unimodular random graph [9,2], which is the natural extension of transitivity to the disordered setting. We examine the generalisations of p c = sup{p : P p (there is an infinite cluster) = 0}, p T = sup {p : E p (|C o |) < ∞} andp c defined by Duminil-Copin and Tassion in [11]. The last quantity was in fact designed to give a simple new proof of p c = p T for transitive graphs, and to address the question of locality of critical percolation: whether the value of p c depends only on the local structure of the graph.
More precisely, Schramm's "locality conjecture", stated first explicitly in [8], says that p c (G n ) → p c (G) holds whenever G n is a sequence of vertex-transitive infinite graphs such that G n converges locally to G (i.e., for every radius r, the r-ball in G n , for n large enough, is isomorphic to the r-ball in G) and sup n p c (G n ) < 1. Typically, however, the natural setting for such locality statements is not the class of transitive graphs, but the class of unimodular random graphs. Indeed, there are several interesting probabilistic quantities, most often related in some way to random walks, which have turned out to possess locality, mostly in the generality of unimodular random graphs: see [9,21,23,10,6,17] for specific examples, and [28,Chapter 14] for a partial overview. Therefore, it is natural to investigate Schramm's conjecture in the setup of unimodular random graphs and see what the proper notion of critical probability may be from the point of view of locality.
The conjecture has been proved for some special transitive graphs: Grimmett and Marstrand [18] proved that p c Z 2 × {−n, . . . , n} d−2 n→∞ −−−→ p c (Z d ). Benjamini, Nachmias and Peres [8] verified that the convergence holds if (G n ) is a sequence of d-regular graphs with large girth and Cheeger constants uniformly bounded away from 0. Martineau and Tassion [25] proved that the convergence holds if (G n ) is a sequence of Cayley graphs of Abelian groups converging to a Cayley graph G of an Abelian group, and p c (G n ) < 1 for all n. The inequality is known for any convergent sequence of transitive graphs; see [28,Section 14.2], and [11].
In Subsection 1.3 we define the generalized critical probabilities p c , p T ,p c ,p T , andp c for unimodular random graphs; somewhat simplistically saying, the first three will be quenched versions of the quantities mentioned above, while the last two will be annealed versions. In Section 2 we examine the relationship between these different generalizations. Our results are summarized in Table 1. The one sentence summary is that p c =p c always holds, but otherwise almost anything can happen, unless the random graph satisfies some very strong uniformity conditions; one that we call "uniformly good" suffices for most purposes.
In Section 3 we investigate the extension of Schramm's conjecture for unimodular random graphs: does p c (G n ) converge to p c (G) if G n → G in the local weak sense (i.e., the laws of the r-balls in G n converge weakly, for every r) and sup p c (G n ) < 1? First we note (Example 3.2) that locality holds for unimodular Galton-Watson trees with bounded degrees, but not in general; this shows that it is natural to restrict one's attention to bounded degree unimodular random graphs. In Subsection 3.2, we give conditions which imply the inequality lim inf p c (G n ) ≥ p c (G). In Subsection 3.3, we show by examples that there are sequences of unimodular random graphs such that G n → G but p c (G) > lim p c (G n ) or p c (G) < lim p c (G n ) < 1.
A corollary to our positive results is that if G is a transitive graph of subexponential volume growth, then there exists a sequence of invariant bi-Lipschitz spanning subgraphs G n such that p c (G n ) → 1. As we will explain in Section 4, this is a strengthening of the simple fact that groups of subexponential growth have cost 1, as defined in [19], studied further in [14,15]. We do not know if this strengthening holds for all groups of cost 1, which class includes, besides all amenable groups, direct products G × Z for any group G, and SL(d, Z) with d ≥ 3. A related question is whether every amenable transitive graph has an invariant random Hamiltonian path. This is the invariant infinite version of what is known as Lovász' conjecture, namely, that every finite transitive graph has a Hamiltonian path, even though he has not conjectured a positive answer. The best general results seem to be [5] and [27].

Notation
Graphs. We always consider locally finite and rooted graphs. The root is denoted by o. We denote by e − and e + the endpoints of the (directed) edge e. When a subgraph S is given (maybe implicitly) and it contains exactly one endpoint of e, then we denote that endpoint by e − . We write x ∼ y if x and y are adjacent vertices in G. We will use dist G (x, y) for the graph distance between the vertices x and y in the graph G. We denote by B G (o, r) the ball around o of radius r in G, i.e., the subgraph induced by the vertex set {x ∈ V (G) : dist G (o, x) ≤ r}. For any subset S of the vertices, let ∂ E S := {e ∈ E(G) : e − ∈ S, e + / ∈ S} be the edge boundary of S, let ∂ in V S := {x ∈ S : ∃y ∼ x, y / ∈ S} be the internal vertex boundary of S, and let ∂ out V S := {x / ∈ S : ∃y ∼ x, y ∈ S} be the outer vertex boundary of S.
Several of our examples will use percolation on Z 2 . The subgraph spanned by the box [−n, n] 2 will be denoted by Q n . We will also use the standard dual percolation on the dual lattice (Z + 1 2 ) 2 . Unimodular random graphs. Let G ⋆ be the space of isomorphism classes of locally finite labeled rooted graphs, and let G ⋆⋆ be the space of isomorphism classes of locally finite labeled graphs with an ordered pair of distinguished vertices, each equipped with the natural local topology: two (doubly) rooted graphs are "close" if they agree in "large" neighborhoods of the root(s). If (G, o) is a random rooted graph, then denote by µ G the distribution of it on G ⋆ , and let E G be the expectation with respect to µ G . We omit the index G from this notation if it is clear what the measure is.
, Definition 2.1). We say that a random rooted graph (G, o) is unimodular if it obeys the Mass Transport Principle: There are several other equivalent definitions; see [28,Definition 14.1]. Also, it is an open question if this class is strictly larger than the class of sophic measures: the closure of the set of finite graphs under local weak convergence.
An important class of unimodular graphs consists of Cayley graphs of finitely generated groups and of invariant random subgraphs of a Cayley graph: Remark 3.3). Let Γ be a Cayley graph of a finitely generated group and let o be a vertex of Γ. If G is a random subgraph of Γ that is invariant under the action of the group, then (G, o) is unimodular.
