Weights of uniform spanning forests on nonunimodular transitive graphs

Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. For a nonunimodular transitive graph on which the wired uniform spanning forest is not the same as the free one, we conjecture that each tree of the free uniform spanning forest is heavy and has branching number bigger than one. We prove this conjecture for grandparent graphs and free products of nonunimodular transitive graphs with one edge.


Introduction
Let G = (V (G), E(G)) be a locally finite, connected infinite graph. The wired uniform spanning forest (WUSF) and the free uniform spanning forest (FUSF) are weak limits of the uniform spanning tree measures on an exhaustion of the graph G, with wired and free boundary condition respectively. The WUSF and FUSF can be disconnected but each component is an infinite tree. Pemantle [19] proved for Z d , the WUSF is the same as FUSF and it is connected iff d ≤ 4. For more background on uniform spanning forests see [2] or Chapter 4 and 10 of [16].
Let Aut(G) denote the automorphism group of G. Suppose Aut(G) has a nonunimodular closed subgroup Γ ⊂ Aut(G) that acts transitively on G. In particular such G must be nonamenable [16,Proposition 8.14] and hence there are infinitely many components for WUSF on G [16,Corollary 10.27]. There is a unique left Haar measure | · | on Γ (up to a multiplicative constant). For each x ∈ V (G), let Γ x := {γ ∈ Γ : γx = x} denote the stabilizer of x and m(x) := |Γ x |. For a cluster C, we define m(C) := x∈C m(x) and call C a Γ-light cluster iff m(C) < ∞. For simplicity we will just say C is a light cluster if Γ is well-understood from the context.
Although each component of the WUSF or FUSF on G is an infinite tree, it may happen that some component is light and has branching number bigger than one. We prove that if there is nonunimodular subgroup Γ ⊂ Aut(G) that acts transitively on G, then each tree of the WUSF on G is Γ-light a.s. Theorem 1.1. Let Aut(G) denote the automorphism group of G. Suppose Aut(G) has a nonunimodular closed subgroup Γ ⊂ Aut(G) that acts transitively on G. Then each tree of the WUSF is Γ-light almost surely.
Benjamini, Lyons, Peres and Schramm gave several equivalent conditions for WUSF = FUSF; see [2,Theorem 7.3]. In particular, WUSF = FUSF for amenable transitive graphs. For the case WUSF = FUSF, we make the following conjecture. Conjecture 1.2. Suppose Aut(G) has a nonunimodular closed subgroup Γ ⊂ Aut(G) that acts transitively on G. Moreover WUSF = FUSF on G. Then each tree in the FUSF is Γ-heavy and has branching number bigger than one almost surely.
However we are only able to prove this for certain examples. Theorem 1.3. Let G be a grandparent graph or a free product of a nonunimodular transitive graph with one edge. Then each connected component of the FUSF on G is heavy and has branching number bigger than one almost surely.
In fact, FUSF on a grandparent graph is connected almost surely. The paper is organized as follows. In Section 2 we review Wilson's algorithm and the tilted mass-transport principle. In Section 3 we consider a toy model for WUSF, namely regular tree with a subgroup of automorphisms that fixes an end. We prove Theorem 1.1 in Section 4 and other related results on tilted volumes. In the remaining two sections we consider FUSF on grandparent graphs and free products of nonunimodular transitive graphs with Z 2 and prove Theorem 1.3.

Uniform spanning forests
If G is a finite connected graph, then it has finitely many spanning trees. The uniform spanning tree (UST) on G is the uniform measure on the set of spanning trees of G and denoted by UST(G). The Aldous-Broder algorithm and Wilson's algorithm are well-known methods to generate the UST on a finite graph G.
Suppose G = (V, E) is a locally finite, connected infinite graph. An exhaustion of G is a sequence of finite connected subgraphs G n = (V n , E n ) of G such that G n ⊂ G n+1 and G = G n . From the graph G, first identify the vertices outside G n to a single vertex, say z n , and then remove loops at z n but keep multiple edges. The graph obtained in this way is denoted by G W n . The weak limits of UST(G n ) and UST(G W n ) exist and they are independent of the choice of the exhaustion; for example see [16,Section 10.1]. We call the weak limits of UST(G n ) and UST(G W n ) free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF) respectively.
Járai and Redig [11] also introduced a v-WUSF on G, which can be roughly understood as a wired spanning forest with v wired to ∞. Suppose v ∈ V (G n ) for every G n in the above exhaustion and let G n be the graph obtained from G W n by identifying v and z n . Then the v-WUSF on G is the weak limit of UST( G n ).
The connected component of v in the v-WUSF is finite almost surely if G is a transient transitive graph [15,

Wilson's algorithm
Next we review Wilson's algorithm for generating WUSF on a transient graph G. This is called Wilson's method rooted at infinity [2,Theorem 5.1]. For finite graphs or recurrent graphs see section 3 of [2] for more details.
For a path w on G that visits every vertex finitely many times, we can define a loop erasure of w, namely let LE(w) be the self-avoiding path obtained by erasing the loops chronologically as they are created.
Suppose G is a transient graph. Fix an arbitrary ordering (v 1 , v 2 , . . .) of the vertices of G. Set F 0 = ∅. Given F n−1 for n ≥ 1, we construct F n as follows. If v n ∈ F n−1 , then let F n = F n−1 . Otherwise start a simple random walk from v n and let τ n denote the first hitting time of F n−1 . In particular, τ n = ∞ if the simple random walk never hits F n−1 . Let P n denote the random walk path stopped at τ n . Loop erase this random walk path and denote it by LE(P n ). Set F n = F n−1 ∪ LE(P n ). Finally set F := n F n . Then F has the law of WUSF on G and it does not depend on the ordering one chose.
Let G be a transient network and let F be a sample of WUSF on G generated by using Wilson's algorithm. Then for every edge e of F, there is a unique orientation such that e is crossed by the loop-erased random walk in that direction. The resulting oriented graph is called oriented wired uniform spanning forest and denoted by OWUSF. From OWUSF one can get WUSF by forgetting the orientation. It is also easy to see that in the OWUSF, every vertex has exactly one edge emanating from it.
Suppose G is a transient transitive graph. Then every tree in the WUSF on G will have only one end almost surely [15,Theorem 7.4]. Let F be a sample of OWUSF on G and let F be the spanning forest obtained from F by forgetting its orientation. Then F has the law of WUSF. For each vertex x ∈ V (G), let T x denote the connected component of x in F. Since T x is one-ended almost surely, there is a unique infinite ray η = (v 0 , v 1 , . . .) starting from v 0 = x representing the unique end of T x . Then the edge (v 0 , v 1 ) is also the unique edge emanating from x in F.
Given a sample F of WUSF on G and u ∈ V (G), we define the future of u to be the unique oriented ray starting from u and denote it by F(u, ∞). We take the convention that u ∈ F(u, ∞). We also define the past of u to the subgraph of F spanned by those vertices v ∈ F such that u ∈ F(v, ∞) and denote it by P(u).
Let F v be a sample of v-WUSF on G. Then one can generate F v by running Wilson's algorithm rooted at infinity but starting with F 0 = {v}, i.e. the forest with a single vertex v and no edge. We can also orient e ∈ F v as the way it is crossed by the loop-erased random walk in the Wilson's algorithm. Then each vertex but v has exactly one edge emanating from it. We can define the past and future for every vertex u in F v according to this orientation and denote them by P v (u) and F v (u, ∞) respectively. In particular, the future of v in F v is the single vertex v itself and the past of v is the connected component of v in F v . Let T v denote the tree containing v in F v . Most notation here coincides with the ones listed on page 15 of [9] for the reader's convenience.
Given a general oriented forest F of G, if there is an oriented path from u to v in F , then u is said to be in the past of v and v is said to be in the future of u. Let past F (v) denote the past of v in the oriented forest F , namely, the set of vertices which lie in the past of v in F . A key lemma we shall need is the following stochastic domination result.
Lemma 2.1 (Lemma 2.1 of [9]). Let G be an infinite network and F be a sample of WUSF on G. For each v ∈ V (G), let F v be an oriented v-WUSF of G. Suppose K is a finite set of vertices of G and define F (K) := u∈K F(u, ∞) and F v (K) := u∈K F v (u, ∞). Then for every u ∈ K and every increasing event A ⊂ {0, 1} E we have and similarly P past Fv\Fv(K) (u) ∈ A | F (K) ≤ P(T u ∈ A ). (2.2)

Nonunimodular transitive graphs and the tilted mass-transport principle
Next we recall the tilted mass-transport principle. We will restrict to transitive graphs. Suppose G is an infinite, locally finite connected graph and Γ ⊂ Aut(G) is a subgroup of automorphisms that acts transitively on G. For x, y ∈ V (G), let |Γ x y| denote the number of vertices in the set {γx : γ ∈ Γ x }, where Γ x is the stabilizer of x. There is a unique (up to a multiplicative constant) nonzero left Haar measure | · | on Γ and we denote by m(x) = |Γ x | the Haar measure of the stabilizer Γ x . Then a simple criterion for unimodularity of Γ is given as follows.
Proposition 2.2 (Trofimov [23]). Suppose Γ ⊂ Aut(G) acts transitively on G. Then Γ is unimodular if and only if for all x, y ∈ V (G), The grandparent graph and the Diestel-Leader graph DL(q, r) with q = r are typical examples of nonunimodular transitive graphs. For more examples see section 3 of [22].
Then from the above lemma 2.3 we know ∆ is a Γ diagonally invariant function, i.e. ∆(x, y) = ∆(γx, γy), ∀γ ∈ Γ. Another important property for the modular function is the cocycle identity : For more background on modular functions defined here see Section 2.1 of [8].
For a cluster K of G, a vertex v ∈ K and a parameter λ ∈ R, Hutchcroft [8] introduced the tilted volume as follows: The tilted mass-transport principle is a useful technique when dealing with nonunimodular transitive graphs. Actually the word 'tilted' can be omitted, and that the mass-transport principle was defined without the word [1]. The 'tilted mass-transport principle' was first used in [8] for a different way of writing the mass-transport principle.
Proposition 2.4 (Proposition 2.2 of [8]). With the same notations as in the above definition of modular function, suppose F : We will often use the following form of the above tilted mass-transport principle. Suppose ω is a Γ-invariant bond percolation process on G = (V, E). Suppose f : Then applying the tilted mass-transport principle with F (x, y) = E[f (x, y, ω)] one has When the percolation ω is clear from the context, we will often write f (x, y) instead of f (x, y, ω) in the above equation. Next we recall some terminology from percolation theory. Suppose G = (V, E) is a locally finite, connected graph. Let 2 E = {0, 1} E be the collection of all subsets η ⊂ E and let F E be the σ-field generated by sets of the form {η : e ∈ η} where e runs over all edges in E. A bond percolation on G is a pair (P, ω), where ω is a random element in 2 E and P is the law of ω. For simplicity sometimes we will just say ω is a bond percolation. The interested reader can refer to [16,Chapter 7 and 8] for more background on percolation theory. If the law P is invariant under a subgroup Γ of automorphisms, then we call (P, ω) a Γ-invariant percolation on G. In particular, WUSF and FUSF can be viewed as Aut(G)-invariant percolation processes on an infinite graph G.
Suppose (P, ω) is an automorphism-invariant percolation on G and E is the corresponding expectation operator. An edge e is called open if ω(e) = 1; otherwise it is called closed. For x ∈ V , the cluster of x is the connected component of x in the subgraph formed by open edges and denoted by C x . In case ω has the law of WUSF or FUSF, we will also denote the connected component of x by T x since each connected component is a tree.
Note the modular function satisfies ∆(y, x) = ∆(x, y) −1 . An immediate application of the tilted mass-transport principle is the following observation at the beginning of section 3 of [8]: Lemma 2.5. Suppose G is an infinite, locally finite connected graph and Γ is a subgroup of automorphisms that acts transitively on G. Suppose (P, ω) is a Γ-invariant percolation on G and x is an arbitrary vertex in G. Then the function f (λ) Note f is also convex, whence f is decreasing on (−∞, 1 2 ] and increasing on [ 1 2 , ∞). Finally we take a look at the level structure of nonunimodular transitive graphs; see also Section 2.2 of [8].
Suppose G is an infinite, locally finite connected graph and Γ is a nonunimodular subgroup of automorphisms that acts transitively on G. For x, y ∈ V (G), if x, y are neighbors then we write x ∼ y. Define If t ≥ s + t 0 and a simple path π := (v 0 , . . . , v n ) in G crosses the slab S s,t (v) in the sense that log ∆(v, v 0 ) ≤ s and log ∆(v, v n ) ≥ t then it must contain some vertex from the slab S s,t (v) by the definition of t 0 . Now we introduce a construction which will help us when applying the tilted masstransport principle. This construction comes from [22] and the set up here is borrowed from Section 2.2 of [8].
Fix an arbitrary vertex v 0 of G, let U v 0 be a uniform [0, 1] random variable independent of the WUSF and v-WUSF we shall consider. For every other v ∈ V (G), let Notice the law of the collection of random variables U : Given the collection of random variables U, the separating layers are defined to be We also call L n (v) the n-th slab relative to v. We will use E, P to denote the expectation operator and probability measure for the joint law of U and WUSF or v-WUSF on G. Note the cocycle identity (∆(x, v) · ∆(v, x) = 1) implies that In particular since 1 − U v has the same law as U x (uniform on [0, 1]) we have is invariant under the diagonal action of Γ, then like (2.3) we have the following form of tilted mass-transport principle (equation (2.2) in [8]) where F is a sample of WUSF and U := {U v : v ∈ V (G)} is defined as above.
If we let {ρ ↔ x} denote the event that ρ and x are in the same connected component in F and set f (ρ, x, F, U) = 1 {ρ↔x,x∈Ln(ρ)} , then where the last equality holds up to a factor of e ±t 0 . The notations we use in this paper are similar to the ones listed on page 15 of [9]. However since we use Γ to denote subgroup of automorphisms, we will use slightly different notations for the future of a vertex and the paths connecting two vertices in the WUSF and v-WUSF. We summarize the notations here for the reader's convenience.  x ↔ y The events that x and y are in the same connected component of F.
T v , T v The connected components of v in the WUSF and v-WUSF respectively.
The future of x in F and F v respectively.
π(x, y), π v (x, y) The path from x to y in F and F v respectively if x, y are in the same connected component; otherwise they are defined to be empty sets.
{X x k } k≥0 A simple random walk on G starting from x. For y = z, we take X y , X z to be independent. σ x y The first visit time of y by a simple random walk starting from x. τ x −n The last visit time of L −n (x) by a simple random walk starting from x.
≍ This denotes an equality that holds up to positive multiplicative constants. More precisely, for two positive functions f, g on (0, ∞), f (R) ≍ g(R) means that there exists R 0 > 0 and c 1 , c 2 > 0 such that c 1 f (R) ≤ g(R) ≤ c 2 f (R) for all R ≥ R 0 , and the implicit constants c 1 , c 2 , R 0 only depend on the graph.
≍ λ Similar to the above, but the implicit constants also depend on λ.

A toy model
Before giving the proof of Theorem 1.1, we first look at the following simple toy model. On the one hand, because of the simple tree structure of the underlying graph, the wired uniform spanning forest is relative easy to analyze in this toy model ( Let G be a regular tree T b+1 of degree b + 1, where b ≥ 2. Given an end ξ of G, let Γ be the subgroup of automorphisms that fixes this end ξ. One can check that Γ acts transitively on G and Γ is nonunimodular by Proposition 2.2. The toy model is just this regular tree G together with the subgroup Γ. The FUSF on T b+1 is trivial, namely it equals the tree itself almost surely. The WUSF on T b+1 can be generated using Wilson's algorithm: For a vertex x ∈ T b+1 , start a simple random walk from x on the tree T b+1 , loop erase this random walk path chronologically to get a ray η starting from x. Let y 1 , . . . , y b denote the neighbors of x not on the ray η. For i = 1, . . . , b start independent simple random walks from the vertex y i and let A i := { the simple random walk starting from y i hits x}. Obviously given η, the events A i are independent. If A i occurs then put the edge connecting y i and x to the WUSF; otherwise y i , x will be in different components in the WUSF. On the event A i , we add only the edge (y i , x) to the WUSF and repeat the process for y i . Obviously P(A i |η) = 1 b . Thus the tree of x is the union of the ray η, and independent random trees attached to the vertices of η. For the root x, the random tree attached is a critical Galton-Watson tree with binomial progeny distribution (b, 1/b). For each other vertex on the ray η, the first generation of the random tree has binomial distribution (b − 1, 1/b) while the subsequent generations has the law of a critical Galton-Watson tree with binomial progeny distribution (b, 1/b). The analysis can be extended to give the whole WUSF easily. This description comes from Section 11 of [2] and it is due to Häggström (1998) [6].
In particular, if one takes λ = 1, one has T x is light almost surely.
We start with a simple lemma. In particular, all moments of X are finite.
Proof. If λ > 1, then the result is trivial since Z n ≤ b n almost surely. For the case λ ∈ (0, 1], we follow a similar strategy as Exercise 5.33 from [16]. Then g 0 (s) = s and conditioning on Z 1 we get is the probability generating function of this Galton-Watson process. Set The claim is trivial for n = 0. Suppose it is true for n = k. Then for n = k + 1, s ∈ [1, s 0 ], one has s 1/b λ ∈ [1, s 0 ] and 1 ≤ s Thus using the recursive relation one has Note the recursive relation also yields g ∞ (s) = sf (g ∞ (s 1/b λ )). Hence g ∞ (s) < ∞ for s ∈ [1, s b λ 0 ]. Repeating this we know g ∞ (s) < ∞ for all s ∈ [1, ∞). This gives the conclusion.
Given a regular tree T b+1 with b ≥ 2, fix a root o ∈ T b+1 and an end ξ. For each vertex v ∈ T b+1 , there is a unique ray η v starting from v and representing ξ. We call the unique neighbor of v on the ray η v the parent of v with respect to ξ, and denote it by v − . For x, y ∈ T b+1 , we define x y (w.r.t. ξ) to be the first intersection vertex of η x and η y . We define the Busemann function h : T b+1 → Z with respect to o, ξ as follows: where d(u, v) is the graph distance between u and v on T b+1 . We also define the horocycles H k with respect to o, ξ as H k = H k (o, ξ) := {x ∈ T b+1 : h(x) = k}. This gives a level structure of T b+1 . Note changing the base o will only change the Busemann function by adding a constant. For two vertices x, y, if x ∈ H k and y ∈ H j for some k < j, then we say that x lies in a low level of the tree relative to y (w.r.t. the fixed end ξ).
For k ∈ {b − 2, b − 1, b} and a vertex x ∈ T b+1 , we denote by T k rooted at x a random subtree of the regular tree T b+1 with a distinguished end ξ that satisfies the following conditions: 1. the root x is the highest vertex in this subtree in the sense that y lies in a low level relative to x for every other vertex y in this subtree; 2. it has the same law as the family tree of a Galton-Watson process with progeny distribution Bin(b, 1 b ) except that the first generation of T k has distribution Bin(k, 1 b ) instead of Bin(b, 1 b ). In particular, T b has the same law as a critical Galton-Watson process with progeny distribution Bin(b, 1 b ). Suppose T b is rooted at x. Then for every λ > 0, | T b | x,λ has the same law as the random variable X in Lemma 3.3.
Fix x ∈ T b+1 , let x 0 denote the highest vertex in the future of x, i.e., the vertex x 0 is the unique one in F x such that ∆(x, Let A k denote the event that ∆(x, x 0 ) = b k for k ≥ 0. Let y denote the highest vertex in the past of x 0 and B k,k ′ be the event that From the description of WUSF on the regular tree at the beginning of this section we have the following observation.
Observation: On the event B k,k ′ , the random tree T x in the WUSF has the same law as the random tree constructed below.
If k = k ′ = 0, then x = x 0 = y and we just attach independently to each vertex in the future of x a random tree with the same law as T b−1 (namely, identifying the root of the T b−1 to the vertex).
If k = 0, k ′ > 0, then x = x 0 = y and we just attach a random tree with law of T b−1 to each vertex in the future of y independently.
If k > 0, k ′ = 0, then x = x 0 = y and we attach a random tree with law of T b−1 to each vertex in the future of x independently except at x and x 0 . At x 0 we attach a random tree with law of T b−2 . At x we attach a random tree with law of T b .
If k > 0, k ′ > 0, then x, x 0 , y are all different and we attach a random tree with law of T b−1 to each vertex in the union of the future of x, y independently except at x, x 0 . At x 0 we attach a random tree with law of T b−2 . At x we attach a random tree with law of T b .
Proof of Proposition 3.1. Let η x = (x 0 , x 1 , x 2 , . . .) be the ray starting from x representing the end ξ. Since the WUSF is invariant under the whole automorphism group of T b+1 , and there are (b + 1)b k−1 vertices on T b+1 with graph distance k to x, by symmetry we have Combining with the fact that P(x 0 ∈ F x ) = 1, one has In particular, we see that the future of x, F x = (v 0 , . . . , v n ) will eventually go down to the lower levels in the sense that ∆(v 0 , v n ) ↓ 0 as n tends to infinity. Thus for λ ≤ 0, s. In the rest proof we assume λ > 0. From the description of the WUSF at the beginning of this section one has Note that for a random tree T rooted at o with law Therefore, on the event B k,k ′ , from the above observation we have .
Therefore we get the upper bound part for (3.1): The lower bound of (3.1) is much easier.
The following is a simple application of the symmetry of T b+1 .
Proposition 3.4. Let P denote the law of WUSF on the regular tree T b+1 , where b ≥ 2. Let x, y ∈ V (G) be two arbitrary vertices and {x ↔ y} denote the event that x, y are in the same tree of WUSF. Set n = dist(x, y) to be the graph distance of x, y. Then Proof. The n = 0 case is trivial. We assume n > 0 in the following. By symmetry we can assume y is the one such that ∆(x, y) = b n . With the same notation as in the proof of proposition 3.1, one has Then simple calculation shows that P[

The lightness for trees in WUSF
Throughout the section we will assume G is an infinite, locally finite connected graph and Γ is a closed nonunimodular subgroup of automorphisms that acts transitively on G. We start with a lemma about simple random walk on G.

Simple random walk on nonunimodular transitive graphs
is a simple random walk on G and λ ∈ R. Then one has then the tilted mass-transport principle and the cocycle identity yield that In particular, taking λ = 0 one has that Since the above two equations are true for all x ∈ V (G), one has Note for λ ∈ (0, 1), the function x → x λ is a concave function on [0, ∞]. By Jensen's inequality one has that Using the cocycle identity and independence, one has . By Jensen's inequality one has that Lemma 4.1 implies that simple random walk on nonunimodular transitive graphs has a drift towards the lower slabs.
is a simple random walk on G starting at x. Let n ∨ 1 = max{1, n} denote the maximum of n and 1. Then there exists a constant c 0 > 0 such that for every n ≥ 0, For an integer n ≥ 0, let τ x −n := sup{k : X k ∈ L −n (x)} be the last visit time of the slab L −n (x). Then there exist constants c 1 , c 2 , c 3 > 0 such that are independent, identically distributed random variables, law of large numbers and Lemma 4.1 then imply that Also there exists a constant c ∈ (0, 1) such that Then the strong Markov property implies that P[N −n ≥ k] ≤ (1 − c) k−1 for every k ≥ 1, whence the first inequality in (4.2) holds for any c 0 ≥ 1 c . Equation (4.4) also implies that τ x −n < ∞ almost surely.
By large deviation principle (e.g. Lemma 2.6.1 and 2.6.3 in [4]) we know there exists constants c 4 , c 5 > 0 such that Combining (4.9) and (4.10) one has that there exists a constant c 8 > 0 such that

First moment of intersections with a slab and the expected tilted volumes
In this subsection we study the expectation of the number of vertices in the intersection of T x with L n (x) and corresponding quantities for T x , F(x, ∞) and P(x). Based on these first moments we can also derive estimates on the tilted volumes. There exist constants c 9 > 0, c 10 > 1 such that for every n ≥ 1 and The following lemma is a simple application of Wilson's algorithm. (4.14) Proof. The inequality (4.13) comes from the proof of Theorem 13.1 in [2] and we recall it for the reader's convenience. From the reversibility of simple random walk and the regularity of G, one has that for any vertices x, y and k ≤ m, By the Wilson's algorithm, If we use Wilson's algorithm to sample F x starting with the simple random walk X y , then we have that x denotes the first visit time of x by the simple random walk X y . The reversibility of simple random walk on G and the regularity of G then imply that . Therefore one has that Proof of Proposition 4.3. We first establish the upper bound in (4.11).
For n ≥ 0, one has that Taking c 10 = 2c 0 one has the upper bounds in (4.11).
≥ c 9 n for n ≥ 1 using a similar strategy as the one used in the proof of Theorem 13.1 of [2].
the Green function for simple random walk on G. Since Γ is a closed subgroup of automorphisms that acts transitively on G, G is nonamenable (Proposition 8.14 of [16]) and hence the spectral radius ρ(G) < 1. Thus by the Varopoulos-Carne bound (see Theorem 13.4 in [16]) there exists a constant C > 0 such that We use Wilson's algorithm to generate the WUSF sample F by starting with a simple random walk. Then the future of x is a subset of X x . By (4.4) one has that 1 ≤ |F(x, ∞) ∩ L −n (x)| < ∞ for every n ≥ 0. Also We shall show that there exists a constant c 9 > 0 such that for any set S ⊂ B(x, n) that contains exactly one vertex at distance k from x for 1 ≤ k ≤ n, the following inequality holds.
where L y (S) to be the total occupation time of S by X y , that is, Since simple random walk X w on G visits S s,t (w) for every t ≤ 0, t − s ≥ t 0 almost surely, one has that y∈L −n (x) g(w, y) ≥ 1, ∀w ∈ S. Thus y∈L −n (x),w∈S g(w, y) ≥ |S| = n. Hence for n ≥ 1, This together with (4.18) implies (4.16) with c 9 = 1 c 11 . Thus for n ≥ 1 by conditioning on F(x, ∞) one has that  The inequalities (4.12) follow from (4.11) and (2.7).
Proposition 4.5. There exists a constant c 12 > 0 such that for every n ≥ 0, There exists constants c 13 , c 14 > 0 such that for every n ≥ 1, Moreover one also has that and Proof. We will prove (4.20), the upper bound in (4.22) and the lower bound in (4.21). The rest will follow from these inequalities and the tilted mass-transport principle.
The upper bound in (4.20) also follows from the above inclusion Next we prove the upper bound in (4.22). By the reversibility of simple random walk and the transitivity of G, one has P[σ x where the last equality holds up a multiplicative constant e ±t 0 .
Taking c 14 = c 0 e t 0 one has the upper bound in (4.22). Finally we prove the lower bound in (4.21). Let y ∈ V and let X y be a simple random walk started at y. Sample F starting with the simple random walk X y . Recall that σ y x denotes the first visit time of x by X y . Let A (y, x) be the event that σ y x < ∞, that the sets {X y m : 0 ≤ m < σ y x } and {X y m : m ≥ σ y x } are disjoint, so that y ∈ P(x) on the event A (y, x) and hence Let Y x be an independent simple random walk also started at x. We also use X x to denote the set of vertices {X x m : m ≥ 0} visited by the random walk X x and write similarly Since X x hits L −n (x) for every n ≥ 0 almost surely, one has that Since G is a nonamenable transitive graph, there exists a positive constant c 13 such that [16,Theorem 10.24]). Thus we have the lower bound in (4.21). Just like (2.7), the tilted mass-transport principle implies that for any n ∈ Z, where the last equality holds up to a factor e ±t 0 . This together with (4.20) implies (4.23); this together with the upper bound in (4.22) implies the upper bound in (4.21) and this together with the lower bound in (4.21) implies the lower bound in (4.22) and Proof. As in the proof of Lemma 4.4, by Wilson's algorithm and the reversibility of simple random walk one has that where σ y x is the first visit time of x by the simple random walk X y . Hence Since simple random walk X x visits L −n (x) for every n ≥ 0 almost surely, one has y∈V 1 {y∈L −n (x)} · 1 {σ x y <∞} ≥ 1 a.s.. Then (4.27) implies that the lower bound in (4.25) holds with c 15 = 1.
On the other hand, Hence the upper bound in (4.25) holds with c 16 = c 0 . By Lemma 4.4 one has P[y ∈ T x ] = P[x ∈ T y ]. Then the tilted mass-transport principle gives the relation between E[|T x ∩ L −n (x)|] and E[|T x ∩ L n (x)|], namely (4.24). This together with (4.25) yields (4.26).
For the intersection of a simple random walk trajectory with a slab, we have the following results.
Proposition 4.7. Let X x denote a simple random walk on the transitive graph G started from x. Then for each k ≥ 1 one has that and In particular, one has that This gives the upper bound of (4.30) and (4.28) in the case k = 1.
This gives the lower bound of (4.28) in the case k = 1. The lower bound of (4.28) in the general case is then immediate since it is increasing in k.
Write N n = ∞ m=0 1 {X x m ∈Ln(x)} for n ∈ Z as in the proof of Corollary 4.2. From the strong Markov property one has that there exists a constant c ∈ (0, 1) . Therefore we have the upper bound of (4.28): By Markov's inequality one has the lower bound of (4.30), namely It remains to show (4.29). For n ≥ 0, the simple random walk X x hits every L −n (x) almost surely. Thus P[N −n > 0] = 1 and Next we extend Corollary 3.2 to all nonunimodular transitive graphs. Proof. If we decompose T x according to its intersection with different slabs, we get Then (4.32) together with (4.11) and (4.12) yields the conclusion.
Similarly Proposition 4.6 and 4.5 yield the following proposition and we omit its proof. Similarly,

Proof of Theorem 1.1
We first extend Lemma 5.2 of [22] to WUSF. It suffices to show that on the event H(x), for all n ∈ Z, T x intersects the slab S s,t (v) in infinitely many vertices.
We sample T x using the Wilson's algorithm. Starting a simple random walk {X n } n≥0 from x. Let η = (v 0 = x, v 1 , v 2 , . . .) denote the loop-erased path of the simple random walk path. Then η = F x . Corollary 4.2 implies that η goes down to the lower direction eventually, i.e. log ∆(v 0 , v n ) → −∞ as n → ∞.
Since η is the future of x and T x has only one end, then T x is formed by attaching finite trees to η at its vertices v i 's.
Set A := sup{∆(x, y) : y ∈ T x }. If A = ∞, since the trees attached to v i are finite and ∆(v 0 , v n ) → 0, there exists a subsequence n k such that the trees attached at v n k will intersect the slab L k (x). In particular, any fixed slab S s,t (v) with t − s ≥ t 0 intersects all the finite trees attached at v n k for sufficiently large k. Hence T x intersects S s,t (v) in infinitely many vertices.
If A < ∞, define N := |{y : y ∈ T x , ∆(x, y) = A}|. If N < ∞, then define the following mass transport, for each v ∈ T x sending mass 1 N from v to each of the N highest vertices in T x . Now the expected mass sent out is at most one while the expected mass received is infinite since T x is heavy with positive probability. This contradicts with the tilted mass-transport principle.
If A < ∞, N = ∞, there exists y ∈ V (G) such that y is one of the highest vertex in T x with positive probability. Let e = (y, z) be an edge in G such that ∆(y, z) > 1. Since y is the highest vertex in T x , e / ∈ T x . By update-tolerance for WUSF ([7, Corollary 3.5]), one has A < ∞, N < ∞ occurs with positive probability and this reduces to the previous case. (The update U(T x , e) adds edge e and the past of z to T x , deletes the edge connecting z to F(z, ∞)\{z}.) We conjecture its counterpart for FUSF of Lemma 4.10 is also true.

The decay of probabilities for certain events
Like the expectation for tilted volumes, we can extend Proposition 3.1 to general nonunimodular transitive graphs (up to some logarithm corrections).
Proposition 4.12. The tilted volume |T x | x,λ has the following properties.  We shall need estimates on high moments of |T x ∩ L n (x)| and |T x ∩ L n (x)|.
Proposition 4.14. There exists c 17 > 0 such that for all n ≥ 0, k ≥ 2, one has and Proof. Let π x (x, y) denote the path connecting x and y in T x . In the case that y / ∈ T x we set π x (x, y) = ∅. Writing A n = A n (x; y 1 , . . . , where the second summation η i :x→y i runs over all the finite simple paths from x to y i for i = 1, . . . , k − 1. Suppose {η i : i = 1, . . . , k−1} is a set of simple paths such that P[∩ k−1 i=1 {π(x, y i ) = η i }] > 0. If we sample the x-WUSF using Wilson's algorithm starting with simple random walks X y 1 , . . . , X y k and the event ∩ k−1 i=1 {π(x, y i ) = η i } occurs, then y k ∈ T x if and only if the simple random walk X y k hits ∪ k−1 i=1 η i . Therefore using union bounds, we have where |η i | + 1 denotes the number of vertices in the simple path η i . Combining the above estimates we have If we sample T x using Wilson's algorithm beginning with a simple random walk {X y m } m≥0 started at X y 0 = y, then |π x (x, y)| + 1 ≤ (1 + σ y x ) · 1 {σ y x <∞} , where σ y x is the first visit time of x by the simple random walk {X x m } m≥0 . The reversibility of simple random walk implies that Then (4.39) implies that where c 18 = 3c 16 e t 0 .
Applying the tilted mass-transport principle one has for n ≥ 0 Taking c 17 = 2c 18 e t 0 c 0 = 6e 2t 0 c 0 c 16 we get (4.35). For (4.36), note that (4.41) also holds for negative n. Hence for n ≥ 0 one has Using the estimates for E[|T x ∩ L n (x)| k ] we can derive upper bounds for E[|T x ∩ L n (x)| k ]. Recall U = {U v : v ∈ V } are the labels we used to define the slabs.
Proof. The proof is similar to the one of Lemma 6.8 in [9]. We present the details for reader's convenience. In the following we fix n ∈ Z. Given the future F(x, ∞) = (x 0 , x 1 , x 2 , . . .) of x and i ≥ 0, we call the connected component of T x \{x 0 , x 1 , . . . , x i−1 , x i+1 , x i+2 , . . .} the i-th bush of T x and denote it by Bush i (x). Denote by N i the number of vertices in L n (x) ∩ Bush i (x). In particular, Bush 0 (x) = P(x). By the lightness of T x , almost surely only finitely many N i 's are nonzero.
Notice that (4.44) For each i ≥ 0, let Y i = {y i,1 , . . . , y i,k i } be a finite (possibly with multiplicity) collection of vertices of G and W i = {w i,1 , . . . , w i,m i } the corresponding set of vertices of Y i without multiplicity. In particular if k i = 0 then m i = 0 and W i is an empty set. Let A i be the event that for every vertex w ∈ W i , w ∈ Bush i (x).
Let {X i,j : i ≥ 0, m i = 0, 1 ≤ j ≤ m i } be a collection of independent simple random walks, independent of F(x, ∞), such that X i,j 0 = w i,j for each i ≥ 0 with m i = 0 and 1 ≤ j ≤ m i .
For each i ≥ 0 such that m i = 0, let B i be the event that, if we sample x i -WUSF using Wilson's algorithm, starting with the random walks X i,1 , . . . , X i,m i , then for every vertex It is easy to see that if we sample F conditional on F(x, ∞) using Wilson's algorithm starting with X 0,1 , . . . , X 0,m 0 , then X 1,1 , . . . , X 1,m 1 , and so on, then we have A i ⊂ B i . Therefore By Proposition 4.6 and 4.14 one has that Let m(x i ) denote the integer m such that x i ∈ L m (x) (if there are two such m's, taking the smaller one). Then using Lemma 4.15 one has that for any n ∈ Z, k ≥ 2 for 1 ≤ t ≤ k. By Proposition 4.5 one has that for n ≥ 0 Similarly if n < 0, then By Hölder's inequality one has that for any t ≥ 2 and nonnegative sequences (a j ) j∈Z , For 2 ≤ t ≤ k, applying the above Hölder's inequality with a j = 1 |j|∨1 and b j = (|j| ∨ 1) × (|n − j| ∨ 1) e t 0 j e t 0 n ∧ 1 |F(x, ∞) ∩ L j (x)| one has that is a finite constant. From the proof of Proposition 4.7 we know that there exists a positive constant c 20 such Hence for n ≥ 0 and 2 ≤ t ≤ k where the constant C t and the details of the last inequality is given in the following. For n ≥ 0 and 2 ≤ t ≤ k where α t = 2 t ∞ j=1 j 2t e −tt 0 j . Similarly, and where β t = 2 t+1 ∞ j=1 j 2t exp(−t 0 j). Combining them one has that for n ≥ 0 and 2 ≤ t ≤ k, with C t := C t c t 20 t!(α t + 1 + β t ). Therefore for n ≥ 0 and k ≥ 2 by (4.48) one has For λ ∈ (0, 1), Proposition 4.8 then implies |T x | x,λ < ∞ almost surely.
For λ ≥ 1, we can write The first summation is finite since T x is light and then it is a summation over finitely many vertex. The second summation is bound above by y∈Tx,∆(x,y)<1 ∆(x, y) 1 2 ≤ |T x | x, 1 2 < ∞. Thus for λ ≥ 1, one also has |T x | x,λ < ∞ almost surely. Then Markov's inequality yields that (4.55) Using the union bounds we have By Proposition 4.3 we have (4.58) Taking a positive integer k = ⌈ 1 λ ⌉ + 1 > 1 λ , by Markov's inequality one has (4.59) By Proposition 4.16 we have Another natural question to consider is the decay of the probability that a vertex connecting to a high slab. Similar question was analyzed for Bernoulli percolation [8,Lemma 5.2].
Proposition 4.17. Let {T x ∩ L n (x) = ∅} denote the event that T x has a nonempty intersection with the slab L n (x). Then for n ≥ 1 (4.61) In particular Proof. On the one hand, for n ≥ 1 one has On the other hand, Markov's inequality implies that n exp(−t 0 n).
Combining the above two inequalities one has the conclusion.
Since the tree T x in the x-WUSF is almost surely finite, P[T x ∩ L n (x) = ∅] and P[T x ∩ L −n (x) = ∅] both decay to zero as n → ∞. Moreover for n ≥ 1, the decay of P[T x ∩ L −n (x) = ∅] is much slower.
Proof. The proof of (4.63) is similar to (4.61) and thus we omit it. By Markov's inequality one has the lower bound in (4.65), namely Let diam int (T x ) denote the intrinsic diameter of the finite tree T x . If T x ∩ L −n (x) = ∅, then diam int (T x ) ≥ n − 1. Hence for n ≥ 1 one has where the last inequality is due to Theorem 7.1 of [9].
For the future F(x, ∞), it intersects every L −n (x), n ≥ 0 almost surely, and the probability P[F(x, ∞) ∩ L n (x) = ∅] decays exponentially. For the past P(x), it is a finite tree and hence P[P(x) ∩ L n (x) = ∅] and P[P(x) ∩ L −n (x) = ∅] both decay to zero as n tending to infinity. We summarize these in the following Proposition.
Proposition 4.19. Suppose n ≥ 0. For the future one has the following asymptotic behavior: For the past one has and The proof of the lower bound of (4.68) is similar to the one of P[T x ∩ L −n (x) = ∅] using Theorem 1.1 in [9] instead.
Since the tree T x in the x-WUSF is almost surely finite, |T x | x,λ < ∞ for all λ ∈ R.
Proposition 4.20. The tail probability P[|T x | x,λ ≥ R] satisfies the following inequalities.
Remark 4.21. As λ → 0, the tilted volume |T x | x,λ → |T x |. So for properly related λ and R, the probabilities P[|T x | x,λ ≥ R] and P[|T x | ≥ R] should be close. This is due to the dependence of λ implicitly in (4.69) and (4.71).
For the future and past of x in the WUSF sample F, one can also consider the tail probability for the titled volumes. We summarize the results in the following proposition and omit the proofs since they are similar to the ones for T x and T x respectively.
If λ < 0, then 4.5 On the asymptotic behavior of T x ∩ L −n (x).
Let B int (x, n) denote the intrinsic ball of radius n centered at x in T x . The size of the intrinsic ball is well-understood; see Section 6 of [9] for more details. In particular, Corollary 6.4, Remark 6.5 and Corollary 6.11 in [9] studied the almost sure asymptotic behavior of |B int (x, n)|.
In the present paper, T x ∩L n (x) plays a similar role as B int (x, n). Since T x is light, almost surely T x ∩ L n (x) = ∅ if n is large. So we are only interested in the behavior of |T x ∩ L −n (x)| when n is large.

Proof.
Let v be the last vertex on the future of x such that ∆(v, x) = 1. In particular, if x is the highest point on its future, then v = x. Denote the path on the future of x from x to v by π(x, v).Let N be the largest number k such that there is some vertex x i in π(x, v)\{v} such that the bush at x i intersects L −k (x). Since all the bushes are finite trees almost surely, N is finite almost surely. For n > N, |T x ∩L −n (x)| = 1 + Z n , where Z n is the size of the n-th generation of a critical branching process with immigration. Indeed the 1 on the right hand side is the contribution of the future and Z n is given as follows: where Y i,j 's are i.i.d. random variables with the progeny distribution Bin(b, 1 b ) and the immigration I i 's are i.i.d. random variables with distribution Bin(b − 1, 1 b ). Since we are only interested in the normalized asymptotic behavior, it suffices to show Zn n converges in distribution to a random variable with Gamma distribution. This is a classical result regarding critical branching process with immigration; for example see Theorem 3 in [17].
For the almost sure result, it suffices to show that log Zn log n → 1 a.s., which is due to Theorem 1.1 of [24].
In view of the above proposition, one might ask whether |Tx∩L −n (x)| n converges almost surely to a random variable with Gamma distribution. However Proposition 4.23 together with the following proposition imply that the limit does not exist. The inequality (4.72) is a direct consequence of Remark 2.2 of [24]. Next we show (4.73).
If I i = t > 0, we write GW i,1 , . . . , GW i,t to be the descendant trees of these t people immigrated in generation i. We use |GW i,j ∩ L −2 k (x)| to denote the contribution of GW i,j to Z 2 k .
For a constant c > 0, define the events A k (c) for k ≥ 2 as follows: We claim that for some small enough constant c, ∞ k=2 P[A k (c)] = ∞. Since there A k 's are independent events, by Borel-Cantelli lemma one has P[A k i.o. ] = 1. Notice on the event A k (c), Z 2 k ≥ c · 2 k log k. Therefore (4.73) holds.
Now it remains to prove the claim that . By the Kolmogorov's estimate (see for example Theorem 12.7 of [16]) one has where Using independence of B i,k (c) for different i's, one has Remark 4.25. This proposition holds for more general critical branching process with immigration if the conditions in Theorem 1.1 of [24] are satisfied with δ = 1 (to use Theorem 3.3 of [18]).

Proposition 4.23 and 4.24 implies that almost surely lim n→∞
|Tx∩L −n (x)| n does not exist. Proposition 4.23 also implies that for the toy model, for every ε > 0, a.s. |T x ∩L −n (x)| n 1−ε . Can one improve this lower bound for the toy model? Question 4.26. What is the asymptotic behavior of |T x ∩ L −n (x)| as n → ∞ for WUSF on general nonunimodular transitive graphs? 5 FUSF on Diestel-Leader graphs and grandparent graphs

FUSF=WUSF on Diestel-Leader graphs
Diestel-Leader graphs are a family of transitive graphs constructed by Diestel and Leader in [3] as possible examples of transitive graphs that are not roughly isometric to any Cayley graph. Later Eskin, Fisher, and Whyte (2012) showed that the Diestel-Leader graph DL(q, r) with q = r is indeed not roughly isometric to any Cayley graph.
Next we give a precise definition of the Diestel-Leader graph DL(q, r). For more details see [26]. Recall the definition of Busemann function h in Section 3. Suppose q ≥ r ≥ 2 are two positive integers and T q+1 , T r+1 are two regular trees with degree q + 1, r + 1 respectively. Fix roots o 1 , o 2 and reference ends ω 1 , ω 2 for T q+1 , T r+1 respectively.
The neighborhood relation is given by x 1 x 2 ∼ y 1 y 2 if and only if x 1 ∼ y 1 and y 1 ∼ y 2 .
A way of visualizing of DL(q, r) is described on page 418 of [26]; see also Figure 2 there for an example DL(2, 2).
A (P -)harmonic function h is the one that satisfies P h = h. Let P 1 , P 2 denote the projection of P on T q+1 and T r+1 respectively: otherwise. Woess proved the following decomposition theorem about positive harmonic functions. Theorem 5.2 (Theorem 2.3 of [26]). If h is a non-negative P -harmonic function on DL(q, r), then there are non-negative P i -harmonic functions h i , i = 1, 2 on T q+1 and T r+1 respectively, such that h(x 1 x 2 ) = h 1 (x 1 ) + h 2 (x 2 ), ∀x 1 x 2 ∈ DL(q, r)

Proposition 5.3. FUSF is the same as WUSF on Diestel-Leader graphs.
Proof. By Theorem 7.3 of [2], it suffices to show that every harmonic Dirichlet fuction on DL(q, r) is a constant function and this is an easy consequence of Theorem 5.2. In fact, suppose there is non-constant harmonic Dirichlet functions on DL(q, r). Then there is non-constant bounded harmonic Dirichlet functions on DL(q, r) (Theorem 3.73 of [20]), whence there is non-constant non-negative harmonic Dirichlet functions on DL(q, r).
Let h be a non-constant non-negative harmonic Dirichlet function on DL(q, r). By Theorem 5.2, there exists non-negative functions h 1 and h 2 on T q+1 and T r+1 such that Since h is not a constant function, at least one of h 1 , h 2 is also not a constant. Without loss of generality, we assume that h 1 is not a constant. Suppose x 1 , y 1 ∈ T q+1 are two neighboring vertices such that y − 1 = x 1 and h 1 (x 1 ) = h 1 (y 1 ). We first show that for any z 1 ∈ T q+1 such that z − 1 = x 1 , one has h 1 (z 1 ) = h 1 (y 1 ). Suppose z 1 = y 1 . Since h is a harmonic Dirichlet function, Since there are infinitely many x 2 ∈ T r+1 such that h(x 2 ) = −h(x 1 ), we have h 1 (y 1 ) = h 1 (z 1 ). From this and the fact that h 1 is P 1 harmonic, one has that h 1 is constant on each horocycle of T q+1 . Similarly h 2 is also a constant on each horocycle of T r+1 , whence h(x 1 x 2 ) only depends on which horocycle x 1 lies in. Then h must be a constant function on DL(q, r) to have finite Dirichlet energy. This contradicts with the choice that h is a non-constant function.
Now by Proposition 5.3, the study of FUSF on DL(qr) reduces to WUSF. In particular we know that each component of FUSF on DL(q, r) with q > r ≥ 2 is one-ended and light almost surely.

FUSF on grandparent graphs
We first recall the definition of grandparent graphs. For more details see Section 7.1 of [16].
Consider a regular tree T b+1 with degree b + 1 ≥ 3. Let ξ be a fixed end of T b+1 . As in the previous subsection, for each v in T b+1 , there is a unique ray η v = (v 0 , v 1 , v 2 , . . .) that represents the end ξ starting at v 0 = v. We call v 2 the ξ-grandparent of v. Throughout this subsection we let G be the graph obtained from T b+1 by adding the edges (v, v 2 ) between v and its ξ-grandparent for all v ∈ T b+1 . It is well known that G is a nonunimodular transitive graph. For two vertices x, y in G, we denote by d G (x, y), d T (x, y) the graph distance of x, y in G and T b+1 respectively.
Fix a base point v and let η v = (v 0 , v 1 , v 2 , . . .) be the unique ray that represents the end ξ starting at v 0 = v. We consider the following exhaustion of G. For n ≥ 1, let G n be the subgraph of G induced by vertices {x : For k, n ≥ 1, let P k,n denote the set of self-avoiding paths that connecting v 0 , v 1 in G n with length k.
We start with an estimate on the size of P k,n .
Proof. For a self-avoiding path π = (w 0 , w 1 , . . . , w k ) in G, if ∆(w i , w i+1 ) = b −2 , then we say the step from w i to w i+1 is downward 2 levels, and denote it by w i −2 → w i+1 . Similarly we define downward 1 level, upward 1 level and upward 2 levels.
→ w i+1 , then the edge e = (w i , w i+1 ) is an edge connecting a vertex to its grandparent, we call it a grandparent edge.
We claim that a self-avoiding path π = (w 0 , w 1 , . . . , w k ) in G connecting v 0 and v 1 has one of the following forms. If k is even, say k = 2t, then exactly one of the following statements holds.
If k is odd, say k = 2t + 1, then exactly one of the following statements holds.
Suppose π = (w 0 , . . . , w k ) is a self-avoiding path connecting v 0 and v 1 with length k. By parity π must use at least one tree edge.
Case one: (First step is downward 2 levels) If the first step of π is downward 2 levels, i.e. w 0 −2 → w 1 , then the next step cannot be upward 2 levels, otherwise it will not be self-avoiding. By the same reason the path can only go downward 2 levels each step before encountering the tree edge. Let t be the number of steps before encountering a tree edge, i.e., w 0 −2 → · · · −2 → w t . Now look at the next step w t → w t+1 , it must be a tree edge by the choice of t. If w t −1 → w t+1 , then the next step must be w t+1 +2 → w t+2 . Since (w t , w t+1 ) is a tree edge, {w t , w t+1 } is a cut set for G. Also v 1 and the descendants of w t+1 are in different connected component of G\{w t , w t+1 }. Thus if w t+2 is a descendant of w t+1 , to come back to v 1 , the path π must use either w t or w t+1 after time t + 2, which contradicts with the fact that π is self-avoiding. Using similar reasoning, if w t +1 → w t+1 , to avoid cycle on π the remaining steps after time t + 1 can only be upward 2 levels until reaching v 1 .
In sum if the first step of π is downward 2 levels, then π must be of the form (a) or (d) depending on the parity of the length of π.
Case two: (First step is upward 2 levels) If the first step of π is upward 2 levels, i.e. w 0 +2 → w 1 , then the next step cannot be downward 2 levels, otherwise it will not be self-avoiding. Let α be the number of steps of π which is upward 2 levels until another type of step is encountering, i.e. w 0 +2 → · · · +2 → w α but w α → w α+1 is not upward 2 levels.
Sub-case 2(a) If the step w α → w α+1 is a tree edge, then similar as in the first case, the remaining steps can only be downward 2 levels until hitting v 1 . Thus in this sub-case the path π must be of the form (b) or (e) depending on the parity of the length of π.
Sub-case 2(b) If the step w α → w α+1 is downward 2 levels, to avoiding cycle, w α+1 must be a different grandchildren of w α other than w α−1 . Similar as in the first case, from the time α the path can only go downward 2 levels each step before encountering a tree edge, say there are β such steps. Now the path looks like w 0 → w α+β → · · · By the choice of β, the next step w α+β → w α+β+1 is a tree edge. Using the same reasoning as in the first case, after time α + β + 1 the path can only be upward 2 levels each step until it hits the ray η v . Then from that hitting vertex, to avoid cycles the path can only go downward 2 levels each step until hitting v 1 . Thus in this sub-case the path π must be of the form (c) or (c') or (f) or (f') depending on the parity of the length of π and whether w α−1 and w α+1 has the same parent.
In particular from the above analysis, we see that each self-avoiding path π that connecting v 0 , v 1 uses exactly one tree edge.
Also from the forms of the path in the above claim, we see that if a self-avoiding path π = (w 0 , . . . , w k ) connecting v 0 and v 1 with length k is one of the forms (a), (c), (d) and (f), then max{d T (w 0 , w i ) : Figure 1: An illustration of G 4 in case b = 2 (with grandparent edges omitted) and a typical path of the form (f). The tree edge on this path is colored red while other edges colored blue.
If π is of the form (c') or (f'), then If π is of the form (b) or (e), then Therefore, if k ≥ n + 2 then P k,n = ∅. By the claim, if k = 2t is even, then the number of paths of the form (a) is b 2(t−1)+1 because for the first t − 1 steps one has b 2 choices for the grandchildren and b choices for w t . Once w 0 , . . . , w t are fixed, the remaining vertices w t+1 , . . . , w 2t are fixed. The number of paths of the form (b) is just 1. The number of paths of the form (c) is t−2 β=1 (b − 1)b 2β (Given α, β, w 0 , . . . , w α are fixed, one has b(b − 1) choices for w α+1 , and b 2 choices for each of w α+2 , . . . , w α+β and b choices for w α+β+1 . Once w 0 , . . . , w α+β+1 are fixed, the remaining path is fixed). Using similar counting argument, we have that the number of paths of the Similarly if k ∈ [1, n) is odd, one also has |P k,n | ≍ b k . If k = n or n + 1, the estimate |P k,n | ≍ b k is also true because we only need to deduct the contributions of cases (b), (c'), (e) and (f') from the previous expression for |P k,n |. Next we recall some notation and results regarding loop-erased random walk on a graph from [12].
Let S(t) be a discrete time Markov chain on a countable state space X with transition probabilities p(x, y). For a subset A of X, define the hitting time τ A := inf{t ≥ 0 : S(t) ∈ A} and the Green function where P x denote the measure of the Markov chain S(t) started from x. Fix a base point o ∈ X and we assume that where A −1 = ∅, A j = {w 0 , . . . , w j } for j ≥ 1 (see Proposition 3.2 of [13]). Proposition 5.6. The FUSF on the grandparent graph G is connected almost surely.
Proof. Recall G n is the sub-graph of G induced by vertices {x : d T (x, v 0 ) ≤ n}. Let T n be a uniform spanning tree on G n . We will show that there exists positive constants c 23 , c 24 such that Start a simple random walk on G n from v 0 and stop at the first hit of v 1 , loop erased this random walk path. Then the self-avoiding path we get has the law of the path from v 0 to v 1 in T n . Let S(t) be the simple random walk on G n , let o be v 0 and B = {v 1 }, by (5.2) (5.4) By Lemma 5.4, (5.3) holds trivially for k ≥ n + 2. In the following we assume k ≤ n + 1.
Notice that in our case the first product in (5.2) equals k−1 j=0 1 deg Gn (w j ) , which is again symmetric for w 0 , . . . , w k−1 . Hence by Proposition 5.5, we have µ(w) is a symmetric function of w 0 , . . . , w k−1 . Using the connection between effective resistance and green function (see Proposition 2.1 of [16]) one has that for any reordering w ′ 0 , . . . , w ′ k−1 of w 0 , . . . , w k−1 For w = (w 0 , . . . , w k ) ∈ P k,n , since w has one of the forms listed in the proof of Lemma 5.4, using a case by case analysis it is easy to see that we can reorder w 0 , . . . , w k−1 as w ′ 0 , . . . , w ′ k−1 such that except at most 8 j ′ s in {0, . . . , k − 1}, one has w ′ j−2 , w ′ j−1 are the grandparent and parent of w ′ j and the grandchildren of w ′ j are also in G n . For such an ordering and an index j such that w ′ j−2 , w ′ j−1 are the grandparent and parent of w ′ j and the grandchildren of w ′ j are in G n , one has that Indeed, from the local structure one has that For the other j's we use trivial estimates Now by (5.5) there exists a constant c 25 > 0 such that max w∈P k,n µ(w) ≤ (k + 1) x y Figure 2: Local structure for estimating R(w ′ j ↔ {w ′ j−2 , w ′ j−1 }), the right half is a network reduction with red edges with conductance 1 + b/2. All blue edges has conductance 1.
This combining with (5.4) and Lemma 5.4 yields (5.3). Since G n is an exhaustion of G, T n converges weakly to the FUSF on G. Thus (5.3) implies that in the FUSF sample F f on G, the probability that d F f (v 0 , v 1 ) ≥ k decays exponentially in k. In particular, v 0 and v 1 are connected almost surely in F f . By transitivity, almost surely any vertex of G is in the same connected component as its parent in F f . Therefore the FUSF F f is connected almost surely. Now we know that the FUSF sample F f on a grandparent graph G is just a tree. Next we consider the branching number of F f . Proposition 5.7. The FUSF sample F f on a grandparent graph G has branching number strictly larger than one.
Proof. From Proposition 5.6 we know that there is at least one tree edge in F f , otherwise there would be at least two trees in F f . For x ∈ V (G), let y be the parent of x and z be the grandparent of x. Let x 1 , . . . , x b be the children of x. Note {x, y} is a cutset for G, and G\{x, y} has b+1 connected components, one containing z and other b ones each containing a children of x. We denote the connected component containing z by K g (x) and the connected components containing Conditioned on the event that the tree edge e = (x, y) ∈ F f , one has the following observation: This can be seen from the definition of FUSF using exhaustion.
Next we show that conditioned on the event e = (x, y) ∈ F f , for each i = 1, . . . , b, almost surely there is another tree edge in F f ∩ K i (x). If not, we define a mass transport as follows: where v − denotes the parent of v.
Then the mass sent out from a vertex is at most one. But if conditioned on the event e = (x, y) ∈ F f , with positive probability there is no other tree edge in F f ∩ K i (x), then x will receive infinite mass with positive probability. This contradicts with the tilted masstransport principle.
Thus we can pick a large constant M > 0 such that the following inequality holds Also observe that conditioned on e = (x, y) ∈ F f and e ′ = (v, v − ) ∈ F f ∩ K i (x), the conditional distributions of F f ∩ K i (v) are independent and are the same as the distribution Conditioned on the tree edge e = (x, y) ∈ F f , we call a component and we also call this edge e ′ = (v, v − ) a good edge for e = (x, y). In a good component we pick an arbitrary good edge. Note the size of good edges form a supercritical Galton-Watson tree in light of (5.7). With positive probability the supercritical Galton-Watson tree has branching number bigger than one [16,Cor. 5.10]. Since in F f two good edges in neighboring generation of the Galton-Watson tree has distance at most M, with positive probability F f also has branching number strictly larger than one. Since the branching number of F f is a constant almost surely (Theorem 10.18 of [16]), it is strictly larger than one almost surely.
Remark 5.8. Proposition 5.7 is non-trivial in the sense that there exist spanning trees of G with branching number equals to one. In fact one can even find recurrent spanning trees of G. It is also of interest to find the exact value of this branching number.
Remark 5.9. The conclusion of Proposition 5.6 and 5.7 also hold for the Cartesian product T b+1 Z 2 , where the Z 2 -edges in T b+1 Z 2 will play the role of the tree edges in the above proofs. We conjecture that for any finite connected graph H, almost surely the FUSF on the Cartesian product T b+1 H is connected and has branching number larger than one.
6 FUSF on free products of nonunimodular transitive graphs with Z 2 The free product of two Cayley graph is well known. More generally one can define the free product of two transitive graphs G 1 , G 2 . For more details, see the description on page 2349 of [22]. Suppose G 0 is a nonunimodular transitive graph and Z 2 is the graph of two vertices connecting by one edge. We now give the detailed definition of the free product G 0 * Z 2 just like [22]. First take a copy of G 0 and countably many copies of Z 2 . Fix a bijection from the vertices of this copy of G 0 to the copies of Z 2 . Identify each vertex of this copy of G 0 with an arbitrary vertex in its image under the bijection. Call the resulting graph H 1 , it is formed by attaching an edge to each vertex of the copy of G 1 . Let I 1 denote the set of vertices on the Z 2 edges that are not identified with a vertex of G 0 . Fix a bijection between I 1 and countably many new copies of G 0 . Identify every vertex of I 1 with an arbitrary vertex in its image by the bijection to obtain a graph H 2 . So H 2 is formed by attaching a G 0 copy to each vertex in I 1 . Continue this process similarly, given H i , and if I i is the set of vertices in H i that were not born by identification in some previous steps, then fix a bijection between I i to a set of infinitely many copies of G 0 if i is odd (or infinitely many copies of Z 2 if i is even). Identify every vertex in I i with an arbitrary vertex in the its image by the bijection to obtain H i+1 . If we view H i as a subgraph of H i+1 , then finally the free product G 0 * Z 2 := H i . It is easy to see G is still a transitive graph. Also we call an edge in the free G 0 * Z 2 a G 0 -edge if its two endpoints lies in the same copy of G 0 in the above construction. Similarly we call the other edges by Z 2 -edges. Definition 6.1. Suppose G 0 is a transitive graph with a closed subgroup Γ of automorphisms that acts transitively on G 0 . Let G = G 0 * Z 2 . Suppose ω 0 is a Γ-invariant percolation process on G 0 . We view ω 0 as a random subgraph of G 0 . For each copy of G 0 in G, we take an independent percolation with the same law as ω 0 . Let w be the union of these independent percolation subgraphs with law as ω 0 and all the Z 2 -edges. We call w the free product percolation of ω 0 on G = G 0 * Z 2 .
Notice the FUSF on G = G 0 * Z 2 is an example of free product percolation of F f (G 0 ) on G, where F f (G 0 ) denote a FUSF sample on G 0 . Thus Theorem 1.3 follows from the combination of Proposition 5.6, 5.7 and a special case of the following proposition. Proposition 6.2. Suppose G 0 is a transitive graph with a closed nonunimodular subgroup Γ of automorphisms that acts transitively on G 0 . Note that Γ * Z 2 acts on G 0 * Z 2 transitively and Γ * Z 2 is also nonunimodular. Suppose ω 0 is a Γ-invariant percolation process on G 0 and w is the free product percolation of ω 0 on G = G 0 * Z 2 . If almost surely every connected component of ω 0 is infinite, then each connected component of w is (Γ * Z 2 )-heavy and has branching number bigger than one.
Proof. The part that each connected component of w has branching number bigger than one is obvious and we omit the details.
The part that each connected component of w is heavy can be proved using comparison to branching random walks.
Write Γ := Γ * Z 2 . Note for x, y in the same G 0 copy of G, and for a Z 2 -edge e = (x, x ′ ), one has |Γ x ′ x| = |Γ x x ′ | = 1. Hence in light of Lemma 2.3 we abuse m(x) to denote |Γ x | when we view x as a vertex in G 0 or | Γ x | when we view x as a vertex in G. In particular for a Z 2 -edge e = (x, x ′ ) one has m(x) = m(x ′ ).
For a fix vertex x ∈ V (G 0 ), since each connected component of ω 0 is infinite, writing K G 0 (x) of the connected component of x in ω 0 , one has Using TMTP and noting y ∈ K G 0 (x) if and only if x ∈ K G 0 (y), one has . For x ∈ V (G), we also use x to denote the vertex in the G 0 copy in the construction of G. There is a unique Z 2 -edge incident to x and we write the other vertex on the Z 2 -edge as x ′ . Let K(x) denote the connected component of x in the free product percolation ω. Let K h (x) denote the connected component of x if we delete the Z 2 -edge e = (x, x ′ ) from K(x).
For a fixed constant M > 0, we truncate K h (x) as follows. First we truncate all the edges e = (y, z) in the G 0 copy of x if max{d G 0 (y, x), d G 0 (z, x)} ≥ M. For all the vertex y in the G 0 copy of x that can connect to x by an ω-open path staying in B G 0 (x, M) (the ball in the G 0 copy of x with center x and radius M), we keep the Z 2 -edge (y, y ′ ) and do the same truncation procedure for y ′ as previous for x, namely we truncate all the edges e = (u, v) in the G 0 copy of y ′ if max{d G 0 (u, y ′ ), d G 0 (v, y ′ )} ≥ M. Keep doing this procedure and in the end we get an infinite random graph K M h (x). This random graph K M h (x) is just the sub-graph of K h (x) induced by those vertices that has an ω-open path to x such that between any two consecutive Z 2 -edges on this path there is at most M other (G 0 )-edges . Now we show that K M h (x) is heavy with positive probability. We first briefly recall the definition of a branching random walk and a related result; see [14] for more details.
Let L := {X i } N i=1 be a random N-tuple of real numbers, where N is also random. We can view the branching random walk as an ordered point process on the real line. An initial point x is located at the origin. It gives birth to N children x 1 , . . . , x N with random displacements X 1 , . . . , X N . Then each x i gives birth to a random number of particles with random displacement relative to the position of x i according to the same law as L and independently of one another and of the initial displacements. This procedure continues forever or until no more particles are born.
For a particle u, let |u| be the generation of u and X(u) for its displacement from its parent, and S(u) for its position (relative to the origin). Denote the initial particle at origin by 0. If u is an ancestor of v, write u < v. Then S(v) = 0<u≤v X(u).
(1) λ(α) < ∞ and λ ′ (α) exists and is finite; (2) E[ α, L log + α, L ] < ∞ and (3) αλ ′ (α)/λ(α) < log λ(α). Now given K M h (x), we construct a corresponding branching random walk in the following manner. Let N + 1 be the number of vertices in the connected component of x in K M h (x) intersecting with the G 0 copy of x, which we denote by K B G 0 (x,M ) (x). Note N ∈ [M, |B G 0 (x, M)|] is a finite random number. Let x 1 , . . . , x N be an arbitrary ordering of the vertices in K B G 0 (x,M ) (x)\{x}. Now for the corresponding branching random walk, we let the initial particle give birth to N children, each with displacement X i = log m(x i ) m(x) . For each x i , let N x i denote the number of vertices in In the corresponding branching random walk, we let the particle corresponding to x i give birth to N x i new particles each with a relative displacement log m(u) Now we verify that the conditions listed above to use Biggin's theorem hold. Conditions (1) and (2) are trivial since in our case X i and N are both bounded. Since α = −1 and λ(−1) > 0, condition (3) is just λ ′ (−1) + λ(−1) log λ(−1) > 0. Since and λ(−1) > e, it suffices to show that λ ′ (−1) + λ(−1) ≥ 0. m(x) tends to infinity because λ(−1) > 1 and W (α) > 0. Using the standard trick (see e.g. Proposition 5.6 in [16]) one has m(K M h (x)) = ∞ almost surely. Therefore K(x) is heavy almost surely. Remark 6.3. If Γ = Aut(G 0 ), Γ * Z 2 might be a closed subgroup of Aut(G 0 * Z 2 ). However by Lemma 2.3, Γ * Z 2 induces the same level structure on G = G 0 * Z 2 as Aut(G 0 * Z 2 ).
Remark 6.4. The free minimal spanning forests on G = G 0 * Z 2 is also a free product percolation that satisfies the condition of Proposition 6.2. Hence each connected component of the free minimal spanning forests on G = G 0 * Z 2 is also heavy and has branching number bigger than one. Interested reader can refer to Chapter 11 of [16] for more background on the free minimal spanning forests.
We conclude with two further open questions on FUSF on nonunimodular transitive graphs. The first question is about the number of trees in the FUSF. Benjamini et al asked whether the number of trees of the FUSF is 1 or ∞ almost surely (Question 15.6 in [2]). Hutchcroft and Nachmias [10] answered this question positively for unimodular transitive graphs and the nonunimodular case remains open. Another question one can consider is about the indistinguishability of the trees in the FUSF on a nonunimodular transitive graph. Since the trees in the WUSF on a nonunimodular transitive graph is light almost surely, they are distinguishable by automorphism-invariant properties, e.g. the sum of degrees of the highest points in the components. So the interesting case is for the trees in the FUSF on nonunimodular transitive graphs with the property that FUSF = WUSF. The techniques from [10] and [21] might be useful for this question.