Universality of highdimensional spanning forests and sandpiles
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Abstract
We prove that the wired uniform spanning forest exhibits meanfield behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of \(\mathbb {Z}^d\), \(d\ge 5\). In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of \(\mathbb {Z}^d\), \(d\ge 5\), some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constantorder multiplicative errors, rather than the polylogarithmicorder multiplicative errors present in that work.
Keywords
Uniform spanning tree Uniform spanning forest Random interlacements Meanfield Critical exponents Anomalous diffusionMathematics Subject Classification
Primary 60K35 Secondary 82B201 Introduction
The uniform spanning forests (USFs) of an infinite graph G are defined as weak limits of the uniform spanning trees of finite subgraphs of G. These limits can be taken with respect to two extremal boundary conditions, yielding the free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF). For transitive amenable graphs such as the hypercubic lattice \(\mathbb {Z}^d\), the free and wired forests coincide and we speak simply of the USF. In this paper we shall be concerned exclusively with the wired forest. Uniform spanning forests have played a central role in the development of probability theory over the last twenty years, and are closely related to several other topics in probability and statistical mechanics including electrical networks [20, 23, 44], looperased random walk [20, 49, 71], the random cluster model [25, 28], domino tiling [23, 42], random interlacements [34, 67], conformally invariant scaling limits [52, 59, 63], and the Abelian sandpile model [24, 39, 40, 56]. Indeed, our results have important implications for the Abelian sandpile model, which we discuss in Sect. 1.6.
Following the work of many authors, the basic qualitative features of the WUSF are firmly understood on a wide variety of graphs. In particular, it is known that every tree in the WUSF is recurrent almost surely on any graph [60], that the WUSF is connected a.s. if and only if two independent random walks on G intersect almost surely [20, 61], and that every tree in the WUSF is oneended almost surely whenever G is in one of several large classes of graphs [2, 20, 33, 35, 53, 61] including all transient transitive graphs. (An infinite tree is oneended if it does not contain a simple biinfinite path.)
 1.
The intrinsic geometry of each tree in \(\mathfrak {F}\) is similar at large scales to that of a critical Galton–Watson tree with finite variance offspring distribution, conditioned to survive forever. In particular, every tree has volume growth dimension 2 (with respect to its intrinsic graph metric), spectral dimension 4/3, and walk dimension 3 almost surely. The latter two statements mean that the nstep return probabilities for simple random walk on the tree decay like \(n^{2/3+o(1)}\), and that the typical displacement of the walk (as measured by the intrinsic graph distance in the tree) is \(n^{1/3+o(1)}\). These are known as the Alexander–Orbach values of these dimensions [4, 45].
 2.
The intrinsic geometry of the past of v in \(\mathfrak {F}\) is similar in law to that of an unconditioned critical Galton–Watson tree with finite variance offspring distribution. In particular, the probability that the past contains a path of length at least n is of order \(n^{1}\), and the probability that the past contains more than n points is of order \(n^{1/2}\). That is, the intrinsic diameter exponent and volume exponent are 1 and 1/2 respectively.
 3.
The extrinsic geometry of the past of v in \(\mathfrak {F}\) is similar in law to that of an unconditioned critical branching random walk on G with finite variance offspring distribution. In particular, the probability that the past of v includes a vertex at extrinsic distance at least n from v depends on the rate of escape of the random walk on G. For example, it is of order \(n^{2}\) for \(G=\mathbb {Z}^d\) for \(d\ge 5\) and is of order \(n^{1}\) for G a transitive nonamenable graph. This is related to the fact that the random walk on the ambient graph G is diffusive in the former case and ballistic in the latter case.
In light of the connections between the WUSF and the Abelian sandpile model, these results imply related results for that model, to the effect that an avalanche in the Abelian sandpile model has a similar distribution to a critical branching random walk (see Sect. 1.6). Precise statements of our results and further background are given in the remainder of the introduction.
The fact that our results apply at such a high level of generality is a strong vindication of universality for highdimensional spanning trees and sandpiles, which predicts that the largescale behaviour of these models should depend only on the dimension, and in particular should be insensitive to the microscopic structure of the lattice. In particular, our results apply not only to \(\mathbb {Z}^d\) for \(d\ge 5\), but also to nontransitive networks that are similar to \(\mathbb {Z}^d\) such as the halfspace \(\mathbb {Z}^{d1}\times \mathbb {N}\) or, say, \(\mathbb {Z}^d\) with variable edge conductances bounded between two positive constants. Many of our results also apply to longrange spanning forest models on \(\mathbb {Z}^d\) such as those associated with the fractional Laplacian \((\Delta )^\beta \) of \(\mathbb {Z}^d\) for \(d\ge 1\), \(\beta \in (0,d/4 \wedge 1)\). Longrange models such as these are motivated physically as a route towards understanding lowdimensional models via the \(\varepsilon \)expansion [72], for which it is desirable to think of the dimension as a continuous parameter. (See the introduction of [66] for an account of the \(\varepsilon \)expansion for mathematicians.)
About the proofs. Our proof relies on the interplay between two different ways of sampling the WUSF. The first of these is Wilson’s algorithm, a method of sampling the WUSF by joining together looperased random walks which was introduced by David Wilson [71] and extended to infinite transient graphs by Benjamini et al. [20]. The second is the interlacement Aldous–Broder algorithm, a method of sampling the WUSF as the set of firstentry edges of Sznitman’s random interlacement process [67]. This algorithm was introduced in the author’s recent work [34] and extends the classical Aldous–Broder algorithm [3, 22] to infinite transient graphs. Generally speaking, it seems that Wilson’s algorithm is the better tool for estimating the moments of random variables associated with the WUSF, while the interlacement Aldous–Broder algorithm is the better tool for estimating tail probabilities.
A key feature of the interlacement Aldous–Broder algorithm is that it enables us to think of the WUSF as the stationary measure of a natural continuoustime Markov chain. Moreover, the past of the origin evolves in an easilyunderstood way under these Markovian dynamics. In particular, as we run time backwards, the past of the origin gets monotonically smaller except possibly for those times at which the origin is visited by an interlacement trajectory. Indeed, the central insight in the proof of our results is that static tail events (on which the past of the origin is large) can be related to to dynamic tail events (on which the origin is hit by an interlacement trajectory at a small time). Roughly speaking, we show that these two types of tail event tend to occur together, and consequently have comparable probabilities. We make this intuition precise using inductive inequalities similar to those used to analyze onearm probabilities in highdimensional percolation [32, 45, 46].
Once the critical exponent results are in place, the results concerning the simple random walk on the trees can be proven rather straightforwardly using the results and techniques of Barlow et al. [12].
1.1 Relation to other work

When G is a regular tree of degree \(k\ge 3\), the components of the WUSF are distributed exactly as augmented critical binomial Galton–Watson trees conditioned to survive forever, and in this case all of our results are classical [13, 43, 54].

In the case of \(\mathbb {Z}^d\) for \(d\ge 5\), Barlow and Járai [11] established that the trees in the WUSF have quadratic volume growth almost surely. Our proof of quadratic volume growth uses similar methods to theirs, which were in turn inspired by related methods in percolation due to Aizenman and Newman [1].

Also in the case of \(\mathbb {Z}^d\) for \(d\ge 5\), Bhupatiraju et al. [21] followed the strategy of an unpublished proof of Lyons et al. [53] to prove that the probability that the past reaches extrinsic distance n is \(n^{2} \log ^{O(1)}n\) and that the probability that the past has volume n is \(n^{1/2} \log ^{O(1)} n\). Our results improve upon theirs in this case by reducing the error from polylogarithmic to constant order. Moreover, their proof relies heavily on transitivity and cannot be used to derive universal results of the kind we prove here.

Peres and Revelle [62] proved that the USTs of large ddimensional tori converge under rescaling (with respect to the Gromovweak topology) to Aldous’s continuum random tree when \(d\ge 5\). They also proved that their result extends to other sequences of finite transitive graphs satisfying a heatkernel upper bound similar to the one we assume here. Later, Schweinsberg [64] established a similar result for fourdimensional tori. Related results concerning looperased random walk on highdimensional tori had previosuly been proven by Benjamini and Kozma [19]. While these results are closely related in spirit to those that we prove here, it does not seem that either can be deduced from the other.

For planar Euclidean lattices such as \(\mathbb {Z}^2\), the UST is very well understood thanks in part to its connections to conformally invariant processes in the continuum [42, 52, 57, 59, 63]. In particular, Barlow and Masson [14, 15] proved that the UST of \(\mathbb {Z}^2\) has volume growth dimension 8/5 and spectral dimension 16/13 almost surely. See also [9] for more refined results.

In [35], the author and Nachmias established that the WUSF of any transient proper plane graph with bounded degrees and codegrees has meanfield critical exponents provided that measurements are made using the hyperbolic geometry of the graph’s circle packing rather than its usual combinatorial geometry. Our results recover those of [35] in the case that the graph in question is also uniformly transient, in which case it is nonamenable and the graph distances and hyperbolic distances are comparable.

A consequence of this paper is that several properties of the WUSF are insensitive to the geometry of the graph once the dimension is sufficiently large. In contrast, the theory developed in [18, 36] shows that some other properties describing the adjacency structure of the trees in the forest continue to undergo qualitative changes every time the dimension increases.

In forthcoming work with Sousi, we build upon the methods of this paper to analyze related problems concerning the uniform spanning tree in \(\mathbb {Z}^3\) and \(\mathbb {Z}^4\).
1.2 Basic definitions
In this paper, a network will be a connected graph \(G=(V,E)\) (possibly containing loops and multiple edges) together with a function \(c:E \rightarrow (0,\infty )\) assigning a positive conductancec(e) to each edge \(e \in E\) such that for each vertex \(v\in V\), the vertex conductance\(c(v):= \sum c(e)< \infty \), taken over edges incident to v, is finite. We say that the network G has controlled stationary measure if there exists a positive constant C such that \(C^{1} \le c(v) \le C\) for every vertex v of G. Locally finite graphs can always be considered as networks by setting \(c(e) \equiv 1\), and in this case have controlled stationary measure if and only if they have bounded degrees. We write \(E^\rightarrow \) for the set of oriented edges of a network. An oriented edge e is oriented from its tail \(e^\) to its head \(e^+\) and has reversal \(e\).
In the limiting construction above, one can also orient the uniform spanning tree of \(G_n^*\) towards the boundary vertex \(\partial _n\), so that every vertex other than \(\partial _n\) has exactly one oriented edge emanating from it in the spanning tree. If G is transient, then the sequence of laws of these random oriented spanning trees converge weakly to the law of a random oriented spanning forest of G, which is known as the oriented wired uniform spanning forest, and from which we can recover the usual (unoriented) WUSF by forgetting the orientation. (This assertion follows from the proof of [20, Theorem 5.1].) It is easily seen that the oriented wired uniform spanning forest of G is almost surely an oriented essential spanning forest of G, that is, an oriented spanning forest of G such that every vertex of G has exactly one oriented edge emanating from it in the forest (from which it follows that every tree is infinite).
1.3 Intrinsic exponents
Let \(\mathfrak {F}\) be an oriented essential spanning forest of an infinite graph G. We define the past of a vertex v in \(\mathfrak {F}\), denoted \(\mathfrak {P}(v)\), to be the subgraph of \(\mathfrak {F}\) induced by the set of vertices u of \(\mathfrak {F}\) such that every edge in the geodesic from u to v in \(\mathfrak {F}\) is oriented in the direction of v, where we also consider v to be included in this set. (By abuse of notation, we will also use \(\mathfrak {P}(v)\) to mean the vertex set of this subgraph.) Thus, a component of \(\mathfrak {F}\) is oneended if and only if the past of each of its vertices is finite. The future of a vertex v is denoted by \(\Gamma (v,\infty )\) and is defined to be the set of vertices u such that v is in the past of u.
In order to quantify the oneendedness of the WUSF, it is interesting to estimate the probability that the past of a vertex is large in various senses. Perhaps the three most natural such measures of largeness are given by the intrinsic diameter, extrinsic diameter, and volume of the past. Here, given a subgraph K of a graph G, we define the extrinsic diameter of K, denoted \(\mathrm {diam}_\mathrm {ext}(K)\), to be the supremal graph distance in G between two points in K, and define the intrinsic diameter of K, denoted \(\mathrm {diam}_\mathrm {int}(K)\), to be the diameter of K. The volume of K, denoted K, is defined to be the number of vertices in K.
It is also expected that each model has an uppercritical dimension, denoted \(d_c\), above which the critical exponents of the model stabilize at their socalled meanfield values. For the uniform spanning forest, the upper critical dimension is believed to be four. Intuitively, above the upper critical dimension the lattice is spacious enough that different parts of the model do not interact with each other very much. This causes the model to behave similarly to how it behaves on, say, the 3regular tree or the complete graph, both of which have a rather trivial geometric structure. Below the upper critical dimension, the geometry of the lattice affects the model in a nontrivial way, and the critical exponents are expected to differ from their meanfield values. The upper critical dimension itself (which need not necessarily be an integer) is often characterised by the meanfield exponents holding up to a polylogarithmic multiplicative correction, which is not expected to be present in other dimensions.
Theorem 1.1
The general form of our result is similar, but has an additional technical complication owing to the need to avoid trivialities that may arise from the local geometry of the network. Let G be a network, let v be a vertex of G, let X and Y be independent random walks started at v, and let q(v) be the probability that X and Y never return to v or intersect each other after time zero.
Theorem 1.2
The presence of q(v) in the theorem is required, for example, in the case that we attach a vertex by a single edge to the origin of \(\mathbb {Z}^d\), so that the past of this vertex in the USF is necessarily trivial. (The precise nature of the dependence on q(v) has not been optimized.) However, in any network G with controlled stationary measure and with \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), there exist positive constants \(\varepsilon \) and r such that for every vertex v in G, there exists a vertex u within distance r of G such that \(q(u)>\varepsilon \) (Lemma 4.2). In particular, if G is a transitive network with \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \) then q(v) is a positive constant, so that Theorem 1.1 follows from Theorem 1.2.
Let us note that Theorem 1.2 applies in particular to any bounded degree nonamenable graph, or more generally to any network with controlled stationary measures satisfying a ddimensional isoperimetric inequality for some \(d>4\), see [47, Theorem 3.2.7]. In particular, it applies to \(\mathbb {Z}^d\), \(d\ge 5\), with any specification of edge conductances bounded above and below by two positive constants (in which case it can also be shown that q(v) is bounded below by a positive constant). A further example to which our results are applicable is given by taking \(G=H^d\) where \(d\ge 5\) and H is any infinite, bounded degree graph.
1.4 Volume growth, spectral dimension, and anomalous diffusion
The theorems concerning intrinsic exponents stated in the previous subsection also allow us to determine exponents describing the almost sure asymptotic geometry of the trees in the WUSF, and in particular allow us to compute the almost sure spectral dimension and walk dimension of the trees in the forest. See e.g. [47] for background on these and related concepts. Here, we always consider the trees of the WUSF as graphs. One could instead consider the trees as networks with conductances inherited from G, and the same results would apply with minor modifications to the proofs.
Theorem 1.3
The values \(d_f=2,d_s=4/3\), and \(d_w=3\) are known as the Alexander–Orbach values of these exponents, following the conjecture due to Alexander and Orbach [4] that they held for highdimensional incipient infinite percolation clusters. The first rigorous proof of Alexander–Orbach behaviour was due to Kesten [43], who established it for critical Galton–Watson trees conditioned to survive (see also [13]). The first proof for a model in Euclidean space was due to Barlow et al. [12], who established it for highdimensional incipient infinite clusters in oriented percolation. Later, Kozma and Nachmias [45] established the Alexander–Orbach conjecture for highdimensional unoriented percolation. See [31] for an extension to longrange percolation, [47] for an overview, and [17] for results regarding scaling limits of a related model.
As previously mentioned, Barlow and Masson [15] have shown that in the twodimensional uniform spanning tree, \(d_f=8/5\), \(d_s=16/13\), and \(d_w=13/5\).
1.5 Extrinsic exponents
We now describe our results concerning the extrinsic diameter of the past. In comparison to the intrinsic diameter, our methods to study the extrinsic diameter are more delicate and require stronger assumptions on the graph in order to derive sharp estimates. Our first result on the extrinsic diameter concerns \(\mathbb {Z}^d\), and improves upon the results of Bhupatiraju et al. [21] by removing the polylogarithmic errors present in their results.
Theorem 1.4
We expect that it should be possible to generalize the proof of Theorem 1.4 to other similar graphs, and to longrange models, but we do not pursue this here.
Theorem 1.5
Note that the hypotheses of this theorem imply that \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \).
Theorem 1.6
Note that the upper bound of Theorem 1.6 is a trivial consequence of Theorem 1.2.
1.6 Applications to the Abelian sandpile model
The Abelian sandpile model was introduced by Dhar [24] as an analytically tractable example of a system exhibiting selforganized criticality. This is the phenomenon by which certain randomized dynamical systems tend to exhibit criticallike behaviour at equilibrium despite being defined without any parameters that can be varied to produce a phase transition in the traditional sense. The concept of selforganized criticality was first posited in the highly influential work of Bak, Tang, and Wiesenfeld [7, 8], who proposed (somewhat controversially [70]) that it may account for the occurrence of complexity, fractals, and power laws in nature. See [38] for a detailed introduction to the Abelian sandpile model, and [41] for a discussion of selforganized criticality in applications.
We define a Markov chain on the set of stable sandpile configurations on K as follows: At each time step, a vertex of K is chosen uniformly at random, an additional grain of sand is placed at that vertex, and the resulting configuration is stabilized. Although this Markov chain is not irreducible, it can be shown that chain has a unique closed communicating class, consisting of the recurrent configurations, and that the stationary measure of the Markov chain is simply the uniform measure on the set of recurrent configurations. In particular, the stationary measure for the Markov chain is also stationary if we add a grain of sand to a fixed vertex and then stabilize [38, Exercise 2.17].
The connection between sandpiles and spanning trees was first discovered by Majumdar and Dhar [56], who described a bijection, known as the burning bijection, between recurrent sandpile configurations and spanning trees. Using the burning bijection, Athreya and Járai [6] showed that if \(d\ge 2\) and \(\langle V_n\rangle _{n\ge 1}\) is an exhaustion of \(\mathbb {Z}^d\) by finite sets, then the uniform measure on recurrent sandpile configurations on \(V_n\) converges weakly as \(n\rightarrow \infty \) to a limiting measure on sandpile configurations on \(\mathbb {Z}^d\). Járai and Werning [40] later extended this result to any infinite, connected, locally finite graph G for which every component of the WUSF of G is oneended almost surely. We call a random sandpile configuration on G drawn from this measure a uniform recurrent sandpile on G, and typically denote such a random variable by \(\mathrm {H}\) (capital \(\eta \)).
We are particularly interested in what happens during one step of the dynamics at equilibrium, in which one grain of sand is added to a vertex v in a uniformly random recurrent configuration \(\mathrm {H}\), and then topplings are performed in order to stabilize the resulting configuration. The multiset of vertices counted according to the number of times they topple is called the Avalanche, and is denoted \({\text {Av}}_v(\mathrm {H})\). The set of vertices that topple at all is called the Avalanche cluster and is denoted by \({\text {AvC}}_v(\mathrm {H})\).
Járai and Redig [39] showed that the burning bijection allows one to relate avalanches to the past of the WUSF, which allowed them to prove that avalanches in \(\mathbb {Z}^d\) satisfy \(\mathbb {P}( v \in {\text {AvC}}_0(\mathrm {H})) \asymp \Vert v\Vert ^{d+2}\) for \(d\ge 5\). (The fact that the expected number of times v topples scales this way is an immediate consequence of Dhar’s formula, see [38, Section 3.3.1].) Bhupatiraju et al. [21] built upon these methods to prove that, when \(d\ge 5\), the probability that the diameter of the avalanche is at least n scales as \(n^{2} \log ^{O(1)} n\) and the probability that the total number of topplings in the avalanche is at least n is between \(c n^{1/2}\) and \(n^{2/5+o(1)}\). Using the combinatorial tools that they developed, the following theorem, which improves upon theirs, follows straightforwardly from our results concerning the WUSF. (Strictly speaking, it also requires our results on the vWUSF, see Sect. 2.2.)
Theorem 1.7
As with the WUSF, our methods also yield several variations on this theorem for other classes of graphs, the following of which are particularly notable. See Sect. 1.5 for the relevant definitions. With a little further work, it should be possible to remove the dependency on v in the lower bounds of Theorems 1.8 and 1.9. The upper bounds of Theorem 1.8 only require that G has polynomial growth, see Proposition 7.8.
Theorem 1.8
Similarly, the following theorem concerning uniformly ballistic graphs can be deduced from Theorems 7.4 and 1.6. Again, we stress that this result applies in particular to any bounded degree nonamenable graph.
Theorem 1.9
Notation
As previously discussed, we use \(\asymp , \preceq \) and \(\succeq \) to denote equalities and inequalities that hold to within multiplication by two positive constants depending only on the choice of network. Typically, but not always, these constants will only depend on a few important parameters such as \(\inf _{v\in V} c(v)\), \(\sup _{v\in V}c(v)\), and \(\Vert {P}\Vert _{\mathrm {bub}}\).

\(\mathfrak {F},\mathfrak {F}_v\) A sample of the wired uniform spanning forest and vwired uniform spanning forest respectively.

\(\mathfrak {T}_v\) The tree containing v in \(\mathfrak {F}_v\).

\(\mathfrak {B}(u,n),\mathfrak {B}_v(u,n)\) The intrinsic ball of radius n around u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\partial \mathfrak {B}(u,n),\partial \mathfrak {B}_v(u,n)\) The set of vertices at distance exactly n from u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\mathfrak {P}(u),\mathfrak {P}_v(u)\) The past of u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\mathrm {past}_{F}(u)\) The past of u in the oriented forest F (which need not be spanning).

\(\mathfrak {P}(u,n),\mathfrak {P}_v(u,n)\) The intrinsic ball of radius n around u in the past of u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\Gamma (u,w), \Gamma _v(u,w)\) The path from u to w in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively, should these vertices be in the same component.

\(\Gamma (u,\infty ), \Gamma _v(u,\infty )\) The future of u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\partial \mathfrak {P}(u,n),\partial \mathfrak {P}_v(u,n)\) The set of vertices with intrinsic distance exactly n from u in the past of u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively.

\(\mathscr {I},\mathscr {I}_v\) The interlacement process and vwired interlacement process respectively.

\(\mathcal {I}_{[a,b]},\mathcal {I}_{v,[a,b]}\) The set of vertices visited by the interlacement process and the vwired interlacement process in the time interval [a, b] respectively.
2 Background
2.1 Looperased random walk and Wilson’s algorithm
Let G be a network. For each \(\infty \le n \le m \le \infty \) we define L(n, m) to be the line graph with vertex set \(\{i \in \mathbb {Z}: n \le i\le m\}\) and with edge set \(\{\{i,i+1\}: n\le i \le m1\}\). A path in G is a multigraph homomorphism from L(n, m) to G for some \(\infty \le n \le m \le \infty \). We can consider the random walk on G as a path by keeping track of the edges it traverses as well as the vertices it visits. Given a path \(w : L(n,m)\rightarrow G\) we will use w(i) and \(w_i\) interchangeably to denote the vertex visited by w at time i, and use \(w(i,i+1)\) and \(w_{i,i+1}\) interchangeably to denote the oriented edge crossed by w between times i and \(i+1\).
Wilson’s algorithm [71] is a method of sampling the UST of a finite graph by recursively joining together looperased random walk paths. It was extended to sample the WUSF of infinite transient graphs by Benjamini et al. [20]. See also [55, Chapters 4 and 10] for an overview of the algorithm and its applications.
Wilson’s algorithm can be described in the infinite transient case as follows. Let G be an infinite transient network, and let \(v_1,v_2,\ldots \) be an enumeration of the vertices of G. Let \(\mathfrak {F}^{0}\) be the empty forest, which has no vertices or edges. Given \(\mathfrak {F}^{n}\) for some \(n\ge 0\), start a random walk at \(v_{n+1}\). Stop the random walk if and when it hits the set of vertices already included in \(\mathfrak {F}^n\), running it forever otherwise. Let \(\mathfrak {F}^{n+1}\) be the union of \(\mathfrak {F}^n\) with the set of edges traversed by the looperasure of this stopped path. Let \(\mathfrak {F}=\bigcup _{n\ge 0} \mathfrak {F}^n\). Then the random forest \(\mathfrak {F}\) has the law of the wired uniform spanning forest of G. If we keep track of direction in which edges are crossed by the looperased random walks when performing Wilson’s algorithm, we obtain the oriented wired uniform spanning forest. The algorithm works similarly in the finite and recurrent cases, except that we start by taking \(\mathfrak {F}^{0}\) to contain one vertex and no edges.
2.2 The vwired uniform spanning forest and stochastic domination
In this section we introduce the vwired uniform spanning forest (vWUSF), which was originally defined by Járai and Redig [39] in the context of their work on the sandpile model (where it was called the WSF\(_o\)). The vWUSF is a variation of the WUSF of G in which, roughly speaking, we consider v to be ‘wired to infinity’. The vWUSF serves two useful purposes in this paper: its stochastic domination properties allow us to ignore interactions between different parts of the WUSF, and the control of the vWUSF that we obtain will be applied to prove our results concerning the Abelian sandpile model in Sect. 9.
The following lemma makes the vWUSF extremely useful for studying the usual WUSF, particularly in the meanfield setting. It will be the primary means by which we ignore the interactions between different parts of the forest. (Indeed, it plays a role analogous to that played by the BK inequality in Bernoulli percolation.) We denote by \(\mathrm {past}_F(v)\) the past of v in the oriented forest F, which need not be spanning. We write \(\mathfrak {T}_v\) for the tree containing v in \(\mathfrak {F}_v\), and write \(\Gamma (u,\infty )\) and \(\Gamma _v(u,\infty )\) for the future of u in \(\mathfrak {F}\) and \(\mathfrak {F}_v\) respectively, as defined in Sect. 1.3.
Lemma 2.1
Note that when K is a singleton, (2.1) follows implicitly from [53, Lemma 2.3]. The proof in the general case is also very similar to theirs, but we include it for completeness. Given a network G and a finite set of vertices K, we write G/K for the network formed from G by identifying all the vertices of K.
Lemma 2.2
Let G be a finite network, let \(K_1 \subseteq K_2\) be sets of vertices of G. For each spanning tree T of G, let \(S(T,K_2)\) be the smallest subtree of T containing all of \(K_2\). Then the uniform spanning tree of G/\(K_1\) stochastically dominates \(T{\setminus }S(T,K_2)\), where T is a uniform spanning tree of G.
Proof
It follows from the spatial Markov property of the UST that, conditional on \(S(T,K_2)\), the complement \(T{\setminus }S(T,K_2)\) is distributed as the UST of the network G/\(S(T,K_2)\) constructed from G by identifying all the vertices in the tree \(S(T,K_2)\), see [35, Section 2.2.1]. On the other hand, it follows from the negative association property of the UST [55, Theorem 4.6] that if \(A \subseteq B\) are two sets of vertices, then the UST of G/A stochastically dominates the UST of G/B. This implies that the claim holds when we condition on \(S(T,K_2)\), and we conclude by averaging over the possible choices of \(S(T,K_2)\). \(\square \)
Proof of Lemma 2.1
The claim (2.1) follows from Lemma 2.2 by considering the finite networks \(G_n^*\) used in the definition of the WUSF, taking \(K_1 = \{u,\partial _n\}\) and \(K_2 = K \cup \{\partial _n\}\), and taking the limit as \(n\rightarrow \infty \).
We now prove (2.2). If \(u=v\) then the claim follows by applying Lemma 2.2 to the finite networks \(G_n^{*v}\), taking \(K_1 = \emptyset \) and \(K_2 = K\), and taking the limit as \(n\rightarrow \infty \). Now suppose that \(u\ne v\). Let \(G/\{u,v\}\) be the network obtained from G by identifying u and v into a single vertex x, and let \(\mathfrak {F}'\) be the xwired uniform spanning forest of \(G/\{u,v\}\). We consider \(\mathfrak {F}'\) as a subgraph of G, and let \(\mathfrak {T}'\) be the component of u in \(\mathfrak {F}'\). It follows from the negative association property of the UST and an obvious limiting argument that \(\mathfrak {F}'\) is stochastically dominated by \(\mathfrak {F}_u\), and hence that \(\mathfrak {T}'\) is stochastically dominated by \(\mathfrak {T}_u\). On the other hand, applying Lemma 2.2 to the finite networks \(G_n^{*v}\), taking \(K_1 = \{u,v\}\) and \(K_2 = K \cup \{v\}\), and taking the limit as \(n\rightarrow \infty \) yields that the conditional distribution of \(\mathrm {past}_{\mathfrak {F}_v{\setminus }F_v(K)}(u)\) given \(F_v(K)\) is stochastically dominated by \(\mathfrak {T}'\) and hence by \(\mathfrak {T}_u\). \(\square \)
3 Interlacements and the Aldous–Broder algorithm
The random interlacement process is a Poissonian soup of doublyinfinite random walks that was introduced by Sznitman [67] and generalized to arbitrary transient graphs by Texeira [68]. Formally, the interlacement process \(\mathscr {I}\) on the transient graph G is a Poisson point process on \(\mathcal {W}^* \times \mathbb {R}\), where \(\mathcal {W}^*\) is the space of biinfinite paths in G modulo timeshift, and \(\mathbb {R}\) is thought of as a time coordinate. In [34], we showed that the random interlacement process can be used to generate the WUSF via a generalization of the Aldous–Broder algorithm. By shifting the time coordinate of the interlacement process, this sampling algorithm also allows us to view the WUSF as the stationary measure of a Markov process; this dynamical picture of the WUSF, or more precisely its generalization to the vWUSF, is of central importance to the proofs of the main theorems of this paper.
See [34, Proposition 3.3] for a limiting construction of the interlacement process from the random walk on an exhaustion with wired boundary conditions.
3.1 vwired variants
In this section, we introduce a variation on the interlacement process in which a vertex v is wired to infinity, which we call the vwired interlacement process. We then show how the vwired interlacement process can be used to generate the vWUSF in the same way that the usual interlacement process generates the usual WUSF.
We will deduce that such a measure exists via the following limiting procedure, which also gives a direct construction of the vrooted interlacement process. Let N be a Poisson point process on \(\mathbb {R}\) with intensity measure \((c(\partial _n)+c(v))\Lambda \). Conditional on N, for each \(t\in N\), let \(W_t\) be a random walk on \(G_n^{*v}\) started at \(\partial _n\) (which is identified with v) and stopped when it first returns to \(\partial _n\), where we consider each \(W_t\) to be an element of \(\mathcal {W}^*\). We define \(\mathscr {I}^n_v\) to be the point process \(\mathscr {I}^n_v:=\left\{ (W_t,t) : t \in N\right\} \).
Proposition 3.1
Let G be an infinite network, let v be a vertex of G, and let \(\langle V_n \rangle _{n\ge 0}\) be an exhaustion of G. Then the Poisson point processes \(\mathscr {I}^n_v\) converge in distribution as \(n\rightarrow \infty \) to a Poisson point process \(\mathscr {I}_v\) on \(\mathcal {W}^* \times \mathbb {R}\) with intensity measure of the form \(Q^*_v \otimes \Lambda \), where \(\Lambda \) is the Lebesgue measure on \(\mathbb {R}\) and \(Q^*_v\) is a \(\sigma \)finite measure on \(\mathcal {W}^*\) such that (3.2) is satisfied for every finite set \(K \subset V\) and every event \(\mathscr {A}\subseteq \mathcal {W}^*\).
The proof is very similar to that of [34, Proposition 3.3], and is omitted.
Corollary 3.2
Let G be an infinite network and let v be a vertex of G. Then there exists a unique \(\sigma \)finite measure \(Q^*_v\) on \(\mathcal {W}^*\) such that (3.2) is satisfied for every finite set \(K \subset V\) and every event \(\mathscr {A}\subseteq \mathcal {W}^*\).
Proof
The existence statement follows immediately from Proposition 3.1. The uniqueness statement is immediate since sets of the form \(\mathscr {A}\cap \mathcal {W}^*_K\) are a \(\pi \)system generating the Borel \(\sigma \)algebra on \(\mathcal {W}^*\). \(\square \)
We call \(\mathscr {I}_v\) the vwired interlacement process. Note that it may include trajectories that are either doubly infinite, singly infinite and ending at v, singly infinite and starting at v, or finite and both starting and ending at v.
We have the following vrooted analogue of [34, Theorem 1.1 and Proposition 4.2], whose proof is identical to those in that paper.
Proposition 3.3
3.2 Relation to capacity
3.3 Evolution of the past under the dynamics
The reason that the dynamics induced by the interlacement Aldous–Broder algorithm are so useful for studying the past of the origin in the WUSF is that the past itself evolves in a very natural way under the dynamics. Indeed, if we run time backwards and compare the pasts \(\mathfrak {P}_0(v)\) and \(\mathfrak {P}_{t}(v)\) of v in \(\mathfrak {F}_0\) and \(\mathfrak {F}_{t}\), we find that the past can become larger only at those times when a trajectory visits v. At all other times, \(\mathfrak {P}_{t}(v)\) decreases monotonically in t as it is ‘sliced into pieces’ by newly arriving trajectories. This behaviour is summarised in the following lemma, which is adapted from [34, Lemma 5.1]. Given a set \(A \subseteq \mathbb {R}\), we write \(\mathcal {I}_{A}\) for the set of vertices that are hit by some trajectory in \(\mathscr {I}_{A}\), and write \(\mathfrak {P}_t(v)\) for the past of v in the forest \(\mathfrak {F}_t\).
Lemma 3.4
Let G be a transient network, let \(\mathscr {I}\) be the interlacement process on G, and let \(\langle \mathfrak {F}_t\rangle _{t\in \mathbb {R}}=\langle \mathsf {AB} _t(\mathscr {I}) \rangle _{t\in \mathbb {R}}\). Let v be a vertex of G, and let \(s<t\). If \(v \notin \mathcal {I}_{[s,t)}\), then \(\mathfrak {P}_s(v)\) is equal to the component of v in the subgraph of \(\mathfrak {P}_t(v)\) induced by \(V{\setminus }\mathcal {I}_{[s,t)}\).
Proof
Suppose that u is a vertex of V, and let \(\Gamma _s(u,\infty )\) and \(\Gamma _t(u,\infty )\) be the futures of u in \(\mathfrak {F}_s\) and \(\mathfrak {F}_t\) respectively. Let \(u=u_{0,s},u_{1,s},\ldots \) and \(u=u_{0,t},u_{1,t},\ldots \) be, respectively, the vertices visited by \(\Gamma _s(u,\infty )\) and \(\Gamma _t(u,\infty )\) in order. Let \(i_0\) be the smallest i such that \(\sigma _s(u_{i,s}) <t\). Then it follows from the definitions that \(\Gamma _s(u,\infty )\) and \(\Gamma _t(u,\infty )\) coincide up until step \(i_0\), and that \(\sigma _s(u_{i,s}) <t\) for every \(i\ge i_0\). (Indeed, \(\sigma _s(u_{i,s})\) is decreasing in i.) On the other hand, if \(v\notin \mathcal {I}_{[s,t)}\) then \(\sigma _s(v)>t\), and the claim follows readily. \(\square \)
Similarly, we have the following lemma in the vwired case, whose proof is identical to that of Lemma 3.4 above. Given \(A \subseteq \mathbb {R}\), we write \(\mathcal {I}_{v,A}\) for the set of vertices that are hit by some trajectory in \(\mathscr {I}_{v,A}\), and write \(\mathfrak {P}_{v,t}(u)\) for the past of u in the forest \(\mathfrak {F}_{v,t}\).
Lemma 3.5
Let G be a network, let v be a vertex of G, let \(\mathscr {I}_v\) be the vwired interlacement process on G, and let \(\langle \mathfrak {F}_{v,t}\rangle _{t\in \mathbb {R}}=\langle \mathsf {AB} _{v,t}(\mathscr {I}_v) \rangle _{t\in \mathbb {R}}\). Let u be a vertex of G, and let \(s<t\). If \(u \notin \mathcal {I}_{v,[s,t)}\), then \(\mathfrak {P}_{v,s}(u)\) is equal to the component of u in the subgraph of \(\mathfrak {P}_{v,t}(u)\) induced by \(V{\setminus }\mathcal {I}_{v,[s,t)}\).
4 Lower bounds for the diameter
In this section, we use the interlacement Aldous–Broder algorithm to derive the lower bounds on the tail of the intrinsic and extrinsic diameter of Theorems 1.1–1.4.
4.1 Lower bounds for the intrinsic diameter
Recall that \(\mathfrak {P}(v)\) denotes the past of v in the WUSF, that \(\mathfrak {T}_v\) denotes the component of v in the vWUSF, and that q(v) is the probability that two independent random walks started at v do not return to v or intersect each other at any positive time.
Proposition 4.1
Note that (4.2) gives a nontrivial lower bound for every transitive network, and can be thought of as a meanfield lower bound. (For recurrent networks, the tree \(\mathfrak {T}_v\) contains every vertex of the network almost surely, so that the bound also holds degenerately in that case.)
Proof
The following lemma shows that the lower bound of (4.1) is always meaningful provided that \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \). We write Open image in new window for the Greens function.
Lemma 4.2
Let G be a network with controlled stationary measure and with \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \). Then there exists a positive constant \(\varepsilon \) such that for every \(v\in V\) there exists \(u\in V\) with \(\mathbf {G}(v,u)\ge \varepsilon \), \(d(v,u) \le \varepsilon ^{1}\), and \(q(u)>\varepsilon \). In particular, if G is transitive then q(v) is a positive constant.
Note that the statement concerning the graph distance may hold degenerately on networks that are not locally finite, but that the Greens function lower bound remains meaningful in this setting.
Proof
4.2 Lower bounds for the extrinsic diameter
In this section we apply a similar method to that used in the previous subsection to prove a lower bound on the tail of the extrinsic diameter. The method we use is very general and, as well as being used in the proof of Theorems 1.4–1.6 and 7.3, can also be used to deduce similar lower bounds for e.g. longranged models.
Proposition 4.3
Proof of Proposition 4.3
Lemma 4.4
Let \(G=(V,E)\) be a network with controlled stationary measure that is dAhlfors regular for some \(d>2\) and satisfies Gaussian heat kernel estimates. Then \(L(r)\preceq r^2\).
Proof
5 The length and capacity of the looperased random walk
In this section, we study the length and capacity of looperased random walk. In particular, we prove that in a network with controlled stationary measure and \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), an nstep looperased random walk has capacity of order n with high probability. The estimates we derive are used extensively throughout the remainder of the paper. In the case of \(\mathbb {Z}^d\), these estimates are closely related to classical estimates of Lawler, see [51] and references therein.
5.1 The number of points erased
Recall that we write \(X^T\) for the random walk ran up to the (possibly random) time T, and use similar notation for other paths such as \(\mathsf {LE}(X)\).
Lemma 5.1
 1.The random variables \(\langle \ell _{n+1}(X)\ell _{n}(X) \rangle _{n\ge 0}\) are independent conditional on \(\mathsf {LE} (X)\), and the estimateholds for every \(n\ge 0\) and \(m \ge 0\).$$\begin{aligned} \mathbf {P}_v\left[ \ell _{n+1}(X)\ell _n(X) 1 =m \mid \mathsf {LE} (X) \right] \le \Vert P^{m}\Vert _{1\rightarrow \infty } \end{aligned}$$(5.1)
 2.If \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), thenand$$\begin{aligned} \mathbf {E}_v \left[ \ell _n(X) \mid \mathsf {LE} \left( X\right) \right] \le \Vert {P}\Vert _{\mathrm {bub}} \, n \end{aligned}$$(5.2)for every \(n\ge 1\) and \(\lambda > 0\).$$\begin{aligned} \mathbf {P}_v \left[ \rho _n(X) \le \lambda ^{1} n \mid \mathsf {LE} (X) \right] \le \Vert {P}\Vert _{\mathrm {bub}} \, \lambda ^{1} \end{aligned}$$(5.3)
Note that, in the other direction, we have the trivial inequalities \(\ell _n \ge n\) and \(\rho _n \le n\).
Proof
For item 2, (5.2) follows immediately from (5.1). Furthermore, \(\rho _n \le \lambda ^{1} n\) if and only if \(\ell _{\lfloor \lambda ^{1} n \rfloor } \ge n\), so that (5.3) follows from (5.2) and Markov’s inequality\(.\square \)
We remark that Lemma 5.1 together with the strong law of large numbers for independent, uniformly integrable random variables [29, Theorem 2.19] has the following easy corollary. Since we do not require the result for the remainder of the paper, the proof is omitted.
Corollary 5.2
The following variation of Lemma 5.1, applying to the looperasure of a random walk stopped upon hitting a vertex v, is proved similarly.
Lemma 5.3
 1.The random variables \(\langle \ell _{n+1}(X^{\tau _v})\ell _{n}(X^{\tau _v}) \rangle _{n=0}^{\gamma 1}\) are independent conditional on the event that \(\tau _v<\infty \) and \(\mathsf {LE} (X^{\tau _v})=\gamma \), and the estimateholds for every vertex \(1 \le n \le \gamma 1\) and every \(m\ge 0\).$$\begin{aligned} \mathbf {P}_u\left( \ell _{n+1}(X^{\tau _v})\ell _n(X^{\tau _v}) 1 =m \mid \tau _v < \infty ,\, \mathsf {LE} (X^{\tau _v})=\gamma \right) \le \Vert P^{m}\Vert _{1\rightarrow \infty } \end{aligned}$$(5.6)
 2.If \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), then$$\begin{aligned} \mathbf {E}_u \left[ \tau _v \mid \tau _v < \infty ,\, \mathsf {LE} (X^{\tau _v})=\gamma \right] \le \Vert {P}\Vert _{\mathrm {bub}} \gamma . \end{aligned}$$(5.7)
Proof
Write \(\ell _n=\ell _n(X^{\tau _v})\). Observe that the conditional distribution of \(\langle X_i \rangle _{i=\ell _n}^{\tau _v}\) given the random variable \(X^{\ell _n}\) and the event \(\ell _n<\tau _v<\infty \) is equal to the distribution of a simple random walk started at \(X_{\ell _n}\) and conditioned to hit v before hitting the set of vertices visited by \(\mathsf {LE}(X)^{n1}\). The rest of the proof is very similar to that of Lemma 5.1. \(\square \)
5.2 The capacity of looperased random walk
Given a transient path X in a network, we define \(\eta _n(X) = \max \{\ell _k(X) : k \ge 0, \ell _k(X) \le n\}\) for each \(n\ge 0\). The time \(\eta _n(X)\) is defined so that \(\mathsf {LE}(X^{\eta _n}) = \mathsf {LE}(X^n)^{\rho _n} = \mathsf {LE}(X)^{\rho _n}\), and in particular, every edge traversed by \(\mathsf {LE}(X^{\eta _n})\) is also traversed by both \(\mathsf {LE}(X)\) and \(\mathsf {LE}(X^n)\). The goal of this subsection is to prove the following estimate, which will play a fundamental role in the remainder of our analysis.
Proposition 5.4
We do not expect these bounds to be optimal.
Lemma 5.5
Proof
Lemma 5.6
Remark 5.7
Lemmas 5.5 and 5.6 give the correct order of magnitude for the capacity of the random walk on \(\mathbb {Z}^d\) for all \(d\ge 3\), which is order \(\sqrt{n}\) when \(d=3\), order \(n/\log n\) when \(d=4\), and order n when \(d\ge 5\). See [5] and references therein for more detailed results.
Lemma 5.6 has the following immediate corollary.
Corollary 5.8
Before proving Lemma 5.6, let us use it, together with Lemma 5.1, to deduce Proposition 5.4.
Proof of Proposition 5.4 given Corollary 5.8
Proof of Lemma 5.6
6 Volume bounds
In this section, we study the volume of balls in both the WUSF and vWUSF. In Sect. 6.1 we prove upper bounds on the moments of the volumes of balls, while in Sect. 6.2 we prove lower bounds on moments and upper bounds on the probability that the volume is atypically small. Together, these estimates will imply that \(d_f(T)=2\) for every component T of \(\mathfrak {F}\) almost surely. The estimates in this section will also be important in Sects. 7 and 8.
6.1 Upper bounds
The goal of this subsection is to obtain tail bounds on the probability that an intrinsic ball in the WUSF contains more than \(n^2\) vertices. The upper bounds we obtain are summarized by the following two propositions, which are generalisations of [11, Theorem 4.1].
Proposition 6.1
We also obtain the following variation of this proposition applying to the vWUSF.
Proposition 6.2
(To prove our main theorems it suffices to have just the first and second moment bounds of Propositions 6.1 and 6.2. We include the exponential moment bounds for future application since they are not much more work to derive.)
Before proving Propositions 6.1 and 6.2, we note the following important corollaries.
Corollary 6.3
Proof
Corollary 6.4
Proof
Remark 6.5
[13, Proposition 2.8]^{2} shows that Corollary 6.4 is sharp in the sense that, when G is a 3regular tree, \(\log \log n\) cannot be replaced with \((\log \log n)^{1\varepsilon }\) for any \(\varepsilon >0\).
We now begin working towards the proof of Propositions 6.1 and 6.2. We begin with a first moment estimate.
Lemma 6.6
Proof
We next use an inductive argument to control the higher moments of \(\mathfrak {B}_v(v,n)\).
Lemma 6.7
Proof
Next, we control the moments of the volume of balls in the WUSF in terms of the moments in the vWUSF.
Lemma 6.8
Proof
Let \(v \in V\) and let \(\Gamma (v,\infty )\) be the future of v in \(\mathfrak {F}\). Let \(v=u_0,\ldots ,u_n\) be the first \(n+1\) vertices in the path \(\Gamma (v,\infty )\). For each \(0\le i \le n\), let \(W_i=\{w_{i,1},\ldots ,w_{i,m_i}\}\) be a finite (possibly empty) collection of vertices of G, and let \(\mathscr {A}_i\) be the event that for every vertex \(w \in W_i\), w is in \(\mathfrak {B}(v,n)\) and that the path connecting w to v first meets \(\Gamma (v,\infty )\) at \(u_i\).
We now prove Proposition 6.1.
Proof of Propositions 6.1 and 6.2
6.2 Lower bounds
In this section, we give lower bounds on the first moment of the volume of the past, and derive upper bounds on the probability that the volume of an intrinsic ball is atypically small.
We begin with the following simple lower bounds on the first moments. We write \(\mathfrak {P}(v,n)\) for the ball of radius n around v in the past of v in \(\mathfrak {F}\), and write \(\partial \mathfrak {P}(v,n)\) for the set of points that are in the past of v in \(\mathfrak {F}\) and have intrinsic distance exactly n from v.
Lemma 6.9
Remark 6.10
If G is a transitive unimodular graph, the masstransport principle yields the exact equality \(\mathbb {E}\partial \mathfrak {P}(v,n) =1\) for every \(n\ge 1\).
Proof
Our next goal is to prove the following lemma.
Lemma 6.11
Lemma 6.11 has the following immediate corollary, which is proved similarly to Corollary 6.4 and which together with Corollary 6.4 establishes that \(d_f(T)=2\) for every component of \(\mathfrak {F}\) almost surely (as claimed in Theorem 1.3). We remark that Barlow and Járai [11] established much stronger versions of Lemma 6.11 and Corollary 6.12 in the case of \(\mathbb {Z}^d\), \(d\ge 5\).
Corollary 6.12
In order to prove Lemma 6.11, we first show that the volume of the tree can be lower bounded with high probability in terms of quantities related to the capacity of the spine, and then show that these quantities are large with high probability.
Lemma 6.13
 1.The estimateholds for every \(n\ge 0\) and \( 0 \le k \le n\).$$\begin{aligned} \mathbb {E}\Big [\mathfrak {B}(v,2n) \;\Big \; \Gamma \Big ] \succeq \, (k+1) \, \mathrm {Cap}_k\bigl (\Gamma ^{n},\Gamma \bigr ) \end{aligned}$$(6.8)
 2.If \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), thenfor every \(1\le k \le n\).$$\begin{aligned} \mathbb {P}\biggl (\mathfrak {B}(v,2n) \le \frac{k\mathrm {Cap}_k(\Gamma ^{n},\Gamma )}{2 \sup _{u\in V} c(u)} \; \bigg  \; \Gamma \biggr ) \preceq \frac{(k+1)(n+1)}{\mathrm {Cap}_k(\Gamma ^{n},\Gamma )^2} \end{aligned}$$(6.9)
Proof
Lemma 6.14
Proof
Proof of Lemma 6.11
7 Critical exponents
In this section we apply the estimates obtained in Sects. 5 and 6 to complete the proofs of Theorems 1.2 and 1.4–1.6.
We will also prove the following extensions of these theorems to the vwired case.
Theorem 7.1
Theorem 7.2
Theorem 7.3
Theorem 7.4
7.1 The intrinsic diameter: upper bounds
Lemma 7.5
Before proving this lemma, let us establish the following corollary of it.
Corollary 7.6
Let G be a network with controlled stationary measure satisfying \(\Vert {P}\Vert _{\mathrm {bub}}<\infty \), and let Q(n) be as above. Then \(Q(n) \preceq n^{1}\) for all \(n\ge 1\).
Proof
We now turn to the proof of Lemma 7.5. We will require the following estimate. Recall that \(\Gamma _v(u,\infty )\) denotes the future of u in \(\mathfrak {F}_v\), which is equal to the path from u to v if \(u\in \mathfrak {T}_v\).
Lemma 7.7
Proof
Proof of Lemma 7.5
Fix \(v\in V\). Let \(\mathscr {I}_v\) be the vwired interlacement process on G, let \(\mathfrak {F}_{v,t} = \langle \mathsf {AB}_{v,t}(\mathscr {I}_v) \rangle _{t\in \mathbb {R}}\), and let \(\mathfrak {B}_{v,t}(v,n)\) denote the ball of radius n about v in \(\mathfrak {F}_{v,t}\) for each \(t\in \mathbb {R}\) and \(n\ge 0\). Recall that for each \(t\in \mathbb {R}\), \(\sigma _t(v) =\sigma _t(v,\mathscr {I}_v)\) is the first time greater than or equal to t such that v is hit by a trajectory of \(\mathscr {I}_v\) at time \(\sigma _t(v)\).
For each two vertices u and v of G, every \(t\in \mathbb {R}\) and \(n \ge 0\), let \(\mathscr {B}_{t,n}(u,v)\) be the event that \(u \in \mathfrak {B}_{v,t}(v,2n){\setminus }\mathfrak {B}_{v,t}(v,n)\), and let \(\mathscr {C}_{t,n}(u,v)\subseteq \mathscr {B}_{t,n}(u,v)\) be the event that \(\mathscr {B}_{t,n}(u,v)\) occurs and that v is connected to \(\partial \mathfrak {B}_{v,t}(v,3n)\) by a simple path that passes through u.
7.2 The volume
Proof of Theorems 7.1 and 1.2
7.3 The extrinsic diameter: upper bounds
In this section we prove our results concerning the tail of the extrinsic diameter of the past.
We begin with the proofs of Theorems 1.6 and 7.4, which are straightforward.
Proof of Theorems Theorems 1.6 and 7.4
The lower bounds follow immediately from Proposition 4.3. Since the extrinsic diameter is bounded from above by the intrinsic diameter, the upper bounds are immediate from Theorems 7.1 and 1.2. \(\square \)
Next, we deduce the upper bounds of Theorems 1.5 and 7.3 from the following more general bound.
Proposition 7.8
Proof
Proof of Theorems 7.3 and 1.5
The lower bounds both follow immediately from Proposition 4.3 and Lemma 4.4, while the upper bounds are immediate from Proposition 7.8. \(\square \)
We now wish to improve this argument and remove the logarithmic correction in the case of \(\mathbb {Z}^d\), \(d\ge 5\). To this end, let \(\mathfrak {F}_0\) be the 0wired uniform spanning forest of \(\mathbb {Z}^d\), and let \(\mathfrak {T}_0\) be the component of 0 in \(\mathfrak {F}_0\). (We do not consider any time parameterised forests in this section, so this notation should not cause confusion.) We write \(\Lambda _m=[m,m]^d\) and \(\partial \Lambda _m = \Lambda _m{\setminus }\Lambda _{m1}\). We say that a vertex \(v\in \partial \Lambda _m\) is a pioneer if it is in \(\mathfrak {T}_0\) and the future of v (i.e. the unique path connecting v to 0 in \(\mathfrak {T}_0\)) is contained in \(\Lambda _m\).
Lemma 7.9
Note that the expectation of \(\partial \Lambda _m \cap \mathfrak {B}_0(0,n)\) is of order \(m e^{\Omega (m^2/n)}\). The point of the lemma is that by considering only pioneers we can reduce the expectation by at least a factor of m.
Before proving Lemma 7.9, let us see how it can be applied to deduce Theorems 7.2 and 1.4.
Proof of Theorems 7.2 and 1.4
The proof of Lemma 7.9 will come down to a few somewhat involved estimates of diagrammatic sums involving random walks on boxes with Dirichlet boundary conditions. We write \(\preceq \) for upper bounds depending only on the dimension d, and, to simplify notation, use the convention that \(0^{\alpha }=1\) for every \(\alpha \in \mathbb {R}\). We also use \(\Omega \) asymptotic notation (\(f=\Omega (g)\) is equivalent to \(f \succeq g\)), where again the implicit constants depend only on d.
We will also use the following estimate.
Lemma 7.10
\(\max _{w \in \partial \Lambda _m} \mathbf {G}^m_\infty (v,w) \preceq \left( m+1\Vert v\Vert _\infty \right) ^{d+1} \) for every \(v \in \Lambda _m\).
Proof
We now turn to the proof of Lemma 7.9.
Proof of Lemma 7.9
8 Spectral dimension, anomalous diffusion
In this section, we apply the estimates of Sect. 6 together with the intrinsic diameter exponent Theorem 7.1 to deduce Theorem 1.3. This will be done via an appeal to the following theorem of Barlow, Járai, Kumagai, and Slade [12], which gives a sufficient condition for Alexander–Orbach behaviour. See [12] for quantitative versions of the theorem, and [48] for generalizations.
Theorem 8.1
The estimate (8.1) has already been established in Corollary 6.3 and Lemma 6.11. Thus, to apply Theorem 8.1, it remains only to prove an upper bound on the probability that the effective conductance is large. The following lemma will suffice.
Lemma 8.2
We begin with the following deterministic lemma. Arguments of the form used to derive this lemma are well known, and a similar bound has appeared in [13, Lemma 4.5].
Lemma 8.3
Proof
Proof of Lemma 8.2
9 Applications to the Abelian sandpile model
Let G be a transient graph and let \(\mathrm {H}\) be a uniform infinite recurrent sandpile on G, as defined in Sect. 1.6. Let \(\mathfrak {F}\) be the wired uniform spanning forest of G, let \(\mathfrak {F}_v\) be the vwired uniform spanning forest of G for each vertex v of G, let \(\mathfrak {T}_v\) be the component of v in \(\mathfrak {F}_v\), and let \(\mathbf {G}\) be the Greens function on G.
 1.Dhar’s formula [24] states that the expected number of times u topples when we add a grain of sand at v is given by the Greens function. That is,See also [38, Section 2.3]. (Note that the right hand side is also the Green’s function for continuous time random walk.)$$\begin{aligned} \mathbb {E}\left[ {\text {Av}}_v(\mathrm {H},u) \right] = \frac{\mathbf {G}(v,u)}{c(u)}. \end{aligned}$$(9.1)
 2.The avalanche cluster at v approximately stochastically dominates the past of v in the WUSF. More precisely, for any increasing Borel set \(\mathscr {A}\subseteq \{0,1\}^V\), we have thatThis follows from the discussion in [21, Section 2.5], see also equation (3.2) of that paper.$$\begin{aligned} \mathbb {P}\left( {\text {AvC}}_v(\mathrm {H}) \in \mathscr {A}\right) \ge \frac{1}{\deg (v)} \mathbb {P}\left( \mathfrak {P}(v) \in \mathscr {A}\right) . \end{aligned}$$(9.2)
 3.The diameter of the avalanche cluster at v is approximately stochastically dominated by the diameter of the component of v in the vWUSF. More precisely, we have thatThis follows from [21, Lemma 2.6].$$\begin{aligned} \mathbb {P}\left( \mathrm {diam}_\mathrm {ext}\left[ {\text {AvC}}_v(\mathrm {H}) \right] \ge r\right) \le \frac{\mathbf {G}(v,v)}{c(v)} \mathbb {P}\left( \mathrm {diam}_\mathrm {ext}\left[ \mathfrak {T}_v \right] \ge r\right) . \end{aligned}$$(9.3)
We now apply these relations to deduce Theorems 1.7–1.9 from the analogous results concerning the WUSF and vWUSF.
Proof
Footnotes
 1.
This terminology arises from the definition of the past in terms of an oriented version of \(\mathfrak {F}\), see Sect. 1.3.
 2.
That work studies the IIC on the 3regular tree, rather than the WUSF. We recall however that the IIC and the component of the origin in the WUSF have the same distribution on a kregular tree, namely that of (the unimodular version of) a critical Binomial Galton–Watson tree conditioned to survive forever.
Notes
Acknowledgements
We thank Martin Barlow, Antal Járai, and Perla Sousi for helpful discussions, and thank Russ Lyons for catching some typos. I also thank the two anonymous referees for their close and careful reading of the paper; their comments and suggestions have greatly improved the paper. Much of this work took place while the author was a Ph.D. student at the University of British Columbia, during which time he was supported by a Microsoft Research Ph.D. Fellowship.
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