The class of unimodular probability measures is convex. A unimodular probability measure is called extremal if it cannot be written as a convex combination of other unimodular probability measures.
Percolation. For simplicity, we will consider only bond percolation processes on unimodular random graphs. For a fixed configuration ω of the random graph G let P ω p be the probability measure obtained by the Bernoulli(p) bond percolation on ω and let E ω p be the expectation with respect to P ω p . The percolation cluster (i.e., the connected component) of the root o will be C o .

Critical probabilities
The long studied critical probabilities p c = sup {p : P p (|C o | = ∞) = 0} first defined by Hammersley and p T = sup {p : E p (|C o |) < ∞} introduced by Temperley have natural generalizations to extremal unimodular random graphs. Let G be an extremal unimodular random graph. In this case the critical probability p c (ω) of a configuration of G is almost surely a constant and the same holds for p T (see [2], Section 6.). Hence one can define It may happen that although E ω p (|C o |) < ∞ for µ-almost every ω, the expectation of these quantities with respect to µ is infinite. This provides a second natural extension of p T to unimodular random graphs:p It follows from the definitions that p c ≥ p T ≥p T . It is known that p c = p T in the case of transitive graphs; see [26,1,4,11]. For unimodular random graphs (even with sub-exponential volume growth), the three critical probabilities can differ; we will present such graphs in Examples 2.7 and 2.9. Duminil-Copin and Tassion [11] introduced the following local quantity for transitive graphs: let G be a rooted graph, S be a finite subgraph containing the root, and define (1.1) They proved that transitive graphs satisfy p c =p c . How to generalize this definition to unimodular random graphs is not a priori clear. The simplest way to define a similar critical probability seems to be a quenched version: find a suitable S ω for almost every configuration ω. For a subgraph S ⊂ ω containing the root, denote by In the original definition (1.1) ofp c , there is no control on what the set S could be, which makes the definition rather ineffective. This becomes particularly problematic in the random graph case (1.3), where a bad neighborhood of o may force S ω to be huge and hard to find. However, it will follow from our Lemma 2.3 that, for transitive graphs, the existence of an S with φ p (S) < 1 is equivalent to the existence of a positive integer r with φ p (B(o, r)) < 1. This provides a second natural extension of the definition ofp c to the random case: we consider the ball of radius r in the random graph ω and we take the expectation of φ ω p (B ω (o, r)) with respect to µ. Then the following critical probability is another extension of the definition ofp c :

Operations preserving unimodularity
Some of our examples arise from Cayley graphs using operations from G ⋆ to G ⋆ . One of this operations is the edge replacement defined in [2], Example 9.8: we replace each edge of a unimodular graph G by a finite graph with two distinguished vertices corresponding to the endpoints of the edge, then we find the correct new distribution for the root that makes the measure unimodular.
If the finite graphs are random, each must have finite expected vertex size. In this section, we define further operations, called vertex replacement and contraction, and we prove that if the initial graph is a unimodular labeled graph with appropriate labels, then the resulting graph by such an operation is also unimodular.
Vertex replacement. Let (Γ, o) be a unimodular random labeled graph with distribution µ, where the labels are in the form (G x , ϕ x ), where G x is a finite graph and ϕ x is a map from which is an isomorphism-invariant Borel function on the subspace of G ⋆⋆ that consists of graphs with labels of the above form. We show that µ ′ obeys the Mass Transport Principle: Contraction. Let (Γ, o) be a unimodular random edge-labeled graph with distribution µ, where the labels of the edges are 0 or 1. We denote by G the random subgraph of Γ spanned by all the vertices and the edges with label 1. For a vertex x of Γ let C x be the connected component of x in G. We define the contracted graph H(Γ): in practice, this is what we get by identifying every vertices in the same component of G. More formally, first we choose (Γ, o, G) with respect to the distribution µ biased by 1 |Co| . The vertices of H(Γ) are the connected components of G and we join two vertices by an edge iff there is an edge in Γ which connects the two components. Let the root o ′ of H(Γ) be the connected component C o . Denote the law of (H(Γ), o ′ ) by µ ′ .
We claim that if µ is unimodular then µ ′ is also unimodular. Let f (ω, u, v) be a Borel function from G ⋆⋆ to [0, ∞] and letf which is an isomorphism-invariant Borel function on the subspace of G ⋆⋆ that consists of graphs with edges labeled by 0 or 1, such that the subgraph defined by the edges with label 1 consists of finite components. We show that µ ′ obeys the Mass Transport Principle:

Relationship of the critical probabilities
We will start by proving in Theorem 2.1 that all bounded degree unimodular graphs satisfy p c =p c . This will be useful in many of our later results.
In the transitive case, the quantity φ p (S) in the definition ofp c can be used to give a short proof (see [11]) of Menshikov's theorem [26]: if Γ is a transitive graph and p < p c (Γ), then there exist a ϕ(p) such that If a graph satisfies this exponential decay for each p < p c and has sub-exponential volume growth, then it is easy to see that p T = p c . In Lemma 2.3, we give a condition for unimodular random graphs that implies (2.1), and we prove in Corollary 2.5 that this condition implies p c = p T =p T if the graph has uniform sub-exponential volume growth. However, in Examples 2.7 and 2.9 we present unimodular random graphs with polynomial volume growth and p T < p c andp T < p T , respectively. This shows that Menshikov's theorem is not true in the generality of unimodular graphs.
The results of this section are summarized in the following table:

Positive results
Our first result is indispensable to the rest of the paper. The first part of the proof depends on new ideas, the second is a slight modification of the proof in [11] for our settings.
Theorem 2.1. If G is a bounded degree unimodular random rooted graph, then p c (G) =p c (G). Proof. We prove first thatp c ≤ p c . Fixing p <p c , we will show that p ≤ p c . We claim that there exists a constant c = c(p) < 1 such that we can find for almost every ω a set S ω that contains the root and satisfies φ ω Recall the definition of φ ω,x p (S ω,x ) from Remark 1.3. Unimodularity implies that almost every ω satisfies the following: for each x ∈ ω there is a set S ω,x containing x such that φ ω,x p (S ω,x ) ≤ c. Fix an S ω,x as above for every ω and x.
Fix ω and denote by T ω the following recursively defined tree: the vertices of the tree are finite sequences of vertices of ω. The root of the tree is (o). If (x 0 , x 1 , . . . , x k ) is a vertex of T ω then its children are the sequences (x 0 , x 1 , . . . , x k , x k+1 ) such that for all j = 1, . . . , k + 1, we have which are disjoint from each other and from the edges {x ′ j , x j }, as j = 1, . . . , k + 1. We say that the union of the above paths and edges is a good path through Let T ω (p) be the random subtree of T ω defined in a similar way but using the random subset of ω obtained by the Bernoulli(p) percolation instead of ω. It is easy to check that in fact T ω (p) ⊆ T ω . Denote by L n (p) the set of vertices of T ω (p) in the nth level. The event that the cluster of the origin in the p-percolation on ω is infinite coincides with the event that there exists an infinite path in the tree T ω (p) (that is, T ω (p) survives).
We claim that for almost every ω the expected number of vertices in L n (p) converges to 0 as n → ∞. More precisely, the expectation of the number of vertices in L n (p) decreases exponentially in n. In the first two inequalities we use the notation for the occurence of events on disjoint edge sets and we apply the BK inequality ( [16], Theorem 2.12). We denote the event It follows by induction that E ω |L n (p)| ≤ c n . Therefore, Note that for any fixed p the sequence (q r (p)) r≥1 converges decreasingly to q(p) as r → ∞, and q(p) > 0 for every p >p c by the definition ofp c .
Fix ω and let H ⊆ ω be any fixed finite subgraph that contains the root. We will use Lemma 1.4. of [11]: ←→ H c for every ω and H, with D being the almost sure bound on the degree of the graph G. The probabilities above depend only on the structure of V H, hence we can use the above inequality to estimate the derivative of the probability µ o ω,p ←→ B ω (o, r) c , as follows. Consider the following sets of finite rooted graphs: Integrate the above inequality on the interval p+pc 2 , p . Using the monotonicity of q(p) and C(p), we get This gives a positive lower bound that is uniform in r. Thus µ o ω,p ←→ ∞ > 0, and p ≥ p c .
One advantage of the definition ofp c for transitive graphs is that it enables one to check whether a certain p is underp c using a finite witness. This characteristic makes the next definition natural.
Definition 2.2. We say that a bounded degree unimodular random graph G is uniformly good if for any p < p c there exists a positive integer r(p) such that µ G ({ω : Uniformly good unimodular graphs satisfy the following exponential decay of φ p (B ω (o, r)) in r, which will imply the coincidence of p c andp c . Proof. If the constants c(p) and R(p) exist, then the sets To prove the other direction, assume that G is uniformly good, and fix p < p c . We can show as in the proof of Theorem 2.1 that there exists a constant c 0 < 1 and a positive integer r 0 such that for almost every ω and every x ∈ ω there exists a finite connected set S ω, Fix an ω and the sets S ω,x as above, a positive integer r and a finite set B ⊇ B ω (o, r). We define the trees T ω and T ω (p) as in the proof of Theorem 2.1. On every directed path in T ω from o to infinity there is a first vertex (x 0 , . . . , x k ) such that x k / ∈ B. Let π be the set of these vertices, which is a minimal set in T ω that separates o from infinity, and let π(p) := π ∩ T ω (p). A similar argument as in the first part of the proof of Theorem 2.1 shows that which is a minimal vertex set that separates the root from infinity. Let R := max{n : L n ∩ π = ∅} < ∞, thus π = π R . Note that each π n is the disjoint union of π n+1 \ L n+1 ⊆ π and π n \ π n+1 ⊆ L n . We estimate F (π, p) by summing over a larger set: the union of π R \ L R and That is, using the bound for the second term in the following estimation, we have A similar argument shows that F (π, p) ≤ F (π n , p) for any n ≤ R. If (x 0 , . . . , x k ) ∈ π, then dist ω (o, x k ) ≥ r, hence the distance between o and π in T ω is at least r r 0 , thus π n = L n for any n ≤ r r 0 . If we apply the above argument for F (π n , p) whith n ≤ r r 0 , then the first term disappear, and the inequality φ Denote byπ the set of the parents of the vertices in π. If for some e ∈ ∂ E B the event o To estimate this inequality, note that by the assumption that the graph is uniformly good. We have with some c < 1 if r is large enough.
Corollary 2.4. If G is a uniformly good unimodular graph, then p c ≤p c .
Proof. Let p < p c , and let c and R(p) be as in We will see in Remark 2.8 that, without the assumption of uniform goodness, the inequality p c ≤p c does not necessarily hold. Also, we will show in Example 2.10 that there are uniformly good graphs with p c <p c .
Corollary 2.5. If G is a uniformly good unimodular graph with uniform sub-exponential volume growth (i.e., for any c < 1 and ε > 0 there is an R such that µ (ω : |B ω (o, r)|c r < ε) = 1 for any r > R), then p c = p T =p T .
Proof. Let p < p c =p c and let c and R(p) be as in Lemma 2.3. Denote by D the maximum degree of G. Let R > R(p) such that µ {ω : |B ω (o, r)|c r/2 < 1} = 1 for any r > R and let It follows that p ≤p T , hencê p T ≥ p c . The other direction follows from the definition ofp T .
Subexponential volume growth will also appear in Corollary 4.1.

Counterexamples
We show in Examples 2.7 and 2.9 that there are unimodular random graphs of uniform subexponential (in fact, quadratic) volume growth, but p T < p c andp T < p T . Both constructions will use Bernoulli percolation on Z 2 as an ingredient; moreover, although we define the graph in the second example as a vertex replacement of Z 2 , it could be defined even as an invariant random subgraph of Z 2 . We further give examples of graphs withp c < p c andp c > p c ; see Examples 2.8 and 2.10, respectively. First we need a lemma that will be useful in our examples. Lemma 2.6. For any ε > 0 there is a probability p 1 < 1 such that for n large enough, the vertices (0, −n), (0, n), (−n, 0), (n, 0) are in the same cluster in Bernoulli(p 1 ) percolation on Q n with probability at least 1 − ε.
Proof. The occurrence of the events in the following two claims implies the occurrence of the event in the statement of the lemma, hence we will be done by a union bound.
Claim 1: For any p > 1/2 and n > n 0 (p, ε) large enough, in Bernoulli(p) percolation on Q n , with probability at least 1 − ε/2, there is a giant cluster with the following properties: it joins all the sides of Q n , while every other cluster in Q n has diameter at most n/5. This was proved in [3, Proposition 2.1].
Claim 2: There exists p 1 < 1 such that for all n and all p > p 1 , On the other hand, for any pair of dual vertices, Moreover, if the distance of x and y is larger than k, then this probability is of course 0, hence for each k there are at most k 2 relevant pairs (x, y). Therefore, for every n, where f (p) converges to 0 as p → 1.
Example 2.7. There is a unimodular graph with polynomial volume growth and p T < p c . In particular, the exponential decay of two-point connection probabilities fails for p ∈ (p T , p c ) on this graph.
Proof. Let T be the 3-regular infinite rooted tree with a distinguished end ξ and a Busemann function (see [29]) h : T → Z that gives the levels w.r.t. to ξ. More precisely, to define h, fix a root o ∈ T. For any vertex x, let (ξ, x) be the unique infinite simple path from x which is in the equivalence class ξ. Denote by o ∧ x the unique vertex in T such that (ξ, Let Λ ⊂ T be the subgraph spanned by the vertices x with h(x) ≥ 0. This tree Λ is called the canopy tree. Denote by L(n) := {x ∈ V (T) : h(x) = n} the n th vertex level and by L E (n) := {e ∈ E(T) : e − ∈ L(n), e + ∈ L(n + 1)} the n th edge level of T, or, for n ≥ 0, of Λ. If we choose the root o of Λ such that P(o ∈ L(n)) = 2 −n−1 , we get a unimodular random graph.
We define the graph G as an edge replacement (see [2], Example 9.8) of the canopy tree: each e ∈ L E (n) is replaced by (Q 2 n (e), (0, −2 n ), (0, 2 n )), where Q 2 n (e) is isomorphic to Q 2 n . It is clear that the volume of B G (o, r), for any root o and radius r, is at most Cr 2 , for some C < ∞. We will now show that p T (G) < p c (G) = 1.  to (0, n) is open. The tree Λ has one end, hence, for any p < 1, That is, p c (G) = 1.
An easy first moment computation (that we omit) shows that p T (Λ) = 1/ √ 2. Now let 0 < ε < 1 − 1/ √ 2. It follows from Lemma 2.6 that there exists p 1 < 1 and some large N such that P p 1 (e ∈ λ) ≥ 1 − ε for all e ∈ L E (n) with n ≥ N . Thus, for o ∈ L(N ), the cluster C o in λ, restricted to the levels n ≥ N , stochastically dominates Bernoulli(1 − ε) percolation on Λ. The latter has infinite expected size, hence the expected size of the cluster in ω of (0, −N ) ∈ Q N (e) for e ∈ L E (N ) is also infinite. That is, p T (G) ≤ p 1 < 1. Proof. It is easy to check that E(φ p (B(o, r))) equals 2p( √ 2p) r if r is even, and equals 3( √ 2p) r+1 /2 if r is odd. Thus it converges to 0 for p < 1/ √ 2, while remains above 1 for p < 1/ √ 2, which implies the claim. Example 2.9. There is a graph with polynomial volume growth andp T < p T .
Proof. Let X be a positive integer valued random variable such that P(X = k) = ck −5/2 for all k ≥ 1. Then EX < ∞ and E(X 2 ) = ∞. We define the graph G as a vertex replacement (see Subsection 1.4) of Z 2 with respect to the following labels as follow. Let {X n , X ′ n : n ∈ Z} be iid copies of X, and for each vertex (m, n) ∈ Z 2 , let G (m,n) be isomorphic to the subgraph of Z 2 spanned by the vertices in [0, 2X m ] × [0, 2X ′ n ], and for the edges going from (m, n) to North, East, South, and West, let the image of ϕ (m,n) be the corresponding midpoint of the box G (m,n) . We can also think of the resulting graph as an invariant random subgraph of Z 2 .
Denote by Y and Y ′ half the length of the sides of the box of o in G, i.e., the law of X 0 and Fix p > 1 2 and let ε > 0. Denote by M (Q n ) the largest cluster in percolation with parameter p in the box Q n , and let where θ(p) = P p (|C o (Z 2 )| = ∞), and ν is chosen as follows: by [16,Theorem 7.61], there is an N = N (p) and ν = ν(p) such that, for any n ≥ N , If o is in the unique infinite cluster of this percolation on Z 2 , then the diameter of C o (Q min{Y,Y ′ } ) is at least ν log min{Y, Y ′ }, hence for n large enough. It follows that there is an N ′ such that To show that p T > 1 2 let e be an edge in Z 2 , and let G e − and G e + be the subgraphs of which is finite if p < p 0 . It follows that for almost every configuration of (G, o) the cluster C o is finite almost surely if p < p 0 , hence p T ≥ p 0 . Example 2.10. There is a quasi-transitive graph withp c > p c .
Proof. Let H k,l be the following finite directed multigraph: the vertex set is {x 0 , x 1 , . . . , x k }, and we have l loops at x 0 , then one edge from x 0 to each x j , j = 1, . . . , k, and one from each x j back to x 0 . Let T k,l be the directed cover of H k,l based at x 0 . Consider two copies of T k,l and connect the roots of them by an edge to get the infinite quasi-transitive graph G k,l , which has vertices of degree 2 and k + l + 1. One can easily compute that to get a unimodular random graph one has to choose the root according to µ On the other hand, the critical probability of a directed cover of a finite graph is p c (T k,l ) = (br(T k,l )) −1 = (growth(T k,l )) −1 = (λ * (H k,l )) −1 , where λ * (H) is the largest positive eigenvalue of the directed adjacency matrix of H k,l ; see [24], Section 3.3 and [20]. One can thus compute that p c (G k,l ) = p c (T k,l ) = 2 l+ √ l 2 +4k . If we set, e.g., k = 3, l = 5, then we have p c (G 3,5 ) = 5 24 < 2

Locality of the critical probability
In this section we examine the question of Schramm's locality conjecture: does p c (G n ) converge to p c (G) if G n → G in the local weak sense? The original question in [8] was phrased for sequences of transitive graphs that converge to a transitive graph in the local sense and satisfy sup p c (G n ) < 1.
First we provide some simple examples of unimodular graphs where the conjecture holds. In Example 3.1, we note that if G n and G are infinite clusters of an independent percolation with appropriate parameters, then the convergence holds. In Example 3.2, we discuss unimodular Galton-Watson trees, and give sufficient and necessary conditions on the offspring distribution to satisfy locality of p c . Then we investigate the inequality lim inf p c (G n ) ≥ p c (G), which is known for transitive graphs; see [11] for a simple proof. We show in Corollary 3.4 and Proposition 3.5 that under certain restrictions on the graphs G and G n the inequality remains true for unimodular random graphs. Examples 3.6 and 3.7 provide graph sequences with lim p c (G n ) < p c (G). These indicate that unimodular graphs do not satisfy Schramm's conjecture in general and show that both of the conditions in Proposition 3.5 are necessary. We show in Example 3.8 a sequence with p c (G) < lim p c (G n ) < 1. In this example G and each G n satisfy the conditions of Corollaries 2.4 and 2.5, thus p c = p T =p T and alsop c (G) < limp c (G n ) < 1. This shows that none of the generalisations of the critical probabilities satisfies the extension of Schramm's conjecture for unimodular graphs in general.

Basic examples
We present now two natural classes of unimodular random graphs that satisfy Schramm's conjecture. The first example is very easy; the proof is left as an exercise. Our second class of examples, unimodular Galton-Watson trees, is less trivial. Let X be a nonnegative integer valued random variable, the offspring distribution of the tree, and let U GW (X) be the unimodular Galton-Watson tree measure on rooted trees: the probability that the root o has k children is for k ≥ 1, while the number of children of each descendant is according to X, independently of the other vertices. This measure is unimodular (see [2], Example 1.1), and if EX > 1, then P(|U GW (X)| = ∞) > 0, thus we can consider the measure U GW ∞ (X) which is U GW (X) conditioned on the event {|U GW (X)| = ∞}. The measure U GW ∞ is also unimodular, being an ergodic component of a unimodular measure.
Example 3.2. Let U GW ∞ (X) be the unimodular Galton-Watson tree with offspring distribution X, conditioned to be infinite. If X n and X are non-negative integer valued random variables with EX n > 1 and EX > 1 or P(X = 1) = 1, then (1) U GW ∞ (X n ) → U GW ∞ (X) in the local weak sense iff X n → X in distribution; (2) p c (U GW ∞ (X n )) → p c (U GW ∞ (X)) iff EX n → EX.
Before the proof, note that this example shows that p c is a continuous function of U GW ∞ (X) when the trees have a uniform bound on their degrees (by the Dominated Convergence Theorem), but not otherwise: if X n → X in distribution, with EX n > 1 and EX > 1, but EX n EX, then the critical probabilities p c (U GW ∞ (X n )) do not converge to where they should. Nevertheless, Fatou's lemma implies that the inequality lim sup p c (U GW ∞ (X n )) ≤ p c (U GW ∞ (X)) does hold without further assumptions. That is, if the trees do not satisfy the locality of p c , then they also fail to satisfy the lower semicontinuity discussed in the next subsection, proved to hold in many cases, including transitive graphs. This suggests that a uniform bound on the degrees is a natural condition when we investigate the locality of p c for unimodular graphs.
For part (1), for any nonnegative integer random variable X, let p k (X) := P(X = k), let f X (t) := ∞ k=0 p k (X)t k be the probability generating function of X, and let q = q(X) := P(|GW (X)| < ∞), which is the smallest non-negative number that satisfies f X (q) = q.
Assume that X n → X in distribution, first with E(X) > 1. From X n → X it follows easily that U GW (X n ) → U GW (X), while, from the uniform convergence of the convex functions f Xn to the strictly convex function f X on [0, 1], we also get q n = q(X n ) → q(X) < 1. Thus U GW ∞ (X n ) → U GW ∞ (X). Now assume that P(X = 1) = 1 and P(X n = 1) → 1 with E(X n ) > 1. Using Bayes' rule and (3.1), We claim that P U GW∞(Xn) (deg o = 2) → 1. If q n converges to some q ∞ < 1, then plugging P(X n = 1) → 1 into (3.2) yields the claim immediately. If q n → 1, then, simplifying the numerator and the denominator of (3.2) by 1 − q n , it becomes Finally, if q n does not converge, we can still apply one of these two arguments to any convergent subsequence, and obtain the claim. Therefore, in the local weak limit, the root has degree 2 almost surely. By unimodularity, this limit must be Z. This is also U GW ∞ (X), thus we have For the other direction of part (1), suppose that there are X n and X such that U GW ∞ (X n ) → U GW ∞ (X), but X n X. The set {X n } of probability distributions must be tight: otherwise, a uniform random neighbour of o in U GW ∞ (X n ), whose offspring distribution stochastically dominates X because of the conditioning on |U GW (X n )| = ∞ , would have arbitrarily large degrees with a uniform positive probability, and thus U GW ∞ (X n ) could not converge to the locally finite graph U GW ∞ (X). It follows from this tightness that there is a subsequence {X k(n) } that converges in distribution to a random variable Y = X.
If we have P(Y = 1) = 1, then the first direction of part (1) implies that U GW ∞ (X k(n) ) → U GW ∞ (Y ) = Z. But we also have U GW ∞ (X k(n) ) → U GW ∞ (X), and it is obvious that U GW ∞ (X) = Z implies that P(X = 1) = 1. That is, X n would in fact converge in distribution to X, a contradiction.
If EY = 1, but P(Y = 1) = 1, then the generating function f Y (t) is strictly convex, hence q(X k(n) ) → q(Y ) = 1. A computation similar to (3.2) and (3.3) gives that the degree distribution of o in U GW ∞ (X k(n) ) converges to that of Y + 1. This must be the degree distribution of o in the local limit U GW ∞ (X). Since P(Y +1 = 2) = 1, we must be in the case EX > 1. However, then we would have p c (U GW ∞ (X)) = 1/EX < 1, while E(deg o) = E(Y + 1) = 2 implies that U GW ∞ (X) is a tree with at most two ends (see [2], Theorem 6.2) hence p c = 1, again a contradiction.
The final case is that EY > 1, for which we can again use the first direction of part (1), saying that U GW ∞ (X k(n) ) → U GW ∞ (Y ). If we prove that the distribution of U GW ∞ (X) determines X, then we must have X = Y , and we are done, as before.
This invertibility follows from the construction in [24], Theorem 5.28, as follows. Let T * := GW (X * ), where the probability generating function of the positive integer valued random variable X * is f * (t) := f X (q+(1−q)t) 1−q , and letT := GW (X), wheref (t) = fX(t) := f (qs) q , and hencē T is almost surely finite. The law of GW (X) conditioned to be infinite equals the law of the tree T constructed as follows: consider the rooted tree T * , and attach to each vertex of T * an appropriate number of independent copies ofT . We get the law of U GW ∞ (X) if we attach to the root an appropriate random number of independent copies of T andT . It follows that the law of U GW ∞ (X) determines (f * ,f ). We get the function f from (f * ,f ) by the transform There is a unique q for which the resulting f (s) has the same second derivative from the left and from the right at s = q. Since f (s) has to be analytic, we see that (f * ,f ) uniquely determines f and hence X.

Lower semicontinuity
The quantity φ p (S) can be used to give a short proof that p c (G) is lower semicontinuous in the local topology of transitive graphs: that is, lim inf p c (G n ) ≥ p c (G) holds; see [11,Section 1.2]. One can show in a similar way that this inequality is also true forp c and unimodular graphs. Proposition 3.3. Let G n and G be unimodular random graphs with uniformly bounded degrees.
If G n converges to G then lim inf n→∞pc (G n ) ≥p c (G).
Proof. Let p <p c (G) and let r be such that E G φ ω p (B ω (o, r)) < 1 − ε with some ε > 0. Let n be large enough to satisfy where D is a uniform bound on the degrees of G n and G and H r is the set of possible rneighbourhoods of the root in graphs with maximum degree D. Any H ∈ H r+1 satisfies φ H p (B ω (o, r)) ≤ D r+1 . We obtain It follows thatp c (G n ) ≥ p thus lim infp c (G n ) ≥p c (G).
Corollary 3.4. Let G n and G be unimodular random graphs with uniformly bounded degrees. If G and G n are uniformly good, then lim inf n→∞ p c (G n ) ≥ p c (G).
Proof. It follows from the conditions on G and G n by Corollary 2.5 that p c (G) =p c (G) and p c (G n ) =p c (G n ). This combined with the statement of Proposition 3.3 gives the inequality.
Proposition 3.5. Let G be a uniformly good unimodular random graph. Furthermore, let G n be unimodular random graphs converging to G in the local weak sense, in a uniformly sparse way: there is a positive integer k such that for each n there is a coupling ν n of µ G and µ Gn such that G ⊆ G n and there is a sequence of positive integers r n → ∞ that satisfies Proof. For the sake of simplicity, we prove the statement for k = 1. It can be proved for general k in a similar way. Let p < p c (G) fixed and let c and R(p) be as in Lemma 2.3. Let n be sufficiently large to satisfy r n /2 > R(p) and c rn/2 < 1 3 . Fix a pair (ω, ω n ) that satisfies the sparseness condition for r n . Then, in the smaller ball B ωn (o, r n /2), there is at most one edge {x, y} ∈ ω n \ω. If this edge exists, let B n := B ωn (o, r n /2)∪B ωn (x, r n /2)∪B ωn (y, r n /2); otherwise, just let B n := B ωn (o, r n /2). Note that B n ⊂ B ωn (o, r n ). Similarly, let B := B ω (o, r n /2) ∪ B ω (x, r n /2) ∪ B ω (y, r n /2), omitting those terms in the union that do not exists in ω. (Note that it may happen that x or y does not exist in ω, but not both, since B ωn (o, r n /2) is connected). We claim that we have φ ωn p (B n ) < 1. if x or y does not exist in ω, all its appearances in the above formulas involving ω can be replaced by the other vertex, and the inequalities remain true. It follows that p <p c (G n ) = p c (G n ).

Counterexamples
Our first example will show that even if we keep the condition of uniformly sparse convergence of G n to G of Proposition 3.5, without G being uniformly good, the conclusion may not hold. Next, Example 3.7 will show that keeping the limit uniformly good but removing the condition of uniform sparseness will make the conclusion false. Finally, Example 3.8 will show that the inequality of Proposition 3.5 may be strict even when the limit is just Z 2 .
Example 3.6. There exists a sequence (G n ) of invariant random subgraphs of a Cayley graph, converging to an invariant subgraph G in a uniformly sparse way, such that lim p c (G n ) < p c (lim G n ).
Proof. The first step is to construct an invariant percolation on a Cayley graph of the lamplighter group all whose clusters are isomorphic to the canopy tree Λ. In more detail: Consider the generators {Rs, R, sL, L} of the lampligher group Z 2 ≀ Z = ⊕ Z Z 2 ⋊ Z, where R := (0, 1), L := (0, −1), and s := (e 0 , 0) ∈ Z 2 ≀ Z with e 0 ∈ {0, 1} Z , (e 0 ) j = δ 0,j . It is well-known (see, e.g., [29]) that the Cayley graph with respect to these generators is the Diestel-Leader graph DL (2,2). This graph can be defined using two trees T 1 and T 2 which both are 3-regular infinite rooted trees with a distinguished end and Busemann functions h i : We call the other two neighbours the children of x. Now consider the following percolation on T 1 : for each vertex x we delete the edge connecting x to one of its two children, independently with equal probabilities. We get a random subgraph of T 1 consisting of infinite simple paths. We then delete the edges in the graph DL(2,2) whose first coordinate is a deleted edge in T 1 . The resulting random subgraph F ⊂ DL(2,2) is invariant under the action of the lamplighter group and it consists of infinitely many components which are all isomorphic to the canopy tree Λ ⊂ T.
The probability that the root is in the n th level of its component in F is clearly 2 −n−1 . The canopy tree with a random root chosen according to this distribution is a unimodular random graph, as it also must be the case by Proposition 1.2. The significance of the canopy tree for this construction (as in Example 2.7) will be that it has one end, thus p c (Λ) = 1, while one can easily compute that p T (Λ) = 1/ √ 2. Now let G be the free product of Z 2 := Z/2Z and the lamplighter group Z 2 ≀Z. Let Γ be the left Cayley graph of G with respect to the generators {a, Rs, R, sL, L} where a is the generator of the free factor Z 2 . Let β : We now define G to be the following random spanning subgraph of Γ: let e be in E(G) iff β(e − ) and β(e + ) are connected by an edge in F. The distribution of G is invariant under the action of G and each component of G is a canopy tree, hence p c (G) = 1.
We define a sequence (G n ) of random subgraphs of Γ converging to G. We choose an element b ∈ {0, 1, . . . n − 1} uniformly at random. For each vertex in L T 1 (b+ kn), k ∈ Z we choose one of its descendants in L T 1 (b + (k + 1)n) uniformly at random and for each vertex in L T 2 (−b + kn), k ∈ Z we choose one of its descendants in L T 2 (−b + (k + 1)n) uniformly at random. Let S n be the set of edges e ∈ E(Γ) such that e is labelled by the generator a and both cordinates of β(e − ) = β(e + ) are chosen vertices in the above procedure. Let G n := G ∪ S n .
We show that p c (G n ) ≤ 1 √ 2 for all n. Let p > 1 √ 2 = p T (Λ), let n be a positive integer and consider Bernoulli(p) percolation on G n . Denote by T (v) the component of the vertex v in G and by C v the component of the vertex v in the percolation on G n . Let s(v) := min{l : L T (v) (l) ∩ S n = ∅}. We define a branching process depending on the percolation on G n . For each vertex v of Γ let Example 3.7. There exists a sequence (G n ) of invariant random subgraphs of a Cayley graph such that lim p c (G n ) < p c (lim G n ) and lim G n is uniformly good.
Proof. Let Γ be a Cayley graph of a finitely generated group such that there exists a random subgraphḠ which satisfies the following: the distribution ofḠ is invariant under the action of the group, it consists of infinitely many infinite components and each component has critical percolation probabilityp < 1. Let G ′ be an invariant random connected subgraph of Γ such that p c (G ′ ) >p. For example, if Γ is amenable, then one can choose G ′ to be an invariant spanning tree of Γ, which always exists and has at most two ends; see [7], Theorem 5.3. Let ε n → 0 be a sequence of positive numbers and let G n be the following random subgraph of Γ: we remove each component ofḠ with probability 1 − ε n and keep it with probability ε n independently for each component. Let G n be the union of G ′ and the remaining components ofḠ. It follows from Proposition 1.2 that G n is unimodular. The sequence (G n ) converges to G ′ , but p c (G n ) ≤p < p c (G ′ ) for each n.
We get a similar example if we set Γ := Z 5 ,Ḡ := y∈Z 2 {y} × Z 3 and G ′ := x∈Z 3 Z 2 × {x}. In this example G ′ is not connected, but each G n is connected almost surely, and p c (G n ) ≤ p c (Z 3 ) < p c (lim G n ) = p c (Z 2 ) < 1 for each n. Proof. We define G n as a vertex and edge replacement (see Subsection 1.4 and [2], Example 9.8) of Z 2 where we replace each vertex x by the graph Q x isomorphic to Q n and we replace each edge by a path of length two that joins the middle points of the neighbouring sides of the boxes corresponding to the endpoints of the edge. The graphs G n can be considered as deterministic subgraphs of Z 2 with a randomly chosen root. The sequence G n converges to Z 2 .
We show that 1 2 < lim p c (G n ) < 1. Denote by G n (p) the subgraph obtained by the Bernoulli(p) percolation on G n , and let H n (p) be the following percolation on Z 2 : let an edge {x, y} open, iff both edges are open in the path that joins the boxes Q x and Q y in G n . The existence of an infinite cluster in G n (p) implies the existence of an infinite cluster in H n (p). The law of H n equals the law of the Bernoulli(p 2 ) percolation on Z 2 , hence p c (G n ) ≥ 1 √ 2 for each n. To show that lim sup p c (G n ) < 1, we define the percolationH n (p) on Z 2 . Denote by A x (n) the event that the vertices (0, −n), (0, n), (−n, 0), (n, 0) are in the same cluster in Bernoulli(p) percolation on the box Q x ⊂ G n . Let an edge {x, y} ∈H n (p), iff {x, y} ∈ H n (p), and both of the events A x (n) and A y (n) occurs. The existence of an infinite cluster inH n (p) implies the existence of an infinite cluster in G n (p). Let ε > 0 satisfy (1 − ε) 4 > 1 2 . Lemma 2.6 implies, that we can find constants 1 − ε < p 1 < 1 and N such that for any p > p 1 , n ≥ N and for any vertex x ∈ V (Z 2 ) P p (A x (n)) ≥ 1 − ε, thus P(e ∈H n (p)) ≥ p 2 1 (1 − ε) 2 ≥ (1 − ε) 4 for any edge e ∈ E(Z 2 ). The events {e 1 ∈H n } and {e 2 ∈H n } are positively correlated, henceH n (p) stochastically dominates Bernoulli((1 − ε) 4 ) percolation. It follows that lim sup p c (G n ) ≤ p 1 < 1.

On transitive graphs of cost 1
The cost of a group G is defined as half of the infimum of the expected degrees of its invariant spanning graphs. (See Subsection 1.1 for references.) The cost of a transitive graph Γ may be defined similarly, over G-invariant random spanning subgraphs, where G ≤ Aut(Γ) is a vertextransitive subgroup of graph-automorphisms that will usually be fixed implicitly. It is not known in general that if we first fix a Cayley graph Γ of G, then the G-cost of Γ is always as small as the cost of G. Nevertheless, cost 1 can be achieved inside Cayley graphs of amenable groups: as proved in [7,Theorem 5.3], an infinite transitive graph Γ has an invariant spanning tree T with at most two ends (hence with expected degree 2 and p c (T ) = 1) iff it is amenable.
The main point of our next proposition is that, under the stronger condition of subexponential decay, we get a spanning tree with the stronger propertyp c (T ) = 1, and using this, we can achieve approximately 1-dimensional percolation behaviour p c (G k ) → 1 via spanning subgraphs that have the same large-scale geometry as Γ. The bi-Lipschitz condition is also natural from the point of view of Elek's combinatorial cost for sequences of finite graphs [13].
Proposition 4.1. If G is a transitive amenable graph, then there is a sequence of invariant random subgraphs G k which satisfies the following: each G k is a bi-Lipschitz (in particular, connected) spanning subgraph of G, the girth of G k tends to infinity and G k locally converges to an invariant random spanning tree T with at most two ends.
If G is a unimodular transitive graph with sub-exponential volume growth then lim p c (G k ) = 1.
Proof. We construct T as in [7], Theorem 5.3: let F n be a sequence of Følner sets such that ∞ n=1 |∂ E Fn| |Fn| < 1. For each n and x ∈ V (G) choose a random g x,n ∈ Aut(G) that takes o to x, and a random bit Z x,n that equals 1 with probability 1 |Fn| . Choose all g x,n and Z x,n independently. Let ω n := E(G) \ x∈V (G),Zx,n=1 ∂ E (g x,n F n ); i.e., we remove all edges in the boundaries of the translates of F n with Z x,n = 1. Letω n = k≥n ω k . Eachω n has only finite components.
To construct T and G k , choose uniform labels L e in [0,1] independently for each e ∈ E(G). For each finite component ofω 1 take the minimal spanning tree of the component with respect to the labels. Denote by T 1 the union of these trees. Let T 2 be the union of T 1 and the edges in ω 2 \ω 1 with minimal labels such that the components of T 2 are spanning trees of the components ofω 2 . Continue inductively, and let T := T n . This is an invariant random spanning tree, which has at most 2 ends (otherwise it would have infinitely many ends, which is impossible, since G is amenable).
To construct G k we define a color for each edge. Let all edges in T be green. In each component ofω 1 do the following: consider the edge with the smallest label which has no color. If there is a path of length at most k between its endpoints consisting of green edges, then color it red, otherwise color it green. Continue inductively for the edges in the component. This procedure defines a color for each edge ofω 1 . If all edges inω n have a color, then continue coloring the edges ofω n+1 \ω n in the same way. Let G k be the union of the green edges. It follows from the construction that G k is invariant, its girth is at least k + 2 and for each edge of G there is a path in G k between its endpoints with length at most k. The sequence G k converges to T .
If G has sub-exponential volume growth, then for each p < 1 there is a positive integer r = r(p) such that φ F p (F ∩ B G (o, r)) ≤ |∂ E B G (o, r)|p r < 1 , for any subgraph F that is a forest in B G (o, r). It follows thatp c (T ) = 1. Moreover, for any p < 1 there is a positive integer k 0 such that G k is almost surely a forest in B G (o, r(p)) for any k ≥ k 0 . The random subsets S ω := G k ∩ B G (o, r(p)) attestp c (G k ) ≥ p. Since p c (G k ) =p c (G k ) holds by Theorem 2.1, we get p c (G k ) → 1.
Our proposition may be thought of as a strengthening of having cost 1: If Γ is a Cayley graph of G, and there exists a sequence of G-invariant spanning subgraphs G k ⊂ Γ with p c (G k ) → 1, then the cost of Γ, hence of G, is 1.
Proof. Take ε k → 0 such that p c (G k ) > 1 − ε k . Then, all clusters of Bernoulli(1 − ε k ) percolation on G k are finite almost surely. Let the set of closed edges be denoted by η k ⊂ G k ⊂ Γ, an invariant percolation itself. In each finite cluster, take a uniform random spanning tree, a subtree of G k . The union of all these finite spanning trees and η k will be ω k . One the one hand, it is clear that ω k is a connected spanning subgraph of G k , hence of Γ. On the other hand, the expected degree of o in ω k is at most E deg η k (o) + 2 ≤ dε k + 2, where deg Γ (o) = d. As k → ∞, we obtain that the cost of Γ is 1.
As we mentioned above, having a G-invariant spanning graph T with p c (T ) = 1 implies that G is amenable, because it is not hard to construct an invariant mean [7,Theorem 5.3]. However, the sequence p c (G k ) → 1 does not imply amenability: Example 4.3. T 3 × Z has a sequence of bi-Lipschitz subgraphs G k with p c (G k ) → 1.
Proof. One can partition the edges of T 3 into 3 disjoint perfect matchings M 1 , M 2 and M 3 in an invariant way. (See, for instance, [22], around Proposition 2.4.) Then, consider the following subgraphs G k ⊆ T 3 ×Z: we keep all the edges in the subgraphs {v}×Z and the edges {e}×{3jk+ik} where e ∈ M i , j ∈ Z. Each G k is clearly bi-Lipschitz equivalent to T 3 × Z. On the other hand, we have p c (G k ) → 1: either from Proposition 3.5, or more directly, by observing that the universal cover T k of G k can be obtained from T 3 by replacing "two thirds" of the edges by a path of length k, for which it is easy to see that p c (T k ) → 1, while p c (T k ) ≤ p c (G k ) holds by [24,Theorem 6.47].
We do not know if the converse of Lemma 4.2 holds: Question 4.4. Does there exist, for any Cayley graph Γ of any group G of cost 1, a sequence of G-invariant bi-Lipschitz spanning subgraphs G k ⊂ Γ with p c (G k ) → 1?
For amenable Cayley graphs Γ, a first step of independent interest could be a positive answer to the following question, mentioned in Subsection 1.1: