Inequalities for critical exponents in $d$-dimensional sandpiles

Consider the Abelian sandpile measure on $\mathbb{Z}^d$, $d \ge 2$, obtained as the $L \to \infty$ limit of the stationary distribution of the sandpile on $[-L,L]^d \cap \mathbb{Z}^d$. When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In $d = 2,$ we show that for any $1 \le k<\infty$, the last $k$ waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the tail of the radius distribution.


Introduction
The Abelian sandpile model is a particle system defined in terms of simple local redistribution events, called topplings, which give rise to non-local dynamical events called avalanches. The model has received a lot of attention in the theoretical physics literature (see [9,3]) due to its remarkable self-organized critical state, conjectured to be characterized by power-law behavior of various quantities related to avalanches. Starting with the seminal work of Dhar, much mathematical progress has been made toward understanding this self-organized critical state. The surveys [33] and [14] collect some of this. However, establishing power law behavior for many fundamental avalanche characteristics on Z d appears difficult in general. The purpose of this paper is to establish new rigorous inequalities, which in high dimensions come close to identifying the correct tail behavior, for these quantities. (1.1) Particles sent to vertices in Z d \ V are lost. It is well-known [8] that given any sandpile η, carrying out all possible topplings in any sequence, results in a uniquely defined stable sandpile η • . The sandpile Markov chain on V is the Markov chain with state space equal to the set of stable sandpiles on V , where at each time step a particle is added at a uniformly chosen vertex of V , and the sandpile is stabilized, if necessary. The unique stationary distribution [8] is denoted ν V .
We will be interested in sandpiles on Z d , where "stable" and "toppling" are defined the same way as for finite V . Athreya and Járai [2] proved that if V (L) = [−L, L] d ∩ Z d , d ≥ 2, then ν L := ν V (L) converges weakly, as L → ∞, to a limit measure ν, called the sandpile measure. Let η : Z d → {0, . . . , 2d − 1} be a sample configuration from the measure ν. Let us add a particle to η at the origin o, and let Av = Av(η) denote the set of vertices that topple, called the avalanche cluster. The set of all topplings, with multiplicity, is called the avalanche. In this paper we study various characteristics of avalanches.
The concept of waves, introduced by Ivashkevich, Ktitarev and Priezzhev [13] in the context of finite graphs, will play an important role. Waves provide a decomposition of an avalanche into smaller sets of topplings: W L,1 , . . . , W L,N ⊂ V (L); see Section 2 for precise definitions. In each wave, every vertex topples at most once, and the union of the waves includes all topplings of the avalanche with the correct multiplicity. The paper [13] analyzed the last wave W N,L in particular when d = 2.
Our first set of results concern the probability that a given vertex topples. Based on an analysis of the last wave, we prove the following rigorous lower bounds on the toppling probability. (i) Let d = 2. Then ν(z ∈ Av) ≥ |z| −3/4+o (1) , as |z| → ∞.
In Theorem 1.1 (ii), ζ can be taken to be any value such that a random walk in Z 3 of length n does not hit the loop-erasure of an independent random walk of length n with probability ≥ c n −ζ for some c > 0. We prefer to write the bound (ii) in this form to emphasize the dependence on this exponent, whose value is of interest in the theory of loop-erased walks. The exponent ζ is known to satisfy the bound ζ < 1/2; see [24,Sections 10.3 and 11.5].
In particular, Theorem 1.1(iii) establishes that θ = 0 when d = 4, with at most a logarithmic correction.
The reason behind the fact that θ = 0 for d ≥ 5 is that in these dimensions loop-erased walk and independent simple random walk do not intersect with positive probability.
The difference in behavior when d ≥ 5 also shows up in our other results, and d = 4 is expected to be the upper critical dimension of the model, in the sense that critical exponents are no longer expected to depend on dimension when d ≥ 5 [32]. We expect that θ is positive in dimensions two and three, in analogy with other statistical physics models below the upper critical dimension. However, it seems difficult to get rigorous upper bounds improving on Dhar's formula, since any such bound would have to control all waves of the avalanche. For the last wave, we have a precise characterization in terms of loop-erased walk; however, we lack a convenient description of the joint distribution of all waves of the avalanche. For similar reasons, we do not expect the bounds coming from last waves in low dimensions to be tight.
Our next set of results concern the radius of the toppled region. Let R = R(η) = sup{|z| : z ∈ Av(η)} be the radius of the avalanche. As we explain below, some of the following inequalities are easy consequences of Theorem 1.1, while some others follow from known results on uniform spanning forests of Z d .
The lower bounds of Theorem 1.2 in dimensions 2, 3, and 4 follow from taking z = re 1 in Theorem 1.1, where e 1 = (1, 0, . . . , 0) ∈ Z d . In dimensions d ≥ 3, upper bounds can be derived from results of Lyons, Morris and Schramm [25]. They analyzed, using the "conductance martingale" of Morris [30], the wired uniform spanning forest measure Aut(G) has a Borel measure which is invariant under the left multiplication by γ ∈ Γ. The group Γ is called unimodular if this measure is also invariant under right multiplication. In addition, we call the graph G unimodular if Aut(G) has some unimodular transitive closed subgroup. In this setting the mass transport principle states that for o ∈ V (G) and a non-negative function f : V × V → [0, ∞], which is invariant under the diagonal action of Γ, we have x∈V f (o, x) = x∈V f (x, o). Let d be a Γ-invariant metric on V , and write diam(A; x) = sup{d(v, x) : v ∈ A}, and let diam(A) = diam(A; o). Write D x (r) = {y ∈ V : d(y, x) ≤ r}. We say that an infinite tree T has one end, if any two infinite self-avoiding paths in T have a finite symmetric difference. Given x ∈ T , we denote by past x the set of vertices y ∈ T such that the unique infinite self-avoiding path in T starting at y contains x. By a percolation on (V, E), we mean a probability measure on subgraphs of (V, E). Given a percolation, we write C x for the connected component of x. When the percolation is supported on spanning forests, we write T x for C x .
Regarding the upper bound in Theorem 1.2 (iv), Lyons, Morris and Schramm state the result WSF(diam(past o ) > r) ≤ r −2 (log r) O(1) ; (1.3) see [25, page 1710]. However, since a proof of (1.3) is not included in [25], and we need a sharpening of (1.2) for our results, we deduce a diameter estimate for T o under  3) due to a stochastic comparison; see [25,Lemma 3.2]). In order to deal with the fact that WSF o is not translation invariant, we restrict attention to T o , which is unimodular; see Section 6.1.
We do not have an upper bound on ν(R ≥ r) in d = 2, and it is an open problem whether ν(R < ∞) = 1. It follows from Theorem 1.2(i) that E ν R = ∞, when d = 2. It may be of independent interest that a short proof of the weaker statement, that E ν L R diverges, can be given without reference to spanning trees or the burning bijection described in Section 2.3. We state this as a separate result. Proposition 1.4. If d = 2, then lim L→∞ E ν L R = ∞.
Our last set of results concern the number of topplings in the avalanche. Let S denote the total number of topplings in the avalanche (that is, elements of Av are counted with multiplicity). Recall that W N,L denotes the last wave in the finite graph V L . Based on the fractal dimension of loop-erased walk and scaling assumptions, Ivashkevich, Ktitarev and Priezzhev [13] derived the exponent ν L (|W N,L | ≥ t) ≈ t −3/8 , in the limit L → ∞.
We prove a rigorous lower bound with the same exponent, which we also extend to higher dimensions. Above the critical dimension, d ≥ 5, we also have an upper bound on the total number of topplings with an exponent which is independent of d. The upper bounds of Theorem 1.2 on the radius in d = 3, 4 provide upper bounds on the size of the avalanche cluster.
Since the size of the avalanche cluster could be measured in two different waysnamely, via |Av| and via S -there are in principle two different possible critical exponents τ S and τ S given (if they exist) by ν(|Av| > t) ≈ t −τ S , ν(S > t) ≈ t −τ S . As in the preceding cases, our theorems give corresponding bounds on the possible values of τ S , τ S . These bounds, as well as the best current bounds on the exponents θ and α described above, are summarized in Table 1.
(ii) Let d = 3. With ζ as in Theorem 1.1, and for some constants C and c > 0, we have There exists C and c > 0 such that We establish the lower bounds in dimensions d = 2, 3 by showing that once a vertex at distance t 1/d from o is in the last wave, at least ct other vertices in its neighbourhood will also be in the last wave. In d = 4, ct is replaced by ct/ log t. Given this, parts (i)-(iii) of Theorem 1.5 can be deduced from parts (i)-(iii) of Theorem 1.1. For the lower bound in d ≥ 5, in Theorem 1.5(iv), we use the following analogue of Theorem 1.3(i). Write The proof of Theorem 1.6 is in Section 7.4 and does not rely on the rest of the paper.
We believe, as it has been argued by Priezzhev [32], that the exponent 1/2 is sharp in Theorem 1.5(iv). But it seems challenging to establish a matching upper bound for S or Av. The main difficulty lies in extracting useful information on the dependence between the waves from the bijection with WSF o . Instead of such an approach, we control the number of waves using our upper bound on the radius in Theorem 1.2(iv), which allows us to use a union bound instead of estimating the dependence. This leads to the upper bounds in Theorem 1.5(ii)-(iv).
All our results have analogues in large finite V (L), or indeed are derived therefrom. Passing to the limit of Z d is not too difficult when d ≥ 3, due to the result of Járai and Redig [15,Theorem 3.11] showing that ν(|S| < ∞) = 1. When d = 2, this is not known.
We bypass this problem with a more technical argument, that we believe is of interest in its own right. We show that for any 1 ≤ k < ∞, the last k-waves (when they exist) have a finite limit as V (L) ↑ Z d . Recall that for η ∈ R L the waves occurring during the stabilization of η + 1 o are denoted W L,1 , . . . , W L,N . Waves can also be defined on Z d , denoted W 1 , W 2 , . . . ; see Section 2.2. On Z d , the number of waves N may take the value infinity.
Let η N −k+1 denote a random configuration on V (L) with law ν L (· | N ≥ k). Given a configuration η on V (L) such that η(o) = 2d, let W(η) denote the set of sites that can be toppled with every site toppling at most once. We extend this also to configurations ξ on Z d such that ξ(o) ≥ 2d. (i) For all k ≥ 1, the law of η N −k+1 converges weakly to the law of a random configuration ξ k in Z 2 . Let ρ k denote the law of ξ k . The law of W N −k+1 = W(η N −k+1 ) converges to the law of W * k := W(ξ k ), that is a.s. finite under ρ k . (ii) For all k ≥ 0 we have lim L→∞ ν L (N = k) = ν(N = k).
See Section 5 for more detailed statements. Our argument in fact gives a power law upper bound on the radii of the last k waves, and leads to the following extension of Theorem 1.2(i). Let R k = sup{|x| : x ∈ W * k }. Theorem 1.8. Assume d = 2.
This is a strengthening of the statement E ν R = ∞, due to a simple comparison proved in Lemma 2.3, and therefore also of Proposition 1.4.
Organization of the paper. In Section 2, we give definitions and background on sandpiles, spanning trees, and random walks; we also prove Proposition 1.4. In Section 2.2, we prove Theorem 1.1 modulo a technical argument required for the two-dimensional case, which we defer to Section 5.
Sections 4, 5, and 6 are devoted to the various radius bounds above. In Section 4, we prove Theorem 1.2 (i) -(iii). Section 5 contains additional arguments for the twodimensional case; here we prove Theorems 1.7, 1.8, and 1.9. In Section 6, we complete the proof of Theorem 1.2 by proving the high-dimensional bounds (Theorem 1.2 (iv) and Theorem 1.3).
Section 7 contains the proofs of the size bounds above: Theorems 1.5 and 1.6.
A note on constants. All our constants will be positive and finite, and they may depend on the dimension d. Other dependence will always be indicated. Constants denoted C and c may change from line to line; those with index (such as c 1 ) stay the same within the same proof.

Definitions and background
In this section, we provide definitions and collect useful facts about the basic objects we use: toppling numbers, waves, spanning trees, bijections and Wilson's algorithm.

Graphs and sandpiles
We will work with finite connected graphs of the form H = (U ∪ {s}, F ), where s is a distinguished vertex, called the sink. We allow multiple edges, so in general H is a multigraph, but we exclude loop-edges. If U = V is a finite subset of Z d , we let G V = (V ∪ {s}, E) denote the wired subgraph induced by V -i.e., where all vertices in Z d \ V are identified to the single vertex s, and loops at s are removed. Of prime importance will be the standard exhaustion V (L) = [−L, L] d ∩ Z d , L ≥ 1; the wired subgraph induced by V (L) will be denoted G L . In general, a subscript L will be shorthand for subscript V (L). We write x ∼ y to denote that vertices x and y of a graph (understood from context) are neighbours. Given a graph H = (U ∪ {s}, F ), we write deg H (x) for the degree of the vertex x ∈ U ∪ {s} in H. When V ⊂ Z d , we write deg V (x) for the degree of vertex x ∈ V in the subgraph of Z d induced by V .
where a xy equals the number of edges connecting x and y. In particular, when V is a finite subset of Z d , we have ∆ V given by We denote the inverse matrix by We write ∆ for the matrix defined as in (2.2), but with x, y ∈ Z d , and g = ∆ −1 when d ≥ 3. Up to a factor (2d) −1 , these matrices are the Green function of simple random walk. Namely, let (S(n)) n≥0 denote a simple random walk in Z d , and let σ A = inf{n ≥ 0 : S(n) ∈ A}.
Then 2d g V (·, ·) = G V (·, ·) and 2d g(·, ·) = G(·, ·), where we define Whenever G appears with arguments, it refers to a Green function as defined above; when it appears without arguments, it refers to graphs as in the notation introduced previously.
Let us fix a finite connected graph H = (U ∪ {s}, F ). A sandpile on H is a function η : U → {0, 1, 2, . . . }. We say that η is unstable at x ∈ U , if η(x) ≥ deg H (x). In this case x is allowed to topple, which means that x sends one particle along each edge incident with it. Particles arriving at s are lost. Toppling x has the effect of subtracting row ∆ H (x, ·) from η(·). It is a basic property of the model that if unstable vertices are toppled in any order until there are no such vertices, the stable sandpile obtained is independent of the order chosen (called the Abelian property) [8]. Hence for any sandpile η there is a well-defined stabilization of η, denoted η • . The sandpile Markov chain is defined as follows. The state space is Ω H = x∈U {0, . . . , deg H (x) − 1}. Given that the current state is η, a single step is defined by choosing a vertex X ∈ U uniformly at random, and moving to state (η + 1 X ) • . The maps a x : are called the addition operators. It follows from the uniqueness of stabilization that a x a y = a y a x , x, y ∈ U . When it is necessary to emphasize the graph H on which the operator a x is applied, we write a x,H for a x ; we also use a x,L for a x,G L . We denote the set of recurrent states of the sandpile Markov chain by R H . It is known that the unique stationary distribution is given by the uniform distribution on R H [8], and that each (restricted) map a x : R H → R H preserves this measure.
In the special case when the graph arises from a finite V ⊂ Z d , we denote the state space by Ω V = {0, . . . , 2d − 1} V , and the set of recurrent states by R V . Now, let η : Z d → {0, 1, 2, . . . } be a sandpile on Z d . If x is unstable in η, we define the toppling of x using the matrix ∆, that is, η → η(·) − ∆(x, ·). Let us call a finite or infinite sequence consisting of topplings of unstable vertices exhaustive, if any vertex that is EJP 22 (2017), paper 85. unstable at some point, is toppled at a later time. It can be shown, similarly to the finite graph case, that for all x ∈ Z d , x topples the same number of times (possibly infinity) in any exhaustive sequence.
We write e i for the unit vector in the i-th positive coordinate direction, | · | for the Euclidean norm, and · for the ∞ norm on Z d . We denote by o the origin in Z d , and When considering A as a subset of G L , we will often use ∂A to denote the boundary restricted to G L -that is, excluding from ∂A any x / ∈ G L -the meaning will be clear in context.
If z 1 , z 2 are two elements of R d , let ang(z 1 , z 2 ) denote the angle between z 1 and z 2 . We will make use of the "little o" notation: a n = n a+o(1) if lim n [ log a n / log n] = a.

Toppling numbers and waves
Given a sandpile η on a finite connected graph H = (U ∪ {s}, F ), we write n(x, y) = n(x, y; η) for the toppling numbers, i.e., number of times y topples when stabilizing η + 1 x . A useful ordering of topplings, introduced by Ivashkevich, Ktitarev and Priezzhev [13], for stabilizing η + 1 x is in terms of waves, which we now define. If η + 1 x is stable, there are no waves. Otherwise, topple x and carry out any further topplings that are possible without toppling x a second time. It is easy to verify that in doing so, every vertex in U topples at most once. We write W 1,H = W 1,H (x; η) for the set of sites toppled so far; this is the first wave. If x is still unstable after the first wave, which happens if and only if all neighbours of o are in W 1,H , topple x a second time, and carry out any further topplings that are possible without toppling x a third time. The set of vertices that topple in doing so is denoted W 2,H = W 2,H (x; η); this is the second wave. We define further waves analogously.
When the graph arises from a finite V ⊂ Z d , we write W 1,V , W 2,V , . . . for the waves. We make similar definitions for sandpiles on Z d . On Z d , the toppling numbers n(x, y; η) are possibly infinite. Waves W 1 , W 2 , . . . are defined analogously to the finite case, and their number may be infinite. The following lemma is straightforward to verify.

Lemma 2.1.
(i) Let H = (U ∪ {s}, F ) be a connected finite graph. The number of waves containing y equals n(x, y; η). In particular, the number of waves is N = N (x; η) := n(x, x; η). (ii) In the case of Z d , the number of waves containing y equals n(x, y; η) (possibly infinite). In particular, the number of waves is N = N (x; η) := n(x, x; η).
The next lemma gives the expected number of topplings in an avalanche.
For cubes in Z d , we have the following deterministic comparison between n(o, o; η) and the radius R(η).  (ii) The same statement holds when η is a sandpile configuration in G L for any L ≥ 1. Proof. (i) In the proof below, it will be convenient to denote n[η] := n(o, o; η). We will also use the notion of "restricted topplings". For L > 0, let η| L denote the restriction of η to V (L); the restricted toppling number n L [η] will denote the number of topplings occurring at o in the stabilization of η| L + 1 o on G L . Note that such a stabilization amounts to taking the configuration η + 1 o on Z d and toppling only sites x ∈ V (L) until all such x are stabilized, neglecting any sites in Z d \ V (L) that may become unstable. In particular, for any stable sandpile configuration η on Z d , n[η] ≥ n L [η] for any L.
Assume η and R < ∞ are as in the statement of the lemma. We claim that the equality n[η] = n R [η] holds. Indeed, by the above observation, n R is exactly the number of topplings at o required to stabilize only the sites of V (R), but by assumption these are the only sites which need to be toppled to stabilize η + 1 o in all of Z d . Therefore, it suffices to show that n R [η] ≤ R.
For this, we will use a special "maximal" configuration φ R from [11,Lemma 4.2]: is proven in [11,Lemma 4.2], and not difficult to see by computing each wave, that n[φ R ] = R. This completes the proof of (i).
(ii) The above proof applies here as well, with only minor changes.

Spanning trees and the burning bijection
In 1990, Dhar [8] introduced a method for checking whether a particular stable configuration η lies in R H , called the "burning algorithm". Application of the burning algorithm provides a bijection ϕ between R H and the set of all spanning trees of H, that we denote by T H ; see [28]. We briefly describe this bijection here. In Section 2.5, we give a version of it for "waves" which will be necessary in our analysis.
Recall that a xy denotes the number of edges between vertices x and y. Note that instead of adding sand as in the lemma, we may initiate the toppling process by placing all x∈U a xs = deg H (s) grains at s, and toppling s first. Suppose we carry out any possible topplings in parallel. We say that x burns at time k, if it is toppled in the k-th parallel toppling step, where we regard s to have burnt at time 0. The bijection is defined as follows. For each y ∈ U , fix an arbitrary ordering ≺ y of the edges adjacent to y. Given η ∈ R H , for each y ∈ U , we adjoin to the tree T an edge connecting y to a neighbour burnt one time step before, chosen as follows. If P y is the EJP 22 (2017), paper 85. number of edges joining y to neighbours burnt before y, and A y is the subset of such edges leading to sites burnt one step before y, then the burning rule implies η(y) = deg H (y) − P y + i for some 0 ≤ i < |A y | .
We add to T the i-th edge in A y in the ordering ≺ y .
The resulting graph T will be a spanning tree (the fact that it spans -i.e., that every site topples in this procedure -is part of the content of Lemma 2.5), and we set ϕ(η) = T . The map ϕ : R H → T H is usually referred to as the "burning bijection", and the toppling procedure used to construct ϕ will be referred to as the "burning procedure".

Intermediate configurations
Now, we give a description of waves in terms of recurrent configurations on an auxiliary graph, as introduced in [13]. As in the previous section, we describe this in an arbitrary finite connected graph H = (U ∪ {s}, F ). Suppose we are interested in waves started by the addition of a particle at a fixed vertex w ∈ U . Consider the graph H obtained from H by adding the edge f := {w, s}. For readability, we denote ∆ H := ∆ H and R H := R H . Let a x = a x,H , x ∈ U denote the addition operators on H . We reserve the notation η • for stabilization on the original graph H. When we need to emphasize that addition is applied on the graph derived from H, we prefer the a x,H notation.
The burning algorithm (Lemma 2.5) implies that R H ⊂ R H . The following lemma compares the sizes of these two sets.
Proof. Let 1 w,w denote the U × U matrix whose only non-zero entry is a 1 at (w, w). We 3) The following immediate corollary will be useful in controlling avalanches. A version of part (i) was proved in [15,Lemma 7.5].
There exists a constant C such that |R L | ≤ C log L |R L | for all L ≥ 2.
Proof. Both statements follow from Lemma 2.6, the equality g L (o, o) = (2d) −1 G L (o, o), and known properties of the Green function G L (o, o); see e.g. [24].
We now describe the interpretation of waves as elements of R H \ R H ; introduced in [13]. Let η ∈ R H , and suppose that η + 1 w is unstable at w in H, i.e. η(w) = deg H (w) − 1. Consider the waves occurring in stabilizing η + 1 w in H. Recall that N = N (η) denotes the number of waves. For 1 ≤ k ≤ N, let η k = a w η k−1 , where η 0 = η. It is straightforward to check that η k is the configuration seen just before the k-th wave is carried out, and a w η = (a w ) N +1 η. Note that the latter statement also holds, trivially, when η + 1 w is stable, in which case N = 0. Definition 2.8. Let η ∈ R H be such that η + 1 w is unstable at w. We call the sequence α(η) := (η 1 , . . . , η N ) the intermediate configurations corresponding to η.
We record here the characterization of R H \ R H ; a similar statement was shown in [16] for a continuous height model.  Proof. Since η k (w) = deg H (w), k = 1, . . . , N , we have η k ∈ R H \ R H , k = 1, . . . , N . This and the relation a w η = (a w ) N +1 η imply that α(η) has distinct entries (the order of a w is at least N + 1). By Dhar's formula, the average number of waves per recurrent configuration is g H (w, w), so g H (w, w)|R H | elements of R H correspond to intermediate configurations.
It is similarly easy to check that η ∈ R H , accounting for another |R H | elements of R H . Comparing this with Lemma 2.6, we see that every element of R H is either an intermediate configuration or recurrent on H, completing the proof.
Given η * ∈ R H \ R H , we denote by W(η * ) the set of vertices that topple in the stabilization a w,H (η * ). It is immediate from this definition that if α(η) = (η 1 , . . . , η N ), then W(η k ) is the k-th wave corresponding to η, i.e. W k,H (w; η). Of particular interest will be the last wave W(η N ). The following corollary follows directly from the definitions.
We will also need the following lemma.
Proof. The upper bound is obvious. To see the lower bound, we assign to η ∈ R H the last intermediate configuration in the stabilization of η + (deg H (w) − η(w))1 w . This map is at most deg H (w) to 1, proving the lower bound.

Bijection for intermediate configurations
In this section, we describe a version of the burning bijection on G V , that will allow us to control topplings occurring in a wave. To the best of our knowledge, such a bijection was first introduced by Ivashkevich, Ktitarev and Priezzhev [13]. See also [32,15,16], where it played a key role. For many of our results, the variant in [13] would suffice. However, a more careful choice of the burning process will be needed in Section 5, so we introduce here the version we need. Our burning process is similar to burning processes introduced in [17] and [10].  For each r ≥ 1 there exists a smallest index J = J(r) ≥ 1 such that Bt (r) J = ∅, and there is a smallest index R ≥ 1 such that J(R + 1) = 1. Then Bt (R+1) 0 = W(η * ) is the set of vertices toppled in the wave represented by η * .
We complete the burning process by sending a sx grains of sand from s to x for each x ∈ V , and follow the usual burning rule. That is, we set: We now define the bijection.
, then there exists a unique pair k . Due to the definition of the burning rule, there exists at least one y ∼ x such that y ∈ Bt We select an edge that connects u to one of these vertices, using the ordering ≺ u , as in Section 2.3. Namely, if P u is the number of edges joining u to neighbours in ∪ <k Bt (r) , and A u is the subset of such edges leading to vertices in Bt there is a unique k ≥ 1 such that u ∈ Bt k , and there exists at least one y ∼ x with y ∈ Bt k−1 . We select an edge to one of these vertices, using the ordering ≺ u as before. Namely, if P u is the number of edges joining u to neighbours in ∪ <k Bt , and A u is the subset of such edges leading to vertices in Bt k−1 , then necessarily We add to T s the i-th edge in A u in the ordering ≺ u . Let ϕ (η * ) := (T o , T s ) denote the two components spanning forest obtained by the above construction. Let us write T V,o for the set of all spanning forests of G V rooted at {s, o}.
(iii) We have the following property: Tracing the burning process to the first time when is a subset of the set of spanning forests of G V rooted at {o, s}. By the matrix-tree theorem applied to G V , the number of spanning so statement (i) follows.
(ii) The burning process that was used to define Bt

Random walk notation and basic facts
Many of our techniques require a detailed analysis of spanning trees via Wilson's algorithm. For this reason, we will often have to consider collections of simple random walks ("SRW") and loop-erased random walks ("LERW") on (subsets of) Z d .
We will denote by S x = (S x (0), S x (1), . . .) an infinite simple random walk on Z d started at x, so that S x (0) = x. We will suppress the subscript when the choice of x is clear or when x = o. In general, S x and S y will be assumed independent when x = y. When multiple independent walks from one site are necessary, we write S x , S x (etc).
If A ⊂ Z d we define the standard stopping times Note that we suppress any dependence of these stopping times on the starting point x; when we write (for instance) S x (ξ A ), we are referring to the location of S x at its first hitting time on A. When the starting vertex may be ambiguous, we use subscripts on the symbol P; for instance, We will abbreviate σ V (n) to σ n , and write ξ o for ξ {o} . The trace of a walk S x between two times a < b (where b can be infinite) will be and similar notation will be used for other intervals-e.g., S(a, b) and so on. We will sometimes abuse notation and treat S x [a, b] as a sequence instead of an unordered set. If A ⊂ Z d is a finite connected set and if x, y ∈ Z d , recall the Green function As before, G V (n) is abbreviated G n . We will use the following standard asymptotics for the Green function inside a large ball and the probability of hitting o before exiting a large ball: Theorem 2.13 (See [23], Prop. 1.5.9 and 1.6.7).
(i) If d = 2, we have uniformly in x ∈ B(n): We also note the following simple observation about G. If K 1 ⊂ K 2 and x, y ∈ K 1 , (2.8) We will need the following result, usually called the Beurling estimate, which gives an upper bound on the probability that a path in Z 2 is not hit by SRW. Lemma 2.14 (Beurling estimate [19], [24, Section 6.8]). Consider Z 2 . There is a constant C such that the following bound holds, uniformly in x, n, and lattice paths α connecting o to ∂B(n):  If A x is a finite subset of Z d , let S A x denote a finite loop-erased random walk killed at the boundary of A: . We will also make use of infinite loop-erased random walks S x on Z d . When d ≥ 3, the definition S x := LS x is unambiguous with probability 1, as noted in the last paragraph.
For d = 2, the loop-erasure of the infinite random walk S x is not well-defined; in this case, S x is defined by taking limits. It is known (see [23,Section 7.4]) that for any finite lattice path γ with |γ| = k, exists, and the extension of this to a measure on infinite paths gives a definition of the distribution of S x . The rate of convergence of S B(n) x to S x is well-controlled; see Lemma 3.4 below. We will refer to all of the processes S x , S A x as loop-erased random walks or LERW. We will also assume as usual (unless stated otherwise) that a LERW is independent of any other walks appearing in a given statement.
We define LERW stopping times ξ, σ analogously to ξ and σ; for instance, As before, we will use x as a subscript on P when considering stopping times to indicate the starting point. When the LERW is finite (i.e., we are considering S A x ), we also will use the superscript A on P, writing (for instance) to avoid confusing the set in which the LERW lives with the set it is hitting. When the LERW is infinite, we omit superscripts altogether. One result important for analyzing LERW is the following "Domain Markov Property" (DMP). This roughly says that the terminal segment of a LERW can be built by starting a SRW at the tip of the initial segment, conditioning it not to hit the initial segment, then erasing loops.
Then for all paths β, Z d for d ≥ 3) and finite LERW on graphs which are not necessarily subsets of Z d .

Wilson's algorithm
Let H = (U ∪ {s}, F ) be a connected finite graph. Since ν H is uniform on R H , the bijection ϕ maps it to the uniform measure on spanning trees of H, which we denote by µ H . In this section, we collect basic facts about Wilson's algorithm, that we will use to analyze µ H . For an in-depth introduction to uniform spanning trees (UST) see the book [26].
We denote a sample from µ H by T H . We will usually identify a spanning tree of H with the set of edges it contains, so T H ⊂ F . Pemantle [31] proved that for x, y ∈ U , the path in T H between x and y is distributed as a LERW from x to y (i.e. as LS x [0, ξ y ]).
Wilson's algorithm [35] provides a method for constructing the full UST (a sample from µ H ) from LERWs.
Wilson's algorithm. Let v 1 , . . . , v n be an enumeration of the vertices in U . We construct a random sequence of tree subgraphs F 0 ⊂ . . . ⊂ F n . Let F 0 have vertex set {s} and empty edge set. If F i−1 has been defined for some 1 ≤ i ≤ n, we let S vi be a simple random walk on H started at v i , and let ξ V (Fi−1) be the first hitting time of the vertex set The output of the algorithm is F n . Wilson's theorem [35] implies that F n is uniform, i.e. distributed as T H .
The measure µ L := µ G L is known as the UST in V (L) with the wired boundary condition. For studying the L → ∞ limit of sandpiles on G L , as well as sandpiles on Z d , it will be useful to consider the weak limit lim L→∞ µ L =: WSF, called the wired spanning forest measure. Existence of the limit is implicit in [31]; see [26] for an in-depth treatment. We denote a sample from WSF by T.
It is well known that WSF concentrates on spanning forests of Z d all whose components are infinite. Pemantle [31] showed that for d ≤ 4, T is a tree WSF-a.s, while for d ≥ 5, T has infinitely many connected components WSF-a.s. This dichotomy is the underlying fact behind mean-field behaviour of the sandpile model for d ≥ 5; which is reflected in some of our results and proofs. We write T x for the component of T containing x ∈ Z d .
It is possible to construct T more directly, using an appropriate extensions of Wilson's algorithm.
, where the stopping time may be finite or infinite. See [6,26] for a proof that ∪ i≥1 F i has the distribution of T given by WSF. This is called Wilson's method rooted at infinity. When d = 2, a method analogous to that in finite volume can be used. We set [6,26].
It will be important for our proofs that WSF-a.s. each component of T has one end, for all d ≥ 2. This means that any two infinite self-avoiding paths lying in the same component of T have a finite symmetric difference. For 2 ≤ d ≤ 4 this was proved by Pemantle [31]. For d ≥ 5 this was first shown by Benjamini, Lyons, Peres and Schramm [6] (who generalized it to a much larger class of infinite graphs). Due to monotonicity, the weak limit lim L→∞ µ L,o =: WSF o exists; see [6,26]. We also use T to denote a sample from WSF o , and write T x for its component containing x ∈ Z d . In particular, T o is the component containing o under the measure WSF o . It was shown by Lyons, Morris and Schramm [25] that for d ≥ 3 (and more generally under a suitable isoperimetric condition), we have WSF o (|T o | < ∞) = 1, and also that this is equivalent to the one-end property of WSF. Járai and Redig [15] used the finiteness of T o to show that ν(S < ∞) = 1 when d ≥ 3.

Toppling probability bounds in low dimensions
In this section, we give a proof of our toppling probability lower bounds stated in Theorem 1.1. These will follow from the following theorem, which gives a lower bound in terms of a non-intersection probability between a loop-erased walk and a simple random walk, and a random walk hitting probability. Our arguments also apply to d ≥ 5 with an identical statement, however, this case is already known from [16, Section 6.2].
Recall that S x and S x are independent simple random walks from the site x.
Moreover, the right hand side of (3.1) is a lower bound on ν L (z ∈ Av) for all L ≥ 4 z .
In Sections 3.1-3.2, we state and prove preliminary results which are useful for establishing Theorem 3.1, and in Section 3.3 we use these to prove the theorem. In Section 3.3, we also give a corollary which will be useful for proofs of later theorems.

Preliminary setup
Our strategy for proving Theorem 3.1 will be to work in large finite volume V (L). That is, given a particular z ∈ Z d , we will choose some L 0 sufficiently large, so that the probability ν L (z ∈ Av) is close to the claimed value for all L ≥ L 0 . We will require L 0 to be on the order of some large multiple of z .
The main idea of the proof is to show a lower bound for the probability that z is in the last wave of the avalanche. By Corollary 2.10, Dividing by |R L |, using Lemma 2.11 and symmetry of V (L), for any fixed e ∼ o we get uniformly in z, L.
We analyze the event Let us apply Wilson's algorithm in the graph G L,o , starting with a walk S e from e, followed by a walk S z from z. This gives that for fixed e ∼ o, the occurrence of A(z, e) is equivalent to a LERW from e to exit V (L) without hitting o, and a SRW from z to hit o before hitting the LERW. We will bound the right-hand side of (3.2) from below by analyzing this random walk event. By time reversal, we will be able to consider the SRW going from o to z. The lower bound then contains two factors: the LERW and SRW avoiding each other near o, and the SRW subsequently hitting z. This leads to the two probabilities in We begin by expressing the probability in (3.2) in terms of the random walk construction specified above (Lemma 3.2). We then lower bound the probability of the resulting walk event by something amenable to analysis by walk intersection techniques that we give in Section 3.2. Let π = LS e [0, σ L ]. Given that the event e / ∈ T L,o has occurred, z will be in T L,o if and only if S z hits o before exiting V (L), and does so avoiding π. This implies statement (i).
In the sequel, we make use of the event Γ z,L , defined as Proof. Using reversibility of the random walk, we can rewrite the probability in the left hand side of (3.5) as follows: In the presence of the indicator, since after each visit to o, the random walk next hits e with probability (2d) −1 . It follows that the right-hand side of (3.6) is at least Using Theorem 2.13 and (2.8), we have Inserting this estimates into (3.7) completes the proof.

SRW and LERW steering
In order to control the spanning tree event in (3.2), we need to give a lower bound on the probability of the event Γ z,L of Lemma 3.3. This will be achieved by "steering" the two walks so that they become well separated as they reach distance of order z , which allows us to arrange that π avoids the box V z ( z /10), and it does not influence very much the probability that S o hits z. In Section 3.2.1 we collect results we need for a single walk. In Section 3.2.2 we prove the required separation estimate. In Section 3.3 we prove a lower bound on P(Γ z,L ) and complete the proof of Theorem 3.1.

Estimates for a single walk
The first lemma we need, shown by Masson, holds for general d, and compares an infinite LERW to a finite LERW, allowing us to control probabilities by restricting to finite balls.
Masson stated this in the case K ⊃ B(4n), however, his proof applies in the slightly more general case we use here.
for all α, all n ≥ 1/δ, and all K ⊃ V (1 + δ)n , where the constants implied by the δ notation only depend on δ and d.
We will need the following "Boundary Harnack inequality", to control a LERW after it has reached the boundary of a box. Estimates of this flavour were proved in [ x · e 1 ≤ n} .
Proof. Let π = π ∩ V x (m). It is shown in [5, Section 3] that This yields Here we used a gambler's ruin estimate and the Harnack principle in the second inequality.

Separation lemma
Separation lemmas for a loop-erased walk and a simple random walk appeared in [29] for d = 2 and in [34] for d = 3. We give here a unified proof that works for all d ≥ 2.
Recall that S x generally denotes a simple random walk independent of the walk S x . We define In the proof of the separation lemma, a basic step is to show that given that nonintersection occurred to distance n/2, there is small probability that separation at distance n is bad.
There exists a function r : (0, 1/2] → (0, ∞) with lim δ→0 r(δ) = 0, and for all δ ∈ (0, 1/2] there exists n 0 (δ) such that we have we have Proof. We condition on π 0 := S L,o [0, σ n/2 ] and S o [0, σ n/2 ], and denote x 1 = S L,o ( σ n/2 ), We bound the probabilities of the two cases separately, showing that each is bounded by a suitable r(δ). Case (a). Let us further condition on x 2 . Since Brownian motion in the cube {u ∈ R d : u ≤ 1} has continuous paths, and the path tends to its exit point, and the probability of any given exit point is 0, there exists r 1 (δ), tending to 0, such that given any boundary point w, the Brownian path intersects {u ∈ R d : u ≤ 1, |u − w| ≤ 2δ} with probability ≤ r 1 (δ). Hence the required bound follows from the invariance principle.
Using the Harnack principle, we have where c 1 < 1. Therefore, we get This completes the proof.
The next step is to show that given a "good" separation at distance n/2, the probability that the paths can be "very well" separated at distance n is at least a constant. We ) the information about the paths up to the exit from V n . Lemma 3.8. For any 0 < δ ≤ 1/2 there exists c(δ) > 0 and n 1 (δ) such that for all n ≥ n 1 (δ) and L ≥ 4n the following hold.
(ii) Moreover, we may further require the event S Proof. We condition on π 0 = S L o [0, σ n/2 ] and S o [0, σ n/2 ]. Let us write x 1 = S L o ( σ n/2 ) and y 1 = S o (σ n/2 ). Due to the conditioning in (3.9), we have |x 1 − y 1 | ≥ δ(n/2). We write X for a random walk starting at x 1 conditioned on the event An application of Lemma 3.5 yields that with probability ≥ cδ, the process X exits V x1 (δn/8) on the face furthest from V (n/2). Also, there is probability ≥ (2d) −1 that S o [σ n/2 , ∞) exits V y1 (δn/8) on the face furthest from V (n/2). Using the Harnack principle for X, and appropriate disjoint corridors of width of order δn for the LERW and the SRW, respectively (see Figure 1 below), there is probability ≥ c(δ) that: In order to further ensure that LX first exits V (n) at a point in H −n , and that A n occurs, we show that for w ∈ ∂V (2n) we have P w σ X L < ξ X V (n) ≥ c > 0. (3.10) This is indeed sufficient, since the events in point (b) and (3.10) imply that the last visit of X to ∂V (n) must occur at a point in H −n . In order to see (3.10), first note that the statement is clear for d ≥ 3, since then The maximum over y is ≤ c 2 < 1, which after rearranging gives For other w ∈ ∂V (2n) we obtain the statement from the Harnack principle.
The construction we gave ensures both the event in (i) and (ii), and thus completes the proof of the lemma. We are ready to prove the Separation Lemma. The argument we give is inspired by [20].
Proof of Lemma 3.6. Let us write f (n) = P A n and g(n) = P A n ∩ {D n ≥ δn} .

Estimate on Γ z,L
In this section, we prove a lower bound on the probability of the event Γ z,L , which yields a finite volume analogue of Theorem 3.1. In the following, let (3.13) Proof. We may assume that z is sufficiently large. Without loss of generality, we also assume that the first coordinate of z is positive and has maximal absolute value among all coordinates. We first require the occurrence of the event B(e) := π ∩ {o} = ∅ .   where δ = δ 0 is the constant chosen in Lemma 3.6. According to that lemma, the event in (3.16) has unconditional probability ≥ cEs L ( z /4) ≥ cEs L ( z ). Due to Z d -symmetry, the conditional probability of (3.16) given S L (1) = e is the same as the unconditional probability.
Let us further require the event in Lemma 3.8 with n = z /2, that is, that the paths extend disjointly to opposite faces of V ( z /2), with the random walk landing on H z /2 , and the LERW landing on H − z /2 . According to Lemma 3.8, this happens with conditional probability ≥ c. It follows from Lemma 3.8, that there is probability bounded away from 0 that the LERW can be further extended to land on H −8 z , in such a way that π is contained in H − 3 z /8 ∪ V (4 z ) c . Since S o (σ z /2 ) ∈ H z /2 , the conditional probability, given π and S o [0, σ z /2 ] that S o hits z before ξ π ∧ σ 4 z is ≥ c |z| 2−d when d ≥ 3, and ≥ (log |z|) −1 when d = 2. Combining the estimates for each part of the construction yields: This completes the proof of the lemma when z is sufficiently large.
The following proposition summarizes the result of the finite L arguments we made in Sections 3.1-3.2.

Proof of Theorem 1.1
In passing to the limit L → ∞, we use the following proposition. We prove only the d ≥ 3 case here; the proof of the more technical d = 2 case is deferred to the end of Section 5; see Lemma 5.10 there.
We can now complete the proof of Theorem 3.1.
Proof of Theorem 3.1. Lemma 3.4 implies that Es L ( z ) in Proposition 3.11 is comparable to the avoidance probability in the theorem, so the statement follows using Proposition 3.11.
For later use we state here a corollary of the construction.
We now turn to the proofs of the explicit bounds for 2 ≤ d ≤ 4.

Proof of Theorem 1.1 when d = 2
By Theorem 3.1, Theorem 1.1(i) will hold once we know The exponent 3/4 + o(1) was first proved by Kenyon [18], who stated it for simple random walk in the half plane. A proof for more general walks was given by Masson

Proof of Theorem 1.1 when d = 3
In this section, we complete the proof of the explicit lower bound in Theorem 1.  This is a simple adaptation of the results of [22]. Indeed, there exists a c 1 > 0 such that (uniformly in m) (3)]. On the other hand, by [22,Lemma 4.7], there exists C 2 , c 2 > 0 such that (uniformly in a, n) Choosing a sufficiently large (relative to c 1 , c 2 ) and m = an 2 completes the proof of (3.19). From this Theorem 1.1(ii) follows.

Proof of Theorem 1.1 when d = 4
In four dimensions, the avoidance probability was determined by Lawler [21], who showed that Es(n) (log n) −1/3 . This and Theorem 3.1 yield Theorem 1.1(iii).

Low-dimensional radius bounds
In this section we prove the radius bounds stated in Theorem 1.2(i)-(iii) for dimensions 2 ≤ d ≤ 4. We start with the lower bounds.
Proof of Theorem 1.2(i)-(iii), lower bounds. Observe that ν (R ≥ r) ≥ ν(re 1 ∈ Av) . Recall that for η ∈ R L we write α(η) = (η 1 , . . . , η N ) for the waves in the stabilization S o (η + 1 o ). Let us extend the notation for the radius to waves and two-component spanning trees in the natural way: Under the bijection of Section 2.5 these two notions coincide. We have η ∈ R L : R(η) > r = η ∈ R L : R(η i ) > r for some 1 ≤ i ≤ N (η) where the last equality uses Lemma 2.6.
Since {R > r} and {R(T L,o ) > r} are both cylinder events, we can take the limit L → ∞ on both sides to get

The last k waves in 2D
In this section we prove that for any 1 ≤ k < ∞, as L → ∞, the last k waves on G L (when they exist) have a weak limit. We introduce some notation for the last k waves. Recall that for η ∈ R L , we denote by N = N (η) = n L (o, o) the number of times o topples during the stabilization of η + 1 o . Recall from Section 2.4, that given η ∈ R L , we write α(η) = (η 1 , . . . , η N ) for the set of intermediate configurations (right before each wave).
Observe that when ξ L k is defined, we have ξ L k−1 = a o,L ξ L k . Analogously to the finite graph case, a o,Z 2 is the operator which adds a particle at the origin and stabilizes the resulting configuration in the graph (Z 2 ) . If there is a path in T o (ξ L k (η)) from o to a vertex x that stays inside B(r), then starting from ξ L k + 1 o there is a sequence of topplings in B(r) that topples x. We write T L,o,k = T o (ξ L k ) and T L,s,k = T s (ξ L k ) for short. When x ∈ T L,o,k , we denote π L,k (x) the unique self-avoiding path in T L,o,k from o to x.
Our control on the size of waves will be in terms of the following random variables.
All quantities in (5.2) are defined to be 0 when N < k. The following lemma states a basic inequality we will need.

Lemma 5.3.
We have: (ii) Write r = P L k for short, and assume for a proof by contradiction, that we had R L in,k ≥ r + 1. This implies that in the stabilization a o,L (ξ L k ) all vertices in B(r + 1) topple, and hence (ξ L k−1 ) B(r) = (ξ L k ) B(r) . Starting from the configuration ξ L k−1 + 1 o , let us topple all sites in B(r) we can. The definition of P L k and property (5.1) imply that all vertices in B(R L in,(k−1) + 1) topple. However, this contradicts the definition of R L in,(k−1) .
In the following proposition we show that the in-radius of the k-th last wave is tight, with a power law upper bound on the tail.
Proof. We prove the statement by induction on k. The case k = 1 holds trivially due to Lemma 5.3(i). Assume k ≥ 2, and that (5.3) holds for k − 1. Let 1 ≤ r 0 < ∞ be fixed, and It is sufficient to bound R L in,k when the event {R L in,(k−1) ≤ r 0 } occurs, and due to Lemma 5.3(ii), it is enough to bound P L k on this event. In what follows, we assume the event For ≥ 1 we are going to bound the probability that r 0 2 < P L k ≤ r 0 2 +1 . Due to Lemma 5.3(ii), this event implies that (T L,o,k , T L,s,k ) belongs to the following event E(x, r 0 , ) for some x ∈ ∂B(r 0 + 1): where π(x) is the path in T o from x to o. Therefore, using Corollary 2.7(ii), we have , r 0 , )). is a simple random walk on Z 2 conditioned on ξ o < ξ B(r02 ) c . On the event E(x, r 0 , ), S cannot hit γ 0 , so we bound the probability that S hits B(r 0 + 1) before γ 0 .
The walk S has to successively cross from ∂B(r 0 2 q ) to B(r 0 2 q−1 ) for q = − 1, . . . , 1. During each crossing, it has a fixed constant probability of hitting γ 0 , since this holds for simple random walk, and the Harnack principle [24] then implies it holds for S . Hence the probability that S reaches B(r 0 + 1) before hitting γ 0 is less than (1 − c 1 ) −1 for some 0 < c 1 < 1. This bounds from above the probability that x ∈ T L,o and π(x) visits B(r 0 2 ) c . Assuming that this event occurs, we now bound the conditional probability that T L,s EJP 22 (2017), paper 85. contains a vertex y ∈ ∂B(r 0 2 +1 ). For this, continue Wilson's algorithm with a walk S y0 starting at any y 0 ∈ ∂B (r 0 2 +1 ) 4 , followed by walks starting at y 1 , . . . , y M , where the latter is an enumeration of all vertices in ∂B(r 0 2 +1 ). Let F j denote the tree generated by the walks S x , S y0 , . . . , S yj . Denote Application of Theorem 2.13(ii) yields that the j = 0 term is at most C log (r Since M ≤ Cr 0 2 +1 , putting the j = 0 and 1 ≤ j ≤ M cases together we get that the sum over 0 ≤ j ≤ M is bounded by C (log r 0 2 +1 )/ log L. Together with the earlier bound on π(x) leaving B o (r 0 2 ) this gives Substituting into (5.5), and summing over ≥ 0 implies, for L sufficiently large that We apply (5.6) with 2 0 = r β 0 , for some β > 0. The expressions in the right hand sides of (5.6) and (5.4) are of equal order (up to logarithms), when in ≤ P (k) , the bounds (5.4) and (5.6) imply, for all large enough L, that Hence we get (5.3) for k with a choice of 0 < α k < α k−1 /(1 + β ).
We next prove that the out-radius of the k-th last wave is also tight, and satisfies a power law upper bound. We are going to need the following lemma.

Lemma 5.5.
There exist constants C such that the following holds. Let 1 ≤ r < r < L, and let K ⊂ V (L) ∪ {s} be a connected set of edges that contains a path connecting B(r) to s. We have Proof. Let us use Wilson's algorithm in the contracted graph G L /K, that is, the edges in K are already present at the start of the algorithm. We let walks start at {x 1 , . . . , x M } = ∂B(r). If T L,o ⊂ B(r ), then at least one of these walks has to reach ∂B(r ) before hitting K. Beurling's estimate (Lemma 2.14) implies that for each x j , this has probability at most C (r /r) −1/2 . Since M = O(r), the statement follows. Proposition 5.6. There exist constants α 1 > α 2 > · · · > 0 and C 1 , C 2 , . . . such that lim sup L→∞ ν L R L out,k > r ≤ C k r −α k , ∀r ≥ 1, ∀k ≥ 1.

(5.7)
In particular, for all 1 ≤ k < ∞ the sequence {R L out,k } L≥1 is tight.
Proof. Fix 1 ≤ k < ∞, and 1 ≤ r 0 < ∞. From Proposition 5.4 we have that there exists L 0 = L 0 (r 0 ) < ∞ such that for all L ≥ L 0 we have Assume the event {R L in,k ≤ r 0 }, which implies that T L,s,k ∩ ∂B(r 0 ) = ∅. We bound the probability that R L out,k > r 1+β 0 , where the parameter β > 0 will be chosen at the end.

(5.9)
Let K be the set of edges in T L,s on the path from x to s. Conditioning on the value K = K, the right hand side of (5.9) equals Choose β so that 2 − β/2 < −α k , so that (5.11) together with (5.8) gives This implies the statement of the proposition with α k = α k /(1 + β).
The next proposition shows that µ L,o (A ⊂ T L,s , B ⊂ T L,o } for fixed finite fixed sets of vertices A and B, satisfies a certain normalization as L → ∞. Tightness of the in-radius established in Proposition 5.4 will allow us to apply this proposition, and subsequently prove Theorem 5.1. We introduce the notation q L := µ L,o (e ∈ T L,o ), where e ∼ o. Due to symmetry, q L does not depend on e. In fact, since q L is the escape probability of random walk from o, we have q L = G L (o, o) −1 . We remark that for the square grid, the quantity G L (o, o) has an explicit formula: , (5.12) where θ = 2π(2 + 1)/(4L + 4), and T L+1 is the degree L + 1 Tchebyshev polynomial.
The formula (5.12) can be derived via contour integration. However, we will not need it, and we omit the proof. We first need the following lemma. In its statement, a(x) is the potential kernel for simple random walk on Z 2 ; see [24].
Conditional on x ∈ T L,o , the law of the path from x to s has a weak limit, as L → ∞.
Proof. (i) We use Wilson's algorithm in the graph G L,o with a walk starting at x. Considering the limit of the bounded martingale {a(S x (t ∧ σ V (L) ∧ ξ {o} ))} t≥0 , and using Lemma Due to Lemma 5.8(i), the probability that the first walk hits s before o is asymptotic to q L a(x 1 )/a(e), as L → ∞. Assuming this event happens, let π L x1 denote the loop-erasure of the walk starting at x 1 , and write π x1 for its weak limit under the conditioning, whose existence is guaranteed by Lemma 5.8(ii). The probability that the walks starting in (A \ {x 1 }) ∪ B terminate before exiting a ball B(r) of large radius r goes to 1 as r → ∞, and L > r, uniformly in the path π L x1 . Since these walks determine the event In the proof of Theorem 5.1 we are going to need the following quantitative bound from [10] on the rate of convergence of ν L to ν. Theorem 5.9 (Theorem 4.1 of [10]). Let d = 2. There exist constants 0 < α < β and C such that if E is any cylinder event depending only on the configuration in B( ), then Proof of Theorem 5.1. (i)-(ii) We showed in Proposition 5.6 that for any fixed 1 ≤ k < ∞, the sequence R L,out , L ≥ 1, is tight. Therefore, we have lim We establish weak convergence of ξ L k . Fix ε > 0, and let and L 0 be such that for all L ≥ L 0 the probability appearing in (5.13) is ≤ ε. Let ζ ∈ R B( ) be a configuration with the following properties:

Inequalities for critical exponents in sandpiles
First observe that the properties of ζ imply that for any η * ∈ R L \ R L , if (η * ) B( ) = ζ then η * is an intermediate configuration, in the graph G L , corresponding to a k-th last wave, and the last k waves all stay in B( ). In particular, using that the k-th last wave stays inside B( ), property (5.1) of the bijection implies that ζ determines a unique pair (A 0 , B 0 ), with A 0 ∪ B 0 = B( ), such that whenever η * ∈ R L \ R L and η * | B( ) = ζ, then we have V (T o (η * )) = B 0 . Therefore, using Lemma 2.6 we can write (5.14) For any η * appearing in the right hand side of (5.14), let η = η * | V L \B0 . Due to the burning process, the conditional distribution of η, given the event {η * | B0 = ζ| B0 , V (T o (η * )) = B 0 }, equals that of a recurrent sandpile in the subgraph G B0 L of G L induced by the set of vertices V (L) ∪ {s} \ B 0 (i.e. with closed boundary condition at B 0 ). Hence the last expression in (5.14) equals Since the wired spanning forest in the subgraph of Z 2 induced by Z 2 \ B 0 is one-ended, we can apply [17,Theorem 3] to deduce that the last factor in (5.15) has a limit as L → ∞. The first factor equals where |T B0 | is the number of spanning trees in the graph induced by B 0 . Due to Proposition 5.7, the product of the second and third factors in (5.16) approaches p A0,B0 , as L → ∞. This implies the existence of the limit Summing over all ζ satisfying (a)-(c), and using the choice of made after (5.13), we get But since the left hand side does not depend on , we have that the limit lim L→∞ ν L (N ≥ k) =: b k exists, proving statement (i). It follows that Statement (ii) follows immediately from this.
(iii) Observe that the proof of parts (i)-(ii) shows that up to a set of measure 0, the support of ρ k can be partitioned into a countable disjoint union of cylinder sets, such that on each element of the partition, the stabilization (a o,Z 2 ) k (ξ k ) takes place within a finite set B( ).
(iv) The countable partition into cylinder sets has the further property that the map ξ k → a o,Z 2 (ξ k ) is measure preserving on each cylinder set of the partition. Hence the claim follows.
For an inequality in the other direction, we write: Due to (5.13), the third term on the right hand side is at most ε( , k) → 0 as → ∞.
Therefore, it is enough to show that We fix 0 < δ < α/β (where α, β are the constants from Theorem 5.9). Let r(i) = L δ/ρ i , i = 0, . . . , k, where the constant 1 < ρ < ∞ will be chosen later. Recall we denote by η 1 , . . . , η k ∈ R L \ R L the first k waves corresponding to η ∈ R L . We define the events H(i) = the i-th wave W(η i ) does not topple any vertices in B(r(i)) after it has reached ∂B(r(i − 1)) , i = 1, . . . , k.  We choose ρ > 9, so that the right hand side of (5.18) goes to 0 as L → ∞. Therefore, in order to prove (5.17), it is enough to bound ν L (N > k, F ,k , H(1) ∩ · · · ∩ H(k)) ≤ ν L N > k, N B( ) = k, H(1) ∩ · · · ∩ H(k) . (5.19) Suppose now that we are given a configuration η ∈ R L . Let us carry out the first wave up to ∂B(r(0)), that is, stop the first wave when a vertex of B(r(0)) c would need to be toppled, if any. Then carry out the second wave up to ∂B(r(1)), the third wave up to ∂B(r(2)), etc. Let F denote the event that during the k-th "partial wave" defined this way, all neighbours of the origin topple. The event in the right hand side of (5.19) implies the event F . This is because the event {N > k}, in the presence of H(1) ∩ · · · ∩ H(k), implies that the origin can be toppled a k + 1-st time after the first k partial waves.
Observe that F is measurable with respect to the pile inside B(r(0)), and r(0) = L δ . Hence, using Theorem 5.9, and ν({N = k}∆{N B( ) = k}) ≤ ε, the right hand side of (5.19) is at most where the last equality follows from the fact that F ⊂ {N > k}. Due to the choice of δ, the second term goes to 0, as L → ∞. Since ε is arbitrary, we obtain statement (v) of the Theorem. We can now complete the proof of Theorem 1.7.
(iii) This follows from Theorem 5.1(ii),(v), since on the event {N = k} we can approximate by cylinder events on which no vertex topples outside a fixed ball.
The following lemma completes the proof of the d = 2 case of Proposition 3.11.
Lemma 5.10. We have Proof. For a sufficiently large number k = k(z), we have the deterministic implication N > k ⇒ z ∈ Av, both in Z 2 , and in V L for L sufficiently large. With such k, we have In the second equality, we applied Theorem 5.1(v) to the first term. In the second term, a.s. finiteness of the last waves allows us to approximate {N = , z ∈ Av} by a cylinder event, and the equality follows.
Proof of Theorem 1.8. (i) The bound follows immediately from the Proposition 5.6.
We now prove Theorem 1.9. The idea of the argument is that, if f (x) := E ν n(o, x) were finite, then by invariance of ν under a o , f would have to be a bounded harmonic function, hence constant. This is in contradiction with the structure of the avalanche. We first give two short lemmas on which this argument will be based. Lemma 5.11. Assuming ν(N < ∞) = 1, we have ν(R < ∞) = 1.
Proof. This follows easily from Theorem 1.8.
Let a o denote the operation on stable sandpiles on Z 2 which maps η to (η + 1 o ) • if a finite stabilization is possible (i.e., if S < ∞). Then the preceding lemma implies if ν(N < ∞) = 1, there exists a set Ω with ν(Ω) = 1 such that a o is defined on Ω. Given such an Ω, the next lemma shows that, similarly to ν L , the infinite-volume measure ν is invariant under a o . Proof of Theorem 1.9. If ν(N = ∞) > 0, there is nothing to prove. We thus assume that ν(N < ∞) = 1. We first note that, under this assumption, the infinite-volume addition operators are well-defined, since n(o, x) ≤ n(o, o). The invariance statement above makes possible a version of the argument underlying Dhar's formula (Lemma 2.2). Assume for the sake of contradiction that E ν N < ∞. In particular, 0 ≤ En(o, x) ≤ En(o, o) < ∞ for all x ∈ Z 2 . Since N < ∞, Lemma 5.11 above shows we can (almost surely) write where the sum above has finitely many nonzero terms and ∆ is the graph Laplacian on Z 2 . In particular, taking expectations and using the invariance above (which implies However, if all vertices topple, the avalanche is infinite, a contradiction.

High-dimensional radius bounds
In this section we prove the bounds for the radius of the avalanche for d ≥ 5. We prove Theorem 1.3 in Section 6.1. We use the results of Section 6.1 in Sections 6.2 and 6.3 to prove the lower and upper bounds on the radius.

Radius bounds on transitive unimodular graphs
See [26,Chapter 8] for background on unimodularity and mass transport.
. Consider the following mass transport. When diam(past x ) > r, let x send unit mass distributed equally among all vertices y ∈ past x ∩ A x (r, 2r). Let us write sent(x) and get(x), respectively, for the amount sent and received by x, respectively. Then E[sent(o)] = µ diam(past o ) > r . On the other hand, using Jensen's inequality, we have and the statement follows. (iii) Although WSF o is not automorphism invariant, essentially the same mass transport can be used as in (ii), due to the fact that the rooted random network (T o , o) is unimodular, in the sense of [7,1]. Let T ⊂ V be a finite tree, and x, y ∈ T . Define the function f (T, x, y) = 1 y∈T |T ∩ A x (r, 4r)| −1 when diam(T ; x) > 4r; 0 otherwise. Let x send the following mass to y: Let γ ∈ Γ. It is clear that f (γT, γx, γy) = f (T, x, y), and shifting Wilson's algorithm by γ shows that WSF x (T x = T ) = WSF γx (T γx = γT ). Therefore, F is invariant under the diagonal action of Γ, and hence This yields the statement.
Remark 6.1. Part (iii) of Theorem 1.3 can also be deduced from part (ii) via the following limiting procedure, the details of which we omit. (This construction was our initial approach to the radius upper bound.) Let ω ⊂ V be independent site percolation on V with density 0 < p 1. Conditional on ω, let WSF ω denote the measure on spanning forests of (V, E) where each vertex in ω is "wired to infinity", in analogy with WSF o . The measure obtained by averaging WSF ω over ω is automorphism invariant. When x ∈ ω, let x send or not send mass according to the same rule as in part (ii), otherwise let x send no mass. Now let p ↓ 0. Conditional on o ∈ ω we have WSF ω ⇒ WSF o , and the mass transport in part (iii) can be recovered in this limit.

Radius lower bound when d ≥ 5
We prove the lower bound using the result of the previous section.
Proof of Theorem 1.2(iv), lower bound. We begin with some terminology. As in [25], let the 'past of a vertex x' in a spanning tree T of G L , be the union of the connected components of T \ {x}, which do not contain s. We denote this object by past x (T ).
This equation holds for any fixed r, and all L. Using the observation that for fixed r the events at both ends are cylinder events, and taking the limit as L ∞, we have, The proof is completed by applying Theorem 1.3(i). Using the fact that in dimensions d ≥ 5 there is probability at least c that two independent simple random walks starting at x do not intersect, we deduce that WSF(x ∈ past o ) ≥ c|x| 2−d . On the other hand, Wilson's algorithm gives Therefore, Theorem 1.3(i) implies WSF(R(past o ) > r) ≥ cr −2 .

Radius upper bound when d ≥ 5
In this section we prove the upper bound in Theorem 1.2(iv). Taking d to be ∞ distance, Theorem 1.3(iii) yields We show that for δ > 0 there exists C 1 = C 1 (δ) such that the expectation in the right hand side is bounded above by C 1 (log r) 3+δ r −4 . This implies the required upper bound on the tail of the diameter. We are going to use the following theorem of [5] on the lower tail of the volume of WSF components. Given D ⊂ Z d , write WSF D c for the wired spanning forest measure on the contracted graph Z d /D c . Theorem 6.3.
[5] Let x, y ∈ Z d be such that y − x ∞ = 4r. Let D ⊂ Z d be such y ∈ ∂D, and V x (4r) ⊂ D. Let x ↔ y denote the event that in WSF D c the path from x to D c reaches D c via an edge incident to y. There exist constants C, c > 0 independent of D and r, such that for all λ > 0 we have We use Theorem 6.3 to establish the following regularity estimate. Fix a positive We show the following lemma. Proof. If diam(T x ) > 4r, then there exists y with y − x ∞ = 4r such that y ∈ T x . Therefore, The term WSF x (y ∈ T x ) ≤ C r 2−d , and there are O(r d−1 ) terms. On the other hand, we . This follows from the fact that LERW from y to x has the same distribution as LERW from x to y, and hence we can use these walks in the first step of Wilson's algorithm on the two sides. See [24,Corollary 11.2.2] for "reversibility" of LERW. Hence Theorem 6.
The statement of the lemma follows.
Proof. When T x is bad, we use that |B x | ≥ 1, and hence due to Lemma 6.4 we have Therefore it is enough to consider the contribution when T x is not bad. When this occurs, we have |B x | ≥ (log r) −3−δ r 4 , and hence This proves the claim.
We can now complete the proof of the upper bound in Theorem 1.2(iv). Lemma 6.5 implies that the right hand side of (6.4) is at most This completes the proof.

Bounds on the size
We now prove Theorem 1.5. Sections 7.1, 7.2 and 7.3 consider low dimensions, and Sections 7.4, 7.5 and 7.6 the case d ≥ 5.

Upper bounds on the size when d = 3, 4
Proof of Theorem 1.5(ii)-(iii), upper bounds. The claimed upper bounds on ν(|Av| ≥ t) follow immediately from Theorem 1.2, via the trivial estimate ν(|Av| ≥ t) ≤ ν(R > c(d) t 1/d ). In order to obtain an upper bound on the tail of S, fix some k ≥ 1. Recalling that N denotes the number of waves, we have ν(S > t) = ν(S > t, N > k) + ν(S > t, N ≤ k). Recalling that d ≥ 3, we can upper bound the first term in the expression above by Next, writing S j for the size of the j-th wave, if S > t and N ≤ k, then we have S j > t/k for some 1 ≤ j ≤ N . Hence, ν(S > t) ≤ C k −1 + ν(S j > t/k, for some j ≤ N ) Due to Theorem 1.2(ii)-(iii), we get the bounds: Optimizing the choice of k yields: C t −1/17 when d = 4.

Lower bounds on the size when 2 ≤ d ≤ 4
We fix z = r e 1 , where r = t 1/d , and write u = z /10. Recall that in the course of the proof of Lemma 3.3 in Section 3, we showed that for L ≥ 100 z , and e a neighbour of the origin, and with π = LS e [0, σ L ], we have (2d) −1 P(Γ z,L ) when d = 3, 4.

(7.2)
Let us write F z,L for the event in the left hand side of (7.2). Our goal will be to show that conditional on the event F z,L (when z is in the last wave), a large number of vertices in V z (u) are also in the last wave, with probability bounded away from 0. We will use a second moment argument to prove this, that is based on Proposition 7.1 below.
Let u = (1 − ε) u, where we are going to choose 0 < ε < 1/4 later. Consider the following partial cycle popping in G L , defined in three stages; see [35] and [26] for background on cycle popping. In the first stage, reveal the LERW π started at e and ending at s. In the second stage, reveal a LERW started at z, ending on hitting π ∪ {o}. In the third stage, pop all cycles that are entirely contained in V z (u) that can be popped. Condition on the event F z,L , that is measurable with respect to the result of the first two stages. Let π = LS z [0, ξ o ], and let π (u ) be the portion of π from z to the first exit from V z (u ). Let For each x ∈ V z (u/2) for which I(x) occurs, let p(x) ∈ π ∩ V z (u) be the point where the revealed path first meets π ∩ V z (u). For technical reasons (that are only required for our argument when d = 4), we also define: The following proposition states bounds on the first and second moments of Y .  There exist 0 < ε < 1/4, c 1 > 0 and C such that the following hold. (i) When d = 2, we have Proof of Theorem 1.5(i)-(iii); lower bounds; assuming Proposition 7.1.
Since on the event F z,L , we have π ⊂ T L,o , we have |T L,o | ≥ Y . Hence for any t > 0 we The upper bounds are immediate from Y u,ε ≤ |V z (u/2)|. For the lower bounds, using the strong Markov property of S x at time ξ π , when S x is at the point p(x), we have P(I(x) ∩ J(x) | F z,L ) = P x ξ π ≤ ξ π (u ) < σ Vz(u) = P x ξ π (u ) < σ Vz(u) .
Let π := LS z [0, σ Vz(u) ], and let π (u ) be the portion of π up to its first exit from V z (u ). Due to Lemma 3.4, there exists c 2 = c 2 (ε), such that the distribution of π (u ) is bounded below by c 2 times the distribution of π (u ). This implies P x ξ π (u ) < σ Vz(u) ≥ c 2 P x ξ π (u ) < σ Vz(u) . (7.5) We lower bound the probability in the right hand side of (7.5) separately in d = 2, 3.
The right hand side in (7.7) is bounded below by a constant c 3 > 0, independent of u. This can be seen by arguments due to Lawler; by adapting the proof of [23,Theorem 3.3.2]. It remains to bound the negative term in (7.6). For this we write Consider independent Brownian motions, starting at x/u and z/u. Since the paths are continuous, and with probability 1 they exit the cube (z/u) + [−1, 1] 3 at different points, the invariance principle implies that the probability in the right hand side of (7.8) goes to 0 uniformly in u ≥ (1/ε), as ε → 0. Therefore, we can fix ε > 0 such that the right hand side of (7.8) is at most c 3 /2, uniformly in u. With such a choice of ε, the first moment is bounded below by c u 3 .

Proof of the first moment bounds
We begin with the first moment lower bound -that is, the lower bound in Proposition 7.1(iii). The line of reasoning leading to (7.6) and (7.7) above holds also for d = 4, and we take these as our starting point. We next establish an appropriate analogue of the constant lower bound given above for (7.6). We restrict to x / ∈ V z (u/4) for simplicity.
Each term above gives a contribution of order log u; we discuss in detail only the first term. By summing first over y 2 and using Theorem 2.13(ii) (taking the n → ∞ limit in this theorem), we get an upper bound of order log u y1∈Vz(u) G(x, y 1 )G(z, y 1 ) .
For each y 1 , either |x − y 1 | or |z − y 1 | is at least u/8 since |x − z| is at least u/4. Thus either G(x, y 1 ) or G(z, y 1 ) is at most Cu −2 . Summing the other factor over y 1 gives a factor of order u 2 , giving the required upper bound. Using the second moment method and noting that S x and S z intersect if J x > 0 completes the proof. It remains to control the negative term of (7.6). This will be accomplished using the following lemma: Lemma 7.3. There exists a constant C 3 > 0 such that, uniformly in 0 < ε < 1/2 and u large (how large depends on ε), and uniformly in y ∈ V z (u) \ V z (u ), Proof of Proposition 7.1(iii), lower bound; assuming Lemma 7.3. Note that The above, combined with Lemma 7.3, allows the choice of an appropriately small ε to give a uniform lower bound of cu 4 / log u for the right-hand side of (7.6), completing the proof of the first moment of Prop. 7.1(iii).
We turn to the proof of Lemma 7.3, which is an adaptation of the proof of [23,
Recall the definition of J x from (7.9). We show that, conditional on {J x > 0}, the expectation of J x is at least c log u. This gives the desired upper bound for P(J x > 0). Lemma 7.5. We can find r > 0 such that, uniformly in 0 < ε < 1/2, u such that εu > u 1/2 , and x ∈ ∂V z (u ) such that x is at least distance u/10 from all but one face of V z (u), we have E[J x | J x > 0] ≥ r log u . On {J x > 0}, there is a lexicographically first intersection in V z (u). Specifically, we can define 1 := inf{j : S x (j) ∈ S z [0, σ Vz(u) ] ∩ V z (u)} and 2 := inf{j : S z (j) = S x ( 1 )} and note that each i is smaller than σ Vz(u) . Using the strong Markov property of S x at time 1 , conditionally on S z [0, σ Vz(u) ] and S x [0, 1 ] the expected value of J x is bounded below by Here we have used the fact that G Vz(u ) (a, S z ( 2 )) ≥ cG(a, S z ( 2 )) for a ∈ V Sz( 2 ) (u 1/2 ) along with translation invariance, and d = denotes equality in distribution.
The conclusion of Lemma 7.5 follows immediately from the above, using the following proposition (along with an a priori power law lower bound for P(J x > 0)): Proof of Lemma 7.3. Comparing Lemma 7.5, and Lemma 7.4, the claim nearly follows, except that y ∈ ∂V z (u ) in (7.10) may be closer than distance u/10 to more than one face of V z (u). However, for such y we can replace V z (u) by a larger box V ⊃ V z (u), whose diameter is still of order u, in such a way that y is at distance εu from the boundary of V , and y is bounded away from the corners of V . Since in V the intersection probability is larger than in V z (u), the claim follows.
The above completes the proof of the lower bound in Proposition 7.1(iii).

Proof of the second moment bounds
Here we prove the second moment bound in Proposition 7.1 (iii). We first introduce some notation. We will work with dyadic cubes 4 i=1 [a i 2 k , (a i + 1)2 k ) ