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I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability and Statistics, DOI 10.1007/978-1-4419-9675-6_23, C Springer Science+Business Media, LLC 2011 701 The Annals of Probability 1999, Vol. 27, No.4, 1809-1836 INDISTINGUISHABILITY OF PERCOLATION CLUSTERS By RUSSELL LYONS 1 AND ODED SCHRAMM 2 Indiana University and Weizmann Institute of Science We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for Pu' 1. Introduction. Grimmett and Newman (1990) showed that if T is a regular tree of sufficiently high degree, then there are p E CO, 1) such that Bernoulli(p) percolation on T X 7L has infinitely many infinite components a.s. Benjamini and Schramm (1996) conjectured that the same is true for any Cayley graph of any finitely generated nonamenable group. (A finitely gener- ated group r is nonamenable iff its Cayley graph satisfies infKlaKl!lKI > 0, where K runs over the finite nonempty vertex sets. See Section 2 for all other definitions.) This conjecture has been verified for planar Cayley graphs of high genus by Lalley (1998) and for all planar lattices in the hyperbolic plane by Benjamini and Schramm (1998). The present paper is concerned with the percolation phase where there are multiple infinite clusters. We show that under quite general assumptions, the infinite clusters are indistinguishable from each other by any invariantly defined property. Let G be a Cayley graph of a finitely generated group (more generally, G can be a transitive unimodular graph). THEOREM 1.1. Consider Bernoulli bond percolation on G with some sur- vival parameter p E (0, 1), and let SIt' be a Borel measurable set of subgraphs of G. Assume that SIt' is invariant under the automorphism group of G. Then either a.s. all infinite percolation components are in SIt', or a.s. they are all outside of SIt'. For example, SIt' might be the collection of all transient subgraphs of G, or the collection of all subgraphs that have a given asymptotic rate of growth or the collection of all subgraphs that have no vertex of degree 5. Received December 1998. 1 Supported in part by Varon Visiting Professorship at the Weizmann Institute of Science and NSF Grant DMS-98-02663. 2 Supported in part by the Sam and Ayala Zacks Professorial Chair. AMS 1991 subject classifications. Primary 82B43, 60B99; secondary 60K35, 60D05. Key words and phrases. Finite energy, Cayley graph, group, Kazhdan, wreath product, uniqueness, connectivity, transitive, nonamenable. 1809

1. Introduction. E CO, - COnnecting REpositoriesLet G be a Cayley graph of a finitely generated group (more generally, G can be a transitive unimodular graph). THEOREM 1.1. Consider

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Page 1: 1. Introduction. E CO, - COnnecting REpositoriesLet G be a Cayley graph of a finitely generated group (more generally, G can be a transitive unimodular graph). THEOREM 1.1. Consider

I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probabilityand Statistics, DOI 10.1007/978-1-4419-9675-6_23, C© Springer Science+Business Media, LLC 2011

701

The Annals of Probability 1999, Vol. 27, No.4, 1809-1836

INDISTINGUISHABILITY OF PERCOLATION CLUSTERS

By RUSSELL LYONS1 AND ODED SCHRAMM2

Indiana University and Weizmann Institute of Science

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for Pu'

1. Introduction. Grimmett and Newman (1990) showed that if T is a regular tree of sufficiently high degree, then there are p E CO, 1) such that Bernoulli(p) percolation on T X 7L has infinitely many infinite components a.s. Benjamini and Schramm (1996) conjectured that the same is true for any Cayley graph of any finitely generated nonamenable group. (A finitely gener­ated group r is nonamenable iff its Cayley graph satisfies infKlaKl!lKI > 0, where K runs over the finite nonempty vertex sets. See Section 2 for all other definitions.) This conjecture has been verified for planar Cayley graphs of high genus by Lalley (1998) and for all planar lattices in the hyperbolic plane by Benjamini and Schramm (1998).

The present paper is concerned with the percolation phase where there are multiple infinite clusters. We show that under quite general assumptions, the infinite clusters are indistinguishable from each other by any invariantly defined property.

Let G be a Cayley graph of a finitely generated group (more generally, G can be a transitive unimodular graph).

THEOREM 1.1. Consider Bernoulli bond percolation on G with some sur­vival parameter p E (0, 1), and let SIt' be a Borel measurable set of subgraphs of G. Assume that SIt' is invariant under the automorphism group of G. Then either a.s. all infinite percolation components are in SIt', or a.s. they are all outside of SIt'.

For example, SIt' might be the collection of all transient subgraphs of G, or the collection of all subgraphs that have a given asymptotic rate of growth or the collection of all subgraphs that have no vertex of degree 5.

Received December 1998. 1 Supported in part by Varon Visiting Professorship at the Weizmann Institute of Science and

NSF Grant DMS-98-02663. 2 Supported in part by the Sam and Ayala Zacks Professorial Chair. AMS 1991 subject classifications. Primary 82B43, 60B99; secondary 60K35, 60D05. Key words and phrases. Finite energy, Cayley graph, group, Kazhdan, wreath product,

uniqueness, connectivity, transitive, nonamenable.

1809

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1810 R. LYONS AND O. SCHRAMM

If .9f is the collection of all transient subgraphs of G, then this shows that almost surely, either all infinite clusters of ware transient (meaning that simple random walk on them is transient), or all clusters are recurrent. In fact, as shown in Proposition 3.11, if G is nonamenable, then a.s. all infinite clusters are transient if Bernoulli percolation produces more than one infinite component. In Benjamini, Lyons and Schramm (1999), it is shown that the same is true if Bernoulli percolation produces a single infinite component.

Theorem 1.1 is a particular case of Theorem 3.3 below, which applies also to some non-Bernoulli percolation processes and to more general .9f.

A collection of subgraphs .9f in G is increasing if whenever H E.9f and He H' c G, we have H' E.9f. Haggstrom and Peres (1999) have proved that for increasing .9f, Theorem 1.1 holds, except possibly for a single value of p E (0,1). They have also proved the following theorem.

THEOREM 1.2 (Uniqueness monotonicity). Let PI < P2 and Pi (i = 1,2) be the corresponding Bernoulli(p) bond percolation processes on G. If there is a unique infinite cluster Pea.s., then there is a unique infinite cluster P2-a.s. Furthermore, in the standard coupling of Bernoulli percolation processes, if there exists an infinite cluster PI-a.s., then a.s. every infinite P2-cluster contains an infinite PI-cluster.

Here, we refer to the standard coupling of Bernoulli(p) percolation for all p, where each edge e E E is assigned an independent uniform [0,1] random variable U(e) and the edges where U(e) ~ p are retained for Bernoulli(p) percolation.

The first part of this theorem partially answers a question of Benjamini and Schramm (1996); the full answer, removing the assumption of unimodu­larity, has been provided by Schonmann (1999a). It follows from Theorem 1.2 that the set of p such that Bernoulli(p) bond percolation on G has more than one infinite cluster a.s. is an interval, called the non uniqueness phase. It is well known that in the nonuniqueness phase, the number of infinite clusters is a.s. infinite [see Newman and Schulman (1981)]. The interval of p such that Bernoulli(p) percolation on G produces a single infinite cluster a.s. is called the uniqueness phase. The infimum of the p in the uniqueness phase is denoted Pu = Pu(G).

To illustrate the usefulness of Theorem 1.1, we now show that it implies Theorem 1.2.

PROOF OF THEOREM 1.2. Suppose that there exists an infinite cluster Pea.s. Let w be the open subgraph of the P 2 process and let YJ be an independent Bernoulli(pdp2) percolation process. Thus, w n YJ has the law of PI and, in fact, (w n YJ, w) has the same law as the standard coupling ofPl

and P 2 • By assumption, w n YJ has an infinite cluster a.s. Thus, for some cluster C of w, we have C n YJ is infinite with positive probability, hence, by Kolmogorov's 0-1 law, with probability 1. By Theorem 1.1, this holds for every infinite cluster C of w. 0

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1811

Our main application of Theorem 1.1 is to prove in Section 4 a sufficient condition for having a unique infinite cluster. Define 7(X, y) to be the probability that x and yare in the same cluster. The process is said to exhibit connectivity decay if

inf{7(x,y): x,yEV} =0.

When a process does not have connectivity decay, it is said to exhibit long-range order. We show that for Bernoulli percolation (and some other percolation processes), nonuniqueness of the infinite cluster is equivalent to connectivity decay. One direction is easy; namely, if a.s. there is a unique infinite percolation cluster, then there is long-range order. This is an immedi­ate consequence of Harris's inequality (that increasing events are positively correlated). Although the other direction seems intuitively obvious as well, its proof is surprisingly difficult (and it is still open for nonunimodular transitive graphs).

We note that connectivity decay in the nonuniqueness phase easily implies Theorem 3.2 of Schonmann (1999a) for the case of unimodular transitive graphs. (Our result also deals with percolation at Pu and extends Schon­mann's theorem to more general percolation processes, but his result extends to Bernoulli percolation on nonunimodular transitive graphs.)

REMARK 1.3. Fix a base vertex 0 E V in G, and consider Bernoulli perco­lation in the nonuniqueness phase. Although inf{7(o, v): v E V} = 0, it may happen that

lim sup{ 7(0, v): dist(o, v) > d} > O. d~oo

For example, take the Cayley graph ofthe free product 7L 2 * 7L2 with the usual generating set, that is, the Cayley graph corresponding to the presentation (a, b, c lab = ba, c = c- 1 ). We have Pu = 1, because the removal of any edge of the Cayley graph corresponding to the generator c disconnects the graph. On the other hand, for p > p/7L 2 ), we have that all pairs of vertices in the same 7L 2 fiber (i.e., pairs g, h such that g-lh is in the group generated by a and b) are in the same cluster with probability bounded away from o.

Benjamini and Schramm (1996) asked for which Cayley graphs G one has Pu(G) < 1. It is easy to show that Pu(G) = 1 when G has more than one end. Babson and Benjamini (1999) showed that Pu(G) < 1 when G is a finitely presented group with one end. Haggstrom, Peres and Schonmann (1999) have shown that Pu(G) < 1 if G is a Cartesian product of infinite transitive graphs. Here, we show that this is also true of certain other classes of groups with one end, namely infinite Kazhdan groups (Corollary 6.6) and wreath products (Corollary 6.8). Presumably, all quasi-transitive (infinite) graphs with one end have Pu < 1.

As another consequence of indistinguishability, we prove in Theorem 6.12 the inequalities Pu(G X H') ~ Pu(G X H) and, in particular, Pu(G) ~ Pu(G X

H), for Cayley graphs (or, more generally, unimodular transitive graphs) G, H', H such that G is infinite and H' c H.

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1812 R. LYONS AND O. SCHRAMM

Several other uses of cluster indistinguishability appear in Benjamini, Lyons, and Schramm (1999).

Crucial techniques for our proofs are the mass-transport principle and stationarity of delayed simple random walk, both explained below. These techniques were introduced in the study of percolation by Haggstrom (1997). In Section 5, we prove an ergodicity property for delayed random walk.

2. Background. We begin with some graph-theoretic terminology. Let G = (V(G), E(G)) be a graph with vertex set V(G) and (symmetric) edge set E(G) ~ V(G) X V(G). When there is an edge in G joining vertices u, v, we say that u and v are adjacent and write u '" v. We always assume that the number of vertices adjacent to any given vertex is finite. The degree deg v =

degG v of a vertex v E V(G) is the number of edges incident with it. A tree is a connected graph with no cycles. A forest is a graph whose connected components are trees. The distance between two vertices u, v E V(G) is denoted by dist(v, u) = distG(v, u) and is the least number of edges of a path in G connecting v and u. Given a set of vertices K c V(G), we let aK denote its edge boundary, that is, the set of edges in E(G) having one vertex in K and one outside of K. A transitive graph G is amenable if inflaKl!lKI = 0, where K runs over all finite nonempty vertex sets K c V(G).

An infinite set of vertices Vo c V(G) is end convergent if for every finite K c V(G), there is a component of G - K that contains all but finitely many vertices of Yo. Two end-convergent sets Yo, V l are equivalent if Vo U V l is end convergent. An end of G is an equivalence class of end-convergent sets. Let g be an end of G. A neighborhood of g is a set of vertices in G that intersects every end-convergent set in g. In particular, when K c V(G) is finite, there is a component of G - K that is a neighborhood of g.

Let r be a finitely generated group and S a finite symmetric generating set for r. Then the (right) Cayley graph G = G(r, S) is the graph with vertices V(G) := r and edges E(G) := {[ v, vs]: v E r, s E S}.

Now suppose that G = (V(G), E(G)) is any graph, not necessarily a Cayley graph. An automorphism of G is a bijection ofV(G) with itself that preserves adjacency; Aut(G) denotes the group of all automorphisms of G with the topology of pointwise convergence. If r c Aut(G), we say that r is (vertex) transitive if for every u, v E V(G), there is ayE r with yu = v. The graph G is transitive if Aut(G) is transitive. The graph G is quasi-transitive if V(G)/ Aut(G) is finite; that is, there is a finite set of vertices Vo such that V(G) = (yv: y E Aut(G), v E Yo}. Note that any finitely generated group acts transitively on any of its Cayley graphs by the automorphisms y: v ~ y v.

It may seem that the most natural class of graphs on which percolation should be studied is the class of transitive (or quasi-transitive) locally finite graphs. At first sight, one might suspect that any theorem about percolation on Cayley graphs should hold for transitive graphs. However, somewhat surprisingly, this impression is not correct. It turns out that theorems about percolation on Cayley graphs "always" generalize to unimodular transitive

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1813

graphs (to be defined shortly), but nonunimodular transitive graphs are quite different.

Recall that on every closed subgroup r c Aut(G), there is a unique (up to a constant scaling factor) Borel measure that, for every y E r, is invariant under left multiplication by y; this measure is called (left) Haar measure. The group r is unimodular if Haar measure is also invariant under right multiplication. For example, when r is finitely generated and G is the (right) Cayley graph of r, on which r acts by left multiplication, then r c Aut(G) is (obviously) closed, unimodular and transitive. The Haar measure in this case is (a constant times) counting measure. A [quasi-]transitive graph G is said to be unimodular if Aut(G) is unimodular. By Benjamini, Lyons, Peres and Schramm (1999) [hereinafter referred to as BLPS (1999)], a transitive graph is unimodular iff there is some unimodular transitive closed subgroup of Aut(G). It is not hard to show that a transitive closed subgroup r c Aut(G) is unimodular iff for all x, y E V(G), we have

I{z E V(G): 3 y E r yx = x and yy = z} I = I {z E V ( G): 3 y E r y y = y and y x = z} I

[see Trofimov (1985)]. Here, unimodularity will be used only in the following form.

THEOREM 2.1 (Unimodular mass-transport principle). Let G be a graph with a transitive unimodular closed automorphism group r c Aut(G). Let o E V(G) be an arbitrary base point. Suppose that cp: V(G) X V(G) ~ [0,00] is invariant under the diagonal action of r. Then

(2.1) L cp(o, v) = L cp(v,o). veV(G) veV(G)

See BLPS (1999) for a discussion of this principle and for a proof. In fact, r is unimodular iff (2.1) holds for every such cpo Also see BLPS (1999) for further discussion of the relevance of unimodularity to percolation.

Given a set A, let 2A be the collection of all subsets 'Y} c A, equipped with the u-field generated by the events {a E 'Y}}, where a E A. A bond percolation process is a pair (P, w), where w is a random element in 2E and P denotes the distribution (law) of w. Sometimes, for brevity, we shall just say that w is a (bond) percolation. A site percolation process (P, w) is given by a probability measure P on 2V(G), while a (mixed) percolation is given by a probability measure on 2V(G)UE(G) that is supported on subgraphs of G. If w is a bond percolation process, then w:= V(G) u w is the associated mixed percolation. In this case, we shall often not distinguish between wand w, and think of w as a subgraph of G. Similarly, if w is a site percolation, there is an associated mixed percolation w:= w U (E(G) n (w X w)), and we shall often not bother to distinguish between wand w.

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1814 R. LYONS AND O. SCHRAMM

If v E V(G) and w is a percolation on G, the cluster (or component) C(v) of v in w is the set of vertices in V(G) that can be connected to v by paths contained in w. We shall often not distinguish between the cluster C(v) and the graph (C(v), (C(v) X C(v» n w) whose vertices are C(v) and whose edges are the edges in w with endpoints in C( v ).

Let p E [0,1]. Then Bernoulli(p) bond percolation (Pp , w) on G is the product measure on 2 E(G) that satisfies Pp[e E w] = p for all e E E(G). Simi­larly, one defines Bernoulli(p) site percolation on 2V(G). The critical probabil­ity p/G) is the infimum over all p E [0,1] such that there is Pp-positive probability for the existence of an infinite connected component in w.

Aizenman, Kesten and Newman (1987) showed that Bernoulli(p) percola­tion in 7L d has a.s. at most one infinite cluster. Burton and Keane (1989) gave a much simpler argument that generalizes from 7L d to any amenable Cayley graph, though this generalization was not mentioned explicitly until Gan­dolfi, Keane and Newman (1992). It follows that Pu(G) = p/G) when G is an amenable Cayley graph. For background on percolation, especially in 7L d , see Grimmett (1989).

Suppose that f is an automorphism group of a graph G. A percolation process (P, w) on G is f-invariant ifP is invariant under each l' E f. This is, of course, the case for Bernoulli percolation.

3. Cluster indistinguishability.

DEFINITION 3.1. Let G be graph and f a closed (vertex-) transitive sub­group of Aut(G). Let (P, w) be a f-invariant bond percolation process on G. We say that P has indistinguishable infinite clusters if for every measurable .9i' c 2V(G) X 2 E(G) that is invariant under the diagonal action of f, almost surely, for all infinite clusters C of w, we have (C, w) E.9i', or for all infinite clusters C, we have (C, w) t/..9i'.

DEFINITION 3.2. Let G = (V(G), E(G» be a graph. Given a set A E 2 E(G) and an edge e E E(G), denote ileA := Au {e}. For.9i' c 2 E(Gl, we write I1 e.9i':= {ileA: A E.9i'}. A bond percolation process (P, w) on G is insertion tolerant if P[I1 e.9i'] > ° for every e E E(G) and every measurable .9i' c 2 E(G)

satisfying P[.9i'] > 0. For example, Bernoulli(p) bond percolation is insertion tolerant when

p E (0,1]. A percolation w is deletion tolerant if P[ II -, e.9i'] > ° whenever e E E( G)

and P[.9i'] > 0, where II -, e W := W - {e}. It turns out that deletion tolerance and insertion tolerance have very different implications. For indistinguisha­bility of infinite clusters, we shall need insertion tolerance; deletion tolerance does not imply indistinguishability of infinite clusters (see Example 3.15 below). A percolation that is both insertion and deletion tolerant is usually said to have "finite energy." Gandolfi, Keane and Newman (1992) use the words "positive finite energy" in place of "insertion tolerance."

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1815

THEOREM 3.3 (Cluster indistinguishability). Let G be a graph with a transitive unimodular closed automorphism group r c Aut( G). Every r -in­variant, insertion-tolerant, bond percolation process on G has indistinguish­able infinite clusters.

Similar statements hold for site and mixed percolations and the proofs go along the same lines. Likewise, the proof extends to quasi-transitive unimod­ular automorphism groups.

Initially, we could only establish the theorem under the assumption of strong insertion-tolerance; that is, P[ lIeJ¥"] ;::: 15 P[J¥"] for some constant 15 > o. We are grateful to Olle Haggstrom for pointing out how to deal with the general case.

REMARK 3.4 (Scenery). For some purposes, the following more general form of this theorem is usefuL Let G be a graph and r a transitive group acting on G. Suppose that X is either V, E, or VuE. Let Q be a measurable space and n := 2E X QX. A probability measure P on n will be called a bond percolation with scenery on G. The projection onto 2E is the underlying percolation and the projection onto QX is the scenery. If (w, q) E n, we set lIe(w, q) := (lIe w, q). We say that the percolation with scenery P is insertion-tolerant if, as before, P[lIe~] > 0 for every measurable ~ c n of positive measure. We say that P has indistinguishable infinite clusters if for every J¥" c 2v X 2E X QX that is invariant under the diagonal action of r, for P-a.e. (w, q), either all infinite clusters C of w satisfy (C, w, q) EJ¥" or they all satisfY (C, w, q) $.J¥". Theorem 3.3 holds also for percolation with scenery (with the same proof as given below).

A percolation with scenery that comes up naturally is as follows. Fix PI < P2 in (0,1). Let z/e E E) be Li.d. with each Ze distributed according to uniform measure on [0,1]. Let Wj be the set of e E E with Ze < Pj' j = 1,2. Then (W2' WI) is an insertion-tolerant percolation with scenery, where WI E 2E is considered the scenery. See Haggstrom and Peres (1999), Haggstrom, Peres and Schonmann (1999), Schonmann (1999a) and Alexander (1995) for exam­ples where this process is studied.

Say that an infinite cluster C is of type J¥" if(C, w) EJ¥"; otherwise, say that it is of type --, J¥". Suppose that there is an infinite cluster C of wand an edge e E E with e $. w such that the connected component C' of w U {e} that contains C has a type different from the type of C. Then e is called pivotal for (C, w).

We begin with an outline of the proof of Theorem 3.3. Assume that the theorem fails. First, we shall show that with positive probability, given that a vertex belongs to an infinite cluster, there are pivotal edges at some distance r of the vertex. By Proposition 3.11, a.s. when there is more than one infinite cluster of w, each such cluster is transient for the simple random walk, and hence for the so-called delayed simple random walk (DSRW). The DSRW on a cluster of w is stationary (in the sense of Lemma 3.13). Fix a base point o E V, and let W be the DSRW on the w-cluster of 0 with W(O) = o. Let n be

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1816 R. LYONS AND O. SCHRAMM

large, and let e be a uniform random edge at distance r from W(n). When DSRW is transient, with probability bounded away from zero, e is pivotal for (C(o), w) and W(j) is not an endpoint of e for any time j < n. On this event, set w' := IIew. Then w' is, up to a controllable factor, as likely as w by insertion tolerance. By transience, e is far from 0 with high probability. Since e is pivotal, the type of C( 0) is different in wand in w'. Since wand w I are the same in a large neighborhood of 0, this shows that the type of C(o) cannot be determined with arbitrary accuracy by looking at w in a large neighborhood of o. This contradicts the measurability of .N and establishes the theorem.

One can say that the proof is based on the contradictory prevalence of pivotal edges. To put the situation in the correct perspective, we point out that there are events depending on LLd. zero-one variables that have in­finitely many "pivotals" with positive probability. For example, let m l , m 2 , •••

be a sequence of positive integers such that Lk2-mk < 00 but Lkmk2-mk = 00,

and let (Xk,j: j E {I, ... , mk}) be i.i.d. random variables with P[ Xk,j = 0] = P[ Xk,j = 1] = 1/2. Let Jf be the event that there is a k such that Xk,j = 1 for j = 1,2, ... , mk' Then with positive probability Jf has infinitely many piv­otals; that is, there are infinitely many (k,j) such that changing Xk,j from 0 to 1 will change from -, Jf to Jf. [This is a minor variation on an example described by Haggstrom and Peres (private communication).]

We now prove the lemmas necessary for Theorem 3.3.

LEMMA 3.5 (Pivotals). Suppose that (P, w) is an insertion-tolerant percola­tion process on a graph G, and let.N c 2V(G) X 2E(G) be measurable. Assume that there is positive probability for coexistence of infinite clusters in .N and -,.N. Then with positive probability, there is an infinite cluster C of the percolation that has a pivotal edge.

PROOF. Let k be the least integer such that there is positive probability that there are infinite clusters of different types with distance between them equal to k. Clearly, k > O. Suppose that')' is a path of length k such that with positive probability, ')' connects infinite clusters of different types; let :9' be the event that')' connects infinite clusters of different types. Given :9', there are exactly two infinite clusters that intersect ,)" by the minimality of k. Let e be the first edge in ')'. When wE:9', let CI and C2 be the two infinite clusters that ')' connects and let C~ and C~ be the infinite clusters of IIe w that contain C I and C2 , respectively. Note that, conditioned on :9', the distance between C~ and C~ is less than k. (The possibility that C~ = C~ is allowed.) Since IIe:9' has positive probability, the definition of k ensures that the type of(C~, IIew) equals the type of(C~, IIew) a.s. Hence, e is pivotal for (CI , w) or for (C2 , w) whenever w E:9'. This proves the lemma. D

LEMMA 3.6. Let f be a transitive closed subgroup of the automorphism group of a graph G. If (P, w) is a f-invariant insertion-tolerant percolation process on an infinite graph G, then almost every ergodic component of w is insertion tolerant.

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REMARK 3.7. A stronger and more general statement is Lemma 1 of Gandolfi, Keane and Newman (1992).

PROOF OF LEMMA 3.6. This is the same as saying that for every f-in­variant event :9' of positive probability, the probability measure P[· I :9'] is insertion tolerant. If :9' is a tail event, then IIe:9' = :9', and insertion tolerance of P[ . I :9'] follows from the calculation

But every f-invariant event is a tail event (mod 0): Write E as an increasing union of finite subsets En. Let :9'n be events that do not depend on edges in En and such that Ln P[:9'n .6.:9'] < 00; such events exist because P and :9' are f-invariant. (Here ?t'.6..%:= (?t'\.%) u (.%\;n is the exclusive or.) Then lim sUPn:9'n is a tail event and equals :9'(mod 0). D

COROLLARY 3.S. Let f be a transitive closed subgroup of the automor­phism group of a graph G. Consider a f-invariant insertion-tolerant percola­tion process (P, w) on G. Then almost surely, the number of infinite clusters of w is 0, 1 or 00.

The proof is standard for the ergodic components; compare Newman and Schulman (1981).

Benjamini and Schramm (1996) conjectured that for Bernoulli percolation on any quasi-transitive graph, if there are infinitely many infinite clusters, then a.s. every infinite cluster has continuum many ends. This was proved by Haggstrom and Peres (1999) in the unimodular case and then by Haggstrom, Peres and Schonmann (1999) in general. For the unimodular case, we give a simpler proof that extends to insertion-tolerant percolation processes. We begin with the following proposition.

PROPOSITION 3.9. Let G be a graph with a transitive unimodular closed automorphism group f c Aut( G). Consider some f -invariant percolation pro­cess (P, w) on G. Then a.s. each infinite cluster that has at least three ends has no isolated ends.

PROOF. For each n = 1,2, ... , let An be the union of all vertex sets A that are contained in some percolation cluster K(A), have diameter at most n in the metric of the percolation cluster and such that K(A) - A has at least three infinite components. Note that if g is an isolated end of a percolation cluster K, then for each finite n, some neighborhood of g in K is disjoint from An. Also observe that if K is a cluster with at least three ends, then K intersects An for some n.

Fix some n ~ 1. Consider the mass transport that sends one unit of mass from each vertex v in a percolation cluster that intersects An and distributes it equally among the vertices in An that are closest to v in the metric of the

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1818 R. LYONS AND O. SCHRAMM

percolation cluster of v. In other words, let K( v) be the set of vertices in C(v) nAn, that are closest to v in the metric of w, and set F(v,w; w):= [K(V)[-l if w E K(v) and otherwise F(v, w; w) := O. Then F(v, w; w), and hence the expected mass f(v, w) := EF(v, w; w) transported from v to w, is invariant under the diagonal r action. If g is an isolated end of an infinite cluster K that intersects An' then there is a finite set of vertices B that gets the mass from all the vertices in a neighborhood of g. But the mass-transport principle tells us that the expected mass transported to a vertex is finite. Hence, a.s. clusters that intersect An do not have isolated ends. Since this holds for all n, we gather that a.s. infinite clusters with isolated ends do not intersect U nAn, whence they have at most two ends. D

PROPOSITION 3.10. Let G be a graph with a transitive unimodular closed automorphism group r c Aut(G). If (P, w) is a r-invariant insertion-tolerant percolation process on G with infinitely many infinite clusters a.s., then a.s. every infinite cluster has continuum many ends and no isolated end.

PROOF. As Benjamini and Schramm (1996) noted, it suffices to prove that there are no isolated ends of clusters. To prove this in turn, observe that if some cluster has an isolated end, then because of insertion tolerance, with positive probability, some cluster will have at least three ends with one of them being isolated. Hence Proposition 3.10 follows from Proposition 3.9. D

PROPOSITION 3.11 (Transience). Let G be a graph with a transitive uni­modular closed automorphism group r c Aut(G). Suppose that (P, w) is a r -invariant insertion-tolerant percolation process on G that has almost surely infinitely many infinite clusters. Then a.s. each infinite cluster is transient.

PROOF. By Proposition 3.10, every infinite cluster of w has infinitely many ends. Consequently, there is a random forest ty c w whose distribution is r -invariant such that a.s. each infinite cluster C of w contains a tree of ty with more than two ends [Lemma 7.4 of BLPS (1999)]. From Remark 7.3 of BLPS (1999), we know that any such tree has Pc < 1. By Lyons (1990), it follows that such a tree is transient. The Rayleigh monotonicity principle [e.g., Lyons and Peres (1998)] then implies that C is transient. D

REMARK 3.12. Examples show that Proposition 3.11 does not hold when r is not unimodular. However, we believe that when r is not unimodular and P is Bernoulli percolation, it does still hold.

Let (P, w) be a bond percolation process on G, and let wE 2E. Let x E V(G) be some base point. It will be useful to consider delayed simple random walk on w starting at x, W = Wxw, defined as follows. Set W(O) := x. If n ~ 0, conditioned on (W(O), ... , Wen»~ and w, let W'(n + 1) be chosen from the neighbors of Wen) in G with equal probability. Set Wen + 1) :=

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1819

W'(n + 1) if the edge [Wen), W'(n + 1)] belongs to w; otherwise, let Wen + 1) := Wen).

Given w, let Wand W* be two independent delayed simple random walks starting at x. Set wen) := Wen) for n ~ 0 and wen) := W*( -n) for n < O. Then w is called two-sided delayed simple random walk. Let Px denote the law of the pair (w, w); it is a probability measure on V7L X 2E. Define Y: V7L ~ V7L by

(Yw)(n) := wen + 1), and let

yew, w) := (Yw, w)

For y E f, we set

y ( w, w) := (y W , yw) ,

where (y w)(n) := y(w(n)). The following lemma generalizes similar lemmas in Haggstrom (1997), in

Haggstrom and Peres (1999) and in Lyons and Peres (1998).

LEMMA 3.13 (Stationarity of delayed random walk). Let G be a graph with a transitive unimodular closed automorphism group f c Aut( G). Let 0 E V( G) be some base point. Let (P, w) be a f-invariant bond percolation process on G. Let Po be the joint law of wand two-sided delayed simple random walk on w, as defined above. Then PoLw] = Po[YSlf] for everyf-invariant Slf c v7L X 2E. In other words, the restriction of Po to the f -invariant (J-field is Y-stationary.

The lemma will follow from two identities. The first is based on the fact that the transition operator for delayed simple random walk on w is symmet­ric, and the second is based on the mass-transport principle, that is, on unimodularity.

PROOF OF LEMMA 3.13. For j E 7L and x E V, set

'Wj:= {(w, w) E V7L X 2E: w(j) = x} and

Let g; c V7L X 2E be measurable. Observe that for all w E 2 E,

(3.1)

where J1,[g; [ w] means LXEvPx[g; [ w]. This follows from the fact that for any j, k E 7L with j < 0 < k and any Vj, vj + 1 , ••• , Vk E V, we have

k-l

".[w.j n w.j + 1 n ... n w.k [ w] = n a· ,..,.. Vj Vj+l Uk .. l'

~~J

where a i := dega(vi )-1 = dega(o)-1 if [Vi' vi + 1] E w, a i := (dega(o) -degw(v))/dega(o) if Vi = vi + 1 and a i := 0 otherwise. By integrating (3.1) over

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1820 R. LYONS AND O. SCHRAMM

w, we obtain (3.2)

which is our first identity. Observe also that JL is r-invariant. Now let .9f c VlL X 2E be r-invariant and measurable. For x, y E V, define

<!>(x,y):= JL[.9fn~l n~O].

Then <!> is invariant under the diagonal action of r on V X V, because JL and .9f are r -invariant. Consequently, the mass-transport principle gives LXEv<!>(X, 0) = LXEv<!>(O, x), which translates to our second identity,

(3.3) JL[.9fn~O] = JL[.9fn~l].

Observe that JL[lF n ~O] = Po[lF] for all measurable IF c VlL X 2E. By using (3.2) with !Jff :=.9f n ~o and then using (3.3) with .9.9f in place of .9f, we obtain finally

Po [.9f] = JL[.9f n ~O] = JL[.9(.9f n ~O)]

D

In Theorem 5.1 below, we show that in an appropriate sense, if w is ergodic and has indistinguishable components, then the delayed simple ran­dom walk on the infinite components of w is ergodic.

REMARK 3.14 (A generalization of Lemma 3.13). Let V be a countable set acted on by a transitive unimodular group r. Let Q be a measurable space, let E := QVxv, and let P be a r-invariant probability measure on E. Suppose that z: Q ~ [0,1] is measurable, and that z(g(x, y» = z(g(y, x» and LvZ(g(X, u» = 1 for all x, y E V and for P-a.e. g. Given 0 E V and a.e. gEE, there is an associated random walk starting at 0 with transition probabilities Pg(x, y) = z( g(x, y». Let Po denote the joint distribution of g ~nd this random walk. Then the above proof shows that the restriction of Po to the r -invariant a-field is Y-invariant.

See Lyons and Schramm (1999) for a still greater generalization.

Let (P, w) be a bond percolation process on an infinite graph G. For every e E E(G), let Y, e be the a-field generated by the events re' E w} with e ' =1= e. Set Z( e) := P[ e E w I Y, e], and call this the conditional marginal of e. Note that insertion tolerance is equivalent to Z(e) > 0 a.s. for every e E E(G).

PROOF OF THEOREM 3.3. Let (P, w) be insertion tolerant and r-invariant. Let 0 E V be some fixed base point of G. Assume that the theorem is false. Then by Corollary 3.8, there is positive probability that w has infinitely many infinite clusters. If we condition on this event, then w is still insertion tolerant, as shown in Lemma 3.6. Consequently, we henceforth assume, with no loss of generality, that a.s. w has infinitely many infinite clusters.

Fix e> O. From Lemma 3.5, we know that there is a positive probability for pivotal edges of clusters of type .9f, or there is a positive probability for

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1821

pivotal edges of clusters of type ---, J¥'. Since we may replace J¥' by its comple­ment, assume, with no loss of generality, that there is a positive probability for pivotal edges of clusters of type J¥'. Fix some r > 0 and 8 > 0 such that with positive probability, C(o) is infinite of type J¥' and there is an edge e at distance r from 0 that is pivotal for (C(o), w) and satisfies Z(e) ~ 8, where Z(e) is the conditional marginal of e, as described above.

Let J¥'o be the event that C(o) is infinite and (C(o), w) EJ¥', and let J¥'; be an event that depends on only finitely many edges such that P[~ ..6. J¥';] < e. Let R be large enough that J¥'; depends only on edges in the ball B(o, R).

Let W: 7L ~ V be two-sided delayed simple random walk on w, with W(O) = o. For n E 7L, let en E E be an edge chosen uniformly among the edges at distance r from Wen). Write P for the probability measure where we choose w according to P, choose W, and choose (en: n E 7L) as indicated.

Given any e E E, let 9'e be the event that w E ~, that e is pivotal for C(o), and that Z(e) ~ 8. Let ~en be the event that en = e and W(j) is not an endpoint of e whenever -00 <j < n. Note that for all t E 2 E, n E 7L, and e E E,

P[~en I w = t] = P[~en I w = IIet].

Thus, for all measurable 91 c 2E with P[91] > 0 and P[Z(e) ~ 8 191] = 1, we have

P[~en n IIe91] = P[~en I IIe91]P[IIe91] = P[~en 191]P[IIe91]

P[ IIe91] A A

P[91] P[~en n91] ~ 8P[~en n91].

In particular,

(3.4) P[~en n IIeJ¥'; n IIe9'e] ~ 8P[~en nJ¥'; n9'e] = 8P[~en nJ¥'; n9'eJ.

Let

~n:= U g'n e , eEE

~R:= U ~en, eEE-B(o,R)

and note that these are disjoint unions. Since IIe9'e c ---, ~ and since IIeJ¥'; C

J¥'; when e $. B(o, R), we may sum (3.4) over all e $. B(o, R) to obtain that

P[J¥'; n ---,~] ~ P[~R nA~ n ---,~] ~ 8P[~R nJ¥'; n9'eJ

~ 8P[ ~R n~ n9'eJ - 8e. (3.5)

Fix n to be sufficiently large that the probability that C(o) is infinite and en E B(o, R) is smaller than e; this can be done by Proposition 3.11. Then :I>[J¥'o n ~n - ~R] :5: e. Hence, we have from (3.5) that

(3.6) e ~ P[J¥'; ..6.~] ~ P[J¥'; n ---,~] ~ 8P[ ~n n~ n9'eJ - 2e8.

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Recall that P[~ fl9'eo] > o. Moreover, conditioned on ~ fl9'eo' transience guarantees a.s. a least m E 7L such that W(m) is at distance r to eo. Consequently, for some m :s: 0,

(3.7)

Let f?8m be the event that Z(em) ;::0: 8, that em is not an endpoint of W(j) for j < m, that C(W(m)) is infinite and of type Sit, and that em is pivotal for C(W(m)). Then f?8m = wm fI~ fl9'e • But f?8m is r-invariant. Therefore, Lemma 3.13 shows that the left-handmside of (3.7) does not depend on m, and it certainly does not depend on 8. Hence, when we take 8 to be a sufficiently small positive number, (3.6) gives a contradiction. This completes the proof of the theorem. D

EXAMPLE 3.15. A deletion-tolerant process that does not have indistin­guishable clusters is obtained as follows. Let X be a 3-regular tree and p E (1/2,1). Let p' E (1/(2p), 1). Begin with Bernoulli(p) percolation on X. Independently for each cluster C, with probability 1/2, intersect it with an independent Bernoulli(p') percolation. The resulting percolation process is clearly deletion tolerant, yet some infinite clusters w have p/w) = 1/(2p), while others have p/w) = 1/(2pp').

REMARK 3.16. Let T be the 3-regular tree, let r be the subgroup of automorphisms of T that fixes an end g of T and let P be Bernoulli(p) bond percolation on T, where p E (1/2,1). Then a.s. each infinite cluster C has a unique vertex Vc "closest" to g. The degree of Vc in the percolation configura­tion distinguishes among the infinite clusters. Hence, Theorem 3.3 does not hold without the assumption that r is unimodular. By a simple modification, a similar example can be constructed where r is the full automorphism group of a graph on which the percolation is performed [compare BLPS (1999)].

However, Haggstrom, Peres and Schonmann (1999) have recently shown that even without the unimodularity assumption, when p > Pc so-called "robust" properties do not distinguish between the infinite clusters of Bernoulli( p) percolation.

QUESTION 3.17. In the case that r is nonunimodular, write /Lx for the Haar measure of the stabilizer of x E V. The infinite clusters C divide into two types: the heavy clusters for which Lx E c /Lx = 00 and the others, the light clusters. It can be that the light clusters are distinguishable: for example, consider a 3-regular tree T with a fixed end g. Let r be the group of automorphisms of T that fix g. Then Bernoulli(2/3) percolation has infinitely many light clusters, which can be distinguished by the degree of the vertex they contain that is closest to g. But is it the case that heavy clusters are indistinguishable for every insertion-tolerant percolation process that is in­variant with respect to a transitive automorphism group?

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1823

4. Uniqueness and connectivity. Our goal is to prove the following theorem.

THEOREM 4.1 (Uniqueness and connectivity). Let G be an infinite graph with a transitive unimodular closed automorphism group r c Aut(G). Let P be a r -invariant and ergodic insertion-tolerant percolation process on G. If P has more than one infinite component a.s., then connectivity decays,

inf{ T ( x, y): x, y E V} = 0.

The intuitive idea behind our proof is that if inf T(X, y) > 0, then each infinite cluster has a positive "density." Since the densities are the same by cluster indistinguishability, there are only finitely many infinite clusters. By Corollary 3.8, there is only one. To make the idea of "density" precise, we use simple random walk X on the whole of G with the percolation sub graph as the scenery, counting how many times we visit each cluster.

For a set C c V, write

1 n

a( C) := lim - L l{x(k)E C)' n~oo n k=l

where the limit exists, for the frequency of visits to C by the simple random walk on G.

LEMMA 4.2 (Cluster frequencies). Let G be a graph with a transitive unimodular closed automorphism group r c Aut(G). There is a r-invariant measurable function freq: 2v ~ [0,1] with the following property. Sup­pose that (P, w) is a r-invariant bond percolation process on G, and let P :=

P X Po, where Po is the law of simple random walk on G starting at the base point o. Then P-a.s. a(C) = freq(C) for every cluster c.

PROOF. Given a set C c V and m, nEll., m < n, let

1 n-l

a~(C) := --- L l{x(k)EC}· n - m k=m

For every a E [0,1], let

X",:= {C c V: lim aa(C) = a Po-a.s.}, n~oo

and set X:= U",E[O,ljXa . Define freq(C):= a when C EX",. If C $.X, put freq(C) := 0, say. It is easy to verify that freq is measurable. Let y E r. To prove that a.s. freq(C) = freq( yC), note that there is an mEN such that with positive probability X(m) = yo. Hence for every measurable A c [0,1] such that a(C) E A with positive probability, we have a( yC) E A with positive probability. This implies that freq is r-invariant.

It remains to prove that P-a.s., every component of w is in X. First observe that the restriction of P to the r -invariant u-field is Y-invariant. This can be

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verified directly, but is also a special case of Remark 3.14: take Q = {-1, 0, 1}, take the value in Q associated to a pair (x, y) E V X V to be 1 if[ x, y] E w, ° if[ x, y] E E - wand -1 if[ x, y] $. E and take z(o) = z(l) = l/degG(o) and z( -1) = 0.

The following argument is modeled on the proof of Theorem 1 of Burton and Keane (1989). Let Fn(j) be the number of times that the j most frequently visited clusters in [0, n - 1] are visited in [0, n - 1]. That is,

Fn(j):= nmax{a~(Cl) + ... +a~(CJ: Cl ,oo"Cj are distinct clusters}.

For each fixed j, it is easy to see that Fn(j) is a subadditive sequence, that is, Fn+k(j) ::::; Fn(j) +ynFk(j). Note that the random variables Fn(j) are in­variant with respect to the diagonal action of r on V7L X 2E. By the subaddi­tive ergodic theorem, limn Fn(j)/n exists a.s. Set a(j):= limn Fn(j)/n -limn Fn(j - l)/n for j ~ 1. We claim that a.s.,

(4.1) lim max{la~(C) - ag(C)I: k, mE 7L, k, m ~ n, C is a cluster} = ° n-->oo

the Cauchy property uniform in C. Indeed, let e> 0. Observe that Lja(j) ::::; 1 and that a(l) ~ a(2) ~ .... Let jl ~ 1 be large enough that a(j) < e/9 for all j ~jl' Let m l be sufficiently large that

IFm(j)/m - Fm(j - l)/m - a(j)1 < e/(9jl + 9)

for all j ::::;jl and all m ~ mI' Set

U:= {x E [0,1]: 3 j ::::;jl Ix - a(j)1 < e/(9jl + 9)} U [0, e/3],

and let U( 8) denote the set of points x E IR within distance 8 of U. For all clusters C that are visited by the random walk and all m = 1,2, ... , there is some j such that a~(C) = Fm(j)/m - Fm(j - l)/m. If j ::::;jl and m ~ ml' it follows that a~(C) E U. The same also holds when j ~ jl and m ~ m l , because the jlth most frequently visited cluster in [0, m - 1], say C', satis­fies a~(C) ::::; a~(C') ::::; a(jl) + e/9 ::::; e/3. Hence a~(C) E U for all clusters C and all m ~ mI' Because -l/m::::; a~(C) - a~+ I(C) ::::; l/m, it follows that for all C and all n ~ m l , the set {a~(C): m ~ n} is contained in some connected component of U(l/n). But when n > max{ml ,9(jl + 1)/ e}, the total length of U(l/n) is less than e, which implies that the diameter of each connected component is less than e. This verifies (4.1).

Since

2max{lag n(C) - a~(C)I: C is a cluster}

= max{1 a;n( C) - a~( C) I: C is a cluster}

has the same law as max{la~(C) - a~n(C)I: C is a cluster} (by the ..?-invari­ance noted above), it follows from (4.1) that a.s. limn-->oo a~(C) = limn-->oo a~n(C) for every cluster C. When the cluster C is fixed, a~(C) and a~n(C) are independent, but both tend to a(C). Hence a(C) is an a.s. constant, which means that P-a.s. we have C E Z for every cluster C. This completes the proof. D

PROOF OF THEOREM 4.1. Let freq be as in Lemma 4.2. Since P is ergodic, Theorem 3.3 implies that there is a constant c E [0,1] such that a.s. freq(C)

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INDISTINGUISHABLE PERCOLATION CLUSTERS 1825

= c for every infinite cluster C. Suppose that there is more than one infinite cluster with positive probability. Then there are infinitely many infinite clusters a.s. Since clearly Lc a(C) = Lcfreq(C) ::::; 1, where the sum is over all clusters, it follows that c = o. Since G is infinite, it is also immediate that freq({v}) = 0 for every v E V. Therefore, freq(C) = 0 a.s. for all clusters, finite or infinite. In particular, freq(C(o» = 0 a.s., where C(O) is the cluster of o.

Let TO := inf{T(x, y): x, y E V}. We have that

E[a~(C(o))] 2 TO,

whence 0 = E[freq(C(o»] = E[ a(C(o»] 2 TO by the bounded convergence theorem, as required. D

QUESTION 4.3. The proof actually shows that (l/n)Lk=JE[ T(O, X(k»] ---70 as n ---7 00 when there are infinitely many infinite clusters in an invariant percolation process that has indistinguishable clusters. Do we have T(O, X(n» ---7 0 a.s.?

EXAMPLE 4.4. We give an example of an ergodic invariant deletion-tolerant percolation process with infinitely many infinite clusters and with T bounded below. The percolation process w will take place on 7L 3. Let {a(n): n E 7L} be independent {O, l}-valued random variables with P[a(n) = 1] = 1/2, and let A be the set of vertices (Xl' X 2 , X 3 ) E 7L 3 with a(x l ) = 1. Let 8> 0 be small, and let Ze be i.i.d. uniformly distributed in [0,1] indexed by the edges of 7L 3

and independent of the a(n)'s. Let Wl be the set of edges e with both endpoints in A such that ze < 1 - 8, and let W 2 be the set of edges e with ze < 8. Let W3 be the set of edges that have no vertex in common with W2 U A and satisfy ze < 1 - 8. Set W := Wl U W2 U W3.

It is immediate to verify that when 8 is sufficiently small, a.s. W has a single infinite component whose intersection with each component of A is infinite, and has a single infinite component in each component of the complement of A. The claimed properties of this example follow easily.

5. Ergodicity of delayed random walk. The following theorem is not needed for the rest of the paper. It is presented here because its proof uses some of the ideas from the proof of Lemma 4.2.

THEOREM 5.1 (Ergodicity of delayed random walk). Let G be a graph with a transitive unimodular closed automorphism group r c Aut( G). Let 0 E V( G) be some base point. Let (P, w) be a r-invariant ergodic bond percolation process on G with infinite clusters a.s. Also suppose that W has indistinguish­able infinite clusters a.s. Let Po be the joint law of wand two-sided delayed simple random walk on w, starting at o. Let sf be an event that is r-invariant and Y-invariant. Then Po[sf I C(o) is infinite] is either 0 or 1.

PROOF. For each m, n E 7L, let .'7':: be the IT-field of r-invariant sets generated by wand the random variables (W(j): j E [m, n]), where W(j) is the location of the delayed random walk at time j.

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Let '?5 be the event that C(o) is infinite. Let 8> O. Then there is an n E N and an event sf' E .'T~n such that PO[sfl .6. sf] < 8 P[ '?5], and therefore

(5.1)

Note that Y'?5 = '?5. Hence we have for all m E 7l..,

Po [ ymsf' .6. sf I '?5] < 8.

Observe also that the two events yn+ ~' and yrn - ~' are independent given w by the Markov property for the delayed random walk. Consequently,

f Po[sf I w]2 dP[ w] ~ f po[yn+lsfl I w]Po[yrn-~1 I w] dP[ w] - 28 wEW wEW

= f po[yn+lsfl,yrn-lsfl I w] dP[ w] - 28 wEW

~ f Po[sf I w] dP[ w] - 48. wEW

Since 8 is arbitrary, it follows that

1 Po[sf I w]2 dP[ w] ~ 1 Po[sf I w] dP[ w], W W

which means that Po[sf I w] is 0 or 1 for almost every w E '?5. Let .£U be the set of w E '?5 such that Po[sf I w] = 1 and let

Ii:= {(C(o),w): WE.£U}.

Finally, let rli denote the orbit of Ii under r; that is, rli:= {(C(yo), yw): w E.£U}. Because ysf =sf and sf is r-invariant, it follows as in the beginning of the proof of Lemma 4.2 that for every y E r and every w such that yo E C(o), we have w E.£U iff y-1w E.£U (after possibly making a measure zero modification of .£U). Therefore,

(5.2)

Since w is ergodic and has indistinguishable infinite clusters, a.s. all infinite clusters of ware in rli or a.s. all infinite clusters of ware not in rli. If the latter is the case, then P[.£U] = 0 and hence Po[sf I '?5] = O. Therefore, assume that

p[(C(o), w) E rli I '?5] = 1.

Hence, by (5.2), P[.£U I '?5] = 1, giving Po[sf I '?5] = 1, as required. D

6. Uniqueness for Bernoulli percolation. We now present some ap­plications of Theorem 4.1 to Bernoulli percolation on Cayley graphs. Later in this section, we prove inequalities relating Pu based on Theorem 3.3.

We first note that the choice of generators does not influence whether Pu < 1.

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THEOREM 6.1. Let 8 1 and 8 2 be two finite generating sets (or a countable group r, yielding corresponding Cayley graphs G1 and G2 • Then Pu(G1) < 1 i{{Pu(G2 ) < 1.

PROOF. Left and right Cayley graphs with respect to a given set of generators are isomorphic via x ~ X-I, so we consider only right Cayley graphs.

Write r;(x) for the probability under Bernoulli(p) percolation that 0 and x lie in the same cluster of G i • Express each element s E 8 1 in terms of a word cp(s) E 8 2 , Let W 2 be Bernoulli(p) percolation on G2 and define WI on G1 by letting [x, xs] E WI iff the path from x to xs in G2 given by cp(s) lies in W 2 •

Then WI is a percolation such that if two edges are sufficiently far apart, then their presence in WI is independent. Thus, Liggett, Schonmann and Stacey (1997) provide a function ((p) E (0,1) such that {(pH 1 when pi 1 and such that WI stochastically dominates Bernoulli(f(p» percolation on G 1• This implies that rAp)(x) ~ r;(x) for each x. If ((p) > Pu(G1), then it follows from this and Theorem 4.1 that p ~ Pu(G2 ), showing that Pu(G1) < 1 implies pU<G2 ) < 1. D

REMARK 6.2. For the situation used in the above proof, and for many similar applications, one does not need the full generality of the theorem of Liggett, Schonmann and Stacey. The following observation suffices. Suppose that (Xi: i E 1) are i.i.d. random variables taking values in {0,1} and with P[Xi = 1] = p. Let (Ij: j E J) be an indexed collection of subsets of I. Set 1) := min{Xi : i E I) and J i := {j E J: i E I). Suppose that n := sup{II): j E

J} and m := sup{1 J;I: i E I} are both finite. Then (1): j E J) stochastically dominates independent random variables (Zj: j E J) with P[Zj = 1] ~ (1 -(1 - p )1/ m)n. Indeed, set X; := max{ZiJ j E Jj and Zj := min{ZiJ i E I), where (ZiJ i E I, j E J) are independent {O, l}-variables with P[Zi,j = 1] =

1 - (1 - p)1!m. Then (Xi: i E 1) stochastically dominates (X;: i E 1), and hence (1): j E J) stochastically dominates (Zj: j E J).

REMARK 6.3. Schonmann (1999a) shows that for every quasi-transitive graph G, Pu = PBB' where PBB is the infimum of all p such that

lim inf P[B(u,r) ~B(u,r)] = 1 r~oo V,UEV

in Bernoulli( p) percolation, where B( u, r) denotes the ball in G of radius r and center u and A ~ A' is the event that there is a cluster C with C n A =1= 0 and C n A' =1= 0. Based on this and the proof of Theorem 6.1, one obtains the following generalization. Suppose that G and G' are quasi-transi­tive graphs and G' is quasi-isometric to G. Then Pu(G) < 1 iff Pu(G') < 1. This observation was also made independently by Y. Peres (private communi­cation).

For our next result regarding Pu' we need the following construction. Let KO be a real-valued random variable and suppose that P is a bond percolation

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process on some graph G. We would like to color the clusters of P in such a way that conditioned on the configuration w, the colors ofthe components are i.i.d. random variables with the same law as KO'

To construct this process, let (v l' V 2' ... ) be an ordering of the vertices in G. Let (Kl' K2"") be i.i.d. random variables with the same law as KO' Given wand v E V, set KjV) := Kj if j is the least integer with Vj E C(v). It is not hard to see that Kw(V) is measurable, as a function on the product of 2E and the sample spaces of the K/S. Let P K denote the law of (w, Kw)' Observe that P K satisfies the description of the previous paragraph, and therefore does not depend on the choice of the ordering of V. Consequently, if ')I is an automor­phism of G and P is ')I-invariant, then pK is ')I-invariant.

LEMMA 6.4 (Ergodicity ofPK). Let [ be a transitive closed subgroup of the automorphism group of a graph G. Suppose that (P, w) is a [-invariant ergodic insertion-tolerant percolation process on G. Let KO be a real-valued random variable, and let pK be as above. Suppose that inf{7'(o, x): x E V(G)} = O. Then pK is [-invariant and ergodic.

PROOF. To prove the ergodicity of P\ let sf be a [-invariant event in 2E(G) X [RV(G) and 8 E (0,1/2). The probability of sf conditioned on w must be a constant, say a, by ergodicity of P. We need to show that a E {O, 1}.

There is a cylindrical event sf' that depends only on the restriction of w and Kw to some ball B(o, r) about 0 and such that PK[sf ..c.sf'] < 8. Then E[lpK[sf' [ w] - al] < 8.

Let 9Jx be the event that some vertex in B(o, r) belongs to the same cluster as some vertex in B(x, r). Since infx T(O, x) = 0 and P is insertion tolerant, there is some x such that P[9Jxl < 8. Indeed, suppose that P[9Jx] ~ 8 for all x. Let gx be the event that all the edges in B(x, r) belong to w, and for A c G, let Y, A denote the LT-field generated by the events {e E w} with e $. A. Then 9Jx is Y,(B(o,r)UB(x,r))-measurable. By insertion tolerance, for every Y,B(o,rfmeasurable event ~ with P[~] > 0, we have P[~ ngo] > O. Consequently, there is some 8> 0 such that

p[P[ go [Y,B(O,r)] > 8] > 1 - 8/2.

It follows that for all Y, B(o, rfmeasurable events ~ with P[~] ~ 8, we have P[go [ ~] > 8/2. In particular, for all x E V, we have P[go [9Jx] > 8/2, which gives P[go n9Jx] > 88/2. Transitivity and insertion tolerance imply that there is a 8' > 0 such that

for all x E V. Hence, for all Y, B(x,rfmeasurable events ~ with P[~] ~ 88/2, we have P[ gx [ ~] > 8' /2. Taking ~:= go n 9Jx gives for all x E V -B(0,2r),

T(O,X) ~ P[go n9Jx ngx ] = P[gx [go n9Jx ]p[go n9Jx ] ~ (8'/2)(88/2),

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which contradicts our assumption, and thereby verifies that P[ f9J < e for some x E V. Fix such an x.

Let 'Yx E r be such that 'Yx 0 = x. Then

E[IPK[ 'YxJ?f' I w] - al] < e.

Since 'YxJ?f' depends only on the colored configuration in B(x, r), we have that on the complement of f9x ,

PK[ J?f' ,'YxJ?f' I w] = PK[ J?f' I W ]PK[ 'YxJ?f' I w],

whence E[lpK[J?f', 'YxJ?f' I w] - a2 1] = O(e). Therefore,

a - a 2 = E[IPK[J?f,'YxJ?f1 w] - a 2 1] = O(e).

Since e was arbitrary, we get that a E {O, I}, as desired. D

We now prove that Pu < 1 for all Cayley graphs of Kazhdan groups. Let r be a countable group and S a finite subset of r. Let zr(2') denote the set of unitary representations of r on a Hilbert space 7f' that have no invariant vectors except o. Set

K(r, S) := max{ e: V 7f' V 7T E zr(2') V v E7f', 3s E S 117T( s)v - vii ~ ellvll}.

Then r is called Kazhdan [or has Kazhdan's property (T)] if K(T, S) > 0 for all finite S. The only amenable Kazhdan groups are the finite ones. Examples of Kazhdan groups include SL(n,1:) for n ~ 3. See de la Harpe and Valette (1989) for background; in particular, every Kazhdan group is finitely gener­ated (page 11), but not necessarily finitely presentable (as shown by examples of Gromov; see page 43). It can be shown directly, but also follows from our Corollary 6.6 below, that every infinite Kazhdan group has only one end. See Zuk (1996) for examples of Kazhdan groups arising as fundamental groups of finite simplicial complexes.

Rather than the definition, we shall use the following characterization of Kazhdan groups: Let P * be the probability measure on subsets of r that is the empty set half the time and all of r half the time. Recall that r acts by translation on the probability measures on 2r.

THEOREM 6.5 [Glasner and Weiss (1997)]. A countable infinite group r is Kazhdan iff P * is not in the weak * closure of the r -invariant ergodic probability measures on 2r.

COROLLARY 6.6. If G is a Cayley graph of an infinite Kazhdan group r, then Pu(G) < 1. Moreover, P[3 a unique infinite cluster] = 0 in Bernoulli(p) percolation.

In an older version of this manuscript, only the first statement appeared. We thank Yuval Peres for pointing out that a modification of the original proof produces the stronger second statement.

Schonmann (1999b) proved that Bernoulli(pu) on T X 7L, where T is a regular tree of degree at least three, does not have a unique infinite cluster,

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and Peres (1999) generalized this result to nonamenable products. On the other hand, Bernoulli(pu) percolation on a planar nonamenable transitive graph has a unique infinite cluster. [See Lalley (1998) for the high genus case, and Benjamini and Schramm (1998) for the general case.] Little else is known, however, about the uniqueness of infinite clusters at Pu' For example, the case of lattices in hyperbolic 3-space is still open.

PROOF OF THEOREM 6.5. Suppose that Pu(G) = 1. Let 0 be the identity in r, regarded as a vertex in G. Write r/x) for the probability under Bernoulli(p) percolation that 0 and x lie in the same cluster. Then by Theorem 4.1, for all p < 1 we have infx r/x) = O. Fix p and let w be the open subgraph of a Bernoulli(p) percolation. Let YJ be the union of the sites of some of the clusters of w, where each cluster is independently put in YJ with probability 1/2. By Lemma 6.4, the law Qp of YJ is r-invariant and ergodic. Furthermore, any fixed finite subset of r either is contained in YJ or is disjoint from YJ with high probability when p is sufficiently close to 1. That is, P * = weak*­limp _d Qp, whence r is not Kazhdan.

To prove the stronger statement, suppose that there is a unique infinite component P-a.s. Let (w, w) be the standard coupling of Bernoulli(p) and Bernoulli(pJ percolation on G. Let YJ be as above. For a vertex x E G, write A(x) for the set of clusters of w that lie in the unique infinite cluster of wand that are closest to x among those with this property (where distance is measured in G). Define YJ' to be the union of YJ with all sites x for which A(x) contains only one cluster and that cluster lies in YJ. Since the law of (w, w, YJ) is ergodic by an obvious extension of Lemma 6.4, so is the law Q~ of its factor YJ'. Again, we obtain P * = weak*-limp --> Pu Q~, whence r is not Kazhdan. D

REMARK 6.7. For probabilists, we believe that the proof we have presented of Corollary 6.6 is the most naturaL For others, we note that one can avoid Lemma 6.4 and Theorem 6.5 by using a theorem of Delorme (1977) and Guichardet (1977) that characterizes Kazhdan groups in terms of positive semidefinite functions. This relies on the fact that rex, y) is positive semidef­inite, as observed by Aizenman and Newman (1984).

Other groups that are not finitely presentable and that have provided interesting examples for probability theory are the so-called lamplighter groups [see Kaimanovich and Vershik (1983) and Lyons, Pemantle and Peres (1996); in these references, these groups are denoted Gd and are amenable, but we shall be interested here in nonamenable examples].

We first give a concrete description of a lamplighter graph and later generalize and use more algebraic language. Suppose that G is a graph. The lamplighter graph LG over G is the graph whose vertices are pairs (A, v), where A c V(G) is finite and v E V(G). We think of A as the locations of the lamps that are on, and consider v as the location of the lamplighter. One neighbor of (A, v) in LG is the vertex (A Ldv}, v) (the lamplighter switches the lamp off or on) and the other neighbors have the form (A, u), where [v, u] E E(G) (the lamplighter walks one step).

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In the algebraic context and language, lamplighter groups are particular wreath products: Let f be a group acting from the left on a set V. Let K be a group; K~ denotes the group of maps f: V ~ K such that f(x) is the identity element idK of K for all but finitely many x E V and with multiplication (fd2)(X):= fl(X)f2(X). Then f acts on K~ by translation: (yf)(x):= f(y-IX). The (restricted) wreath product K \ f is the set K~ X f with the multiplica­tion

(fl'YI)(f2'Y2):= (fl(yd2)'YIY2)·

If f and K are finitely generated and f acts transitively on V, then K \ f is finitely generated. To see this, let YI' ... ' Ys generate f and k l , ... , k t

generate K. Write idr for the identity element of f and idv for the identity element of K~. Let 0 E V be fixed. Write Fj for the element of K~ defined by Fj(o) := k j and Fj(x) := idK for all x =1= o. Set

Sr:= {(idv'Yi): 1:::;; i:::;; s},

SK:= {(Fj,idr): 1 :::;;j:::;; t}. Then S := Sr U SK is a finite generating set for K \ f. Indeed, let (f, y) E

K \ f. For x E V, choose Yx E f such that Yxo = x and write hx E K~ for the function hx(o) := f(x) and hx(Y) := idK for y =1= o. Then

(6.1) (f,y) = ( n (idv, Yx)(hx,idr)(idv, y;I))(idV,Y), xEV

f(x)'''id K

where the product over x is taken in any order. If we then write each (h x , id r ) as a product of (Fj, id r ), each (idv, Yx ) as a product of (idv, y), each (idv, y;l) as a product of (idv, Yi)' and (idv, y) as a product of (idv, y), we obtain a representation as a product of elements of S.

The lamplighter groups Gd are those where f = V = 71. d and K = 71. 2 • [By a theorem of Baumslag (1961), the groups Gd are not finitely presentable.]

COROLLARY 6.8. Let K be a finite group with more than one element. Let f be a finitely generated group acting transitively on an infinite set V. If G is any Cayley graph of K \ f, then Pu(G) < 1.

To prove this, we borrow a technique from Benjamini, Pemantle and Peres (1998): If </> and l/J are two paths, denote by [</> n l/J [ the number of edges they have in common (as sets of edges).

LEMMA 6.9. Let G be a graph and x, y be any two vertices in G. Let () E (0,1) and c > 0 be constants. Suppose that IL is a probability measure on (possibly self-intersecting) paths l/J joining x to y such that

(IL X IL){( </>, l/J): [</> n l/J[ = n} :::;; c()n

for all n E N. Let () < P < 1. Then for Bernoulli(p) percolation on G, we have

r(x, y) 2 c- l (l - ()jp).

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PROOF. Define the random variable

Z := L 11-( rf;) 1{I/J .is open} •

of! P[ rf; IS open]

Then E[Z] = 1 and

E[ Z2] = L 11-( cfJ) 11-( rf;) P[ ~, rf; are both .open] cP, of! P[ cfJ IS open]P[ rf; IS open]

= L l1-(cfJ)I1-(rf;)p+pnI/J1 cP,of!

= L p - n L 11-( cfJ) 11-( rf; ) n lc/>nof!l=n

:::; LP-nCon = c(1 - Ojp) -1. n

Therefore, the Cauchy-Schwarz inequality yields

T(X,y) ~ P[Z > 0] ~ E[Z]2jE[Z2] ~ c- I (I- Ojp). D

PROOF OF COROLLARY 6.8. Since K is assumed to be finite, we take the generating set {k l , ... , k t } to be all of K. We use the Cayley graph given by the generating set S. It suffices to exhibit a measure on paths connecting (idv , id r) to ({, y) that satisfies the condition of Lemma 6.9 with 0 and c not depending on (f, y), since then Theorem 4.1 implies that Pu :::; o.

Define edges joining pairs in V by

E := {[ x, yt x]: x E V, 1 :::; i :::; s, B = ± I}.

The graph G' := (V, E) will be called the base graph. Note that the base graph is not the same as the Cayley graph, G = (V(G), E(G». Let 7T: V(G) ~ V be the projection 7T(g,f3):=f3o. Let VI 'V 2' ... and UI ,U2, ... be infinite simple paths in G' starting at VI = ° and u 1 = yo, respectively, that are disjoint, except that VI may equal U I • Because G' is an infinite, connected, transitive graph, it is easy to show that such paths exist. For each j = 1,2, ... , fix an aj E Sr such that ajvj = Vj+1 and fix a f3j E Sr such that f3jUj = Uj+l.

Given a word W = WIW2 ... Wn with letters from S, let W- 1 denote the word W;IW;.\ ... w 1\ and let W(j), j E {O, 1, ... , n}, denote the group ele­ment W I W2 ... Wj.

Let Wy be a word in Sr representing (idv, y) with the property that for every V E V such that ((v) =1= idK , there is a j such that 7TW(j) = v. Let n be the length of Wy. Let WIT be the word a l a 2 ... an' and let Wf3 be the word f3I f32 ... f3n· Let W be the concatenation

W := WIT W,,-I Wy Wf3 Wf3-I Wy- I WIT W,,-I Wy Wf3 Wf3-I ,

and let N be the length of W. Let the letters in W be W = W IW 2 ••• wN •

Consider words of the form cfJ(X):= W1X1W2X2 ... WNXN' where X =

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(Xl"'" X N ) E sff = (SK)N. For any v E V, let J v be the set of j E

{I, 2, ... , N} such that 7TW(j) = v. Let 7TK : K \ r ~ K~ be the projection onto the first coordinate. The uniform measure 11-0 on the set of X E sff such that 4>(X) is a word representing (f, y) can be described as follows: DjE J v 7TK X/O) = f(v) (where the order of multiplication is the order of J v as a subset of~) and for every jl E J v , the random variables (Xj : j E J v , j =1= jl) are independent, uniform in SK and independent of (Xj : j t:/. J). Note that 4>(X) can also be thought of as a random path in G from (idv, id r ) to (f, y).

Let X and Y be i.i.d. with law 11-0' We want to bound the probability that there are k edges shared by the paths 4>(X) and 4>(Y). In fact, we shall bound the probability that these paths share at least k vertices. Given any j E {l, ... , N}, let H(j) be the set of v E V such that min J v <j < max J v ,

and let h(j) := I H(j)I. The choice of W ensures that

h(j) ~ min{j, N - j, n} - 1.

Because N = O(n), this gives

h(j) ~ C l min{j, N - j} - 1

for some universal constant c l > O. If (g, 8) is the element of V represented by the word W 1 X 1W 2 ••• WjXj' then (g(v): v E H(j)), are Li.d. uniform in K. Consequently, the probability that W 1 X 1W 2 •.• WjXj = w l Y l w 2 ••• Wj'~" as elements of K \ r, is at most IKI-max{h(j),h(j')}. If there are more than 8k

vertices common to 4>(X) and 4>(Y), then there must be a pair of indices j,j' E {k, k + 1, ... , N - k} such that W 1X 1W 2 ... WjXj = W 1Y 1W 2 ... wj'~" as elements of K \ r, or with a similar equality when the rightmost letter is dropped from either or both sides. The probability for that is at most

N-k N-k N-k N-k 4 L L IKI-max{h(j),h(j')} ~ 4 L L IKI-(h(j)+h(j'))/2 ~ c2 IKI- Cl k ,

j=k j'=k j=k j'=k

where C2 is a constant depending only on IKI. Consequently, as appeal to Lemma 6.9 completes the proof. D

Benjamini and Schramm (1996) conjectured that for any nonamenable Cayley graph G, we have Pc(G) < Pu(G). This is still open, but using our main result on cluster indistinguishability, we may show that a comparable statement fails for invariant percolation processes.

COROLLARY 6.10. There is an invariant deletion-tolerant percolation pro­cess (P, w) on a nonamenable Cayley graph that has only finite components a.s., but such that for every e> 0, the union of w with an independent Bernoulli(e) percolation process produces a unique infinite cluster a.s.

PROOF. Recall that Bernoulli(p) bond percolation on 7L 2 produces a.s. no infinite cluster iff p ~ 1/2 [see Grimmett (1989)]. Let 1F2 be the Cayley graph of the free group on two letters with the usual generating set. Let G := IF 2 X

7L 2 . Each edge of G joins vertices (xl> Yl) to (X 2 , Y2)' whether either Xl = X 2

and Yl - Y2' or Xl - X 2 and Yl = Y2' The former type of edge will be called a

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71. 2 edge, while the latter will be called an 1F2 edge. For each x E 1F2' the set of all edges [( x, Yl)' (x, Y2)] is called the 71. 2 fiber on x. Let w be Bernoulli(1/2) bond percolation on all the 71. 2 edges; it contains no 1F2 edge. Then w is invariant and deletion tolerant. When we take the union with an independent Bernoulli(e) process 7], the intersection of w U 7] with each 71. 2 fiber contains a.s. a unique infinite cluster. These clusters a.s. connect to each other in G. Thus, w U 7] contains a unique infinite cluster whose intersection with each 71. 2 fiber has an infinite component. Since w U 7] is insertion tolerant, cluster indistinguishability of w U 7] implies that it has no other infinite cluster. D

QUESTION 6.11. Is there an insertion-tolerant process with the property exhibited in Corollary 6.10?

The same line of thought is also used to prove the following theorem, which is an answer to a question posed by Yuval Peres (private communication) and motivated by the work of Salzano and Schonmann (1997, 1999) on contact processes.

THEOREM 6.12. Let G, H and H' be unimodular transitive graphs. As­sume that Hie Hand G is infinite. Then

In particular, when H' consists of a single vertex of H we get

PROOF. Let w be Bernoulli(p) bond percolation on G X H, where p > Pu(G X H'), and let r be the product of a unimodular transitive automor­phism group on G with a unimodular transitive automorphism group on H. Then r is a unimodular transitive automorphism group on G X H.

Let Z := h(G X H'): l' En. By Theorem 1.2 applied to G X H', for every Z E Z a.s. there is a unique infinite cluster, say Qz, of w Ii Z. Now let Z, Z' E Z. We claim that a.s. Qz and Qz' are in the same cluster of w. Indeed, since G is infinite, there are infinite sequences of vertices Vj E Z, vj E Z' such that dist(vj' vj) = inf{dist(v, Vi): v E Z, Vi E Z'} for all j. Because infj P[vj E Qz, vj E Qz'] > 0, with positive probability Vj E Qz and vj E Qz' for infinitely many j, and by Kolmogorov's 0-1 law this holds a.s. It is then clear that for some such j there is a connection in w between Vj and vj. It follows that a.s. there is a unique cluster Q of w that contains every Qz, Z E Z.

Let d' be the set of all pairs (C, W) E 2V(GXH) X fE(GXH) such that C meets an infinite component of Z Ii W for every Z E Z. We know that Q is the only cluster of w with (Q, w) Ed'. By Theorem 3.3, it follows then that Q is the only infinite cluster of w. D

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REMARK 6.13. The same theorem holds without the assumption of uni­modularity and with "transitive" replaced by "quasi-transitive." This follows similarly from the indistinguishability of robust properties proved by Haggstrom, Peres and Schonmann (1999).

REMARK 6.14. There are infinite Cayley graphs H' c H with PU(H') < Pu(H). For example, one may take H' = 7L 2 and let H be the free product 7L 2 * 7L 2 , as in Remark 1.3.

The work of Schonmann (1999a) was motivated by an analogy between percolation and contact processes. More specifically, the property of having complete convergence with survival for a contact process is closely analogous to having uniqueness of the infinite cluster for percolation. However, Remark 6.14 describes an instance where this analogy fails, because complete conver­gence with survival is a property which is monotone in the graph [Salzano and Schonmann (1997)].

Acknowledgments. We are grateful to Itai Benjamini, Olle Haggstrom, Yuval Peres and Roberto Schonmann for fruitful conversations and helpful advice.

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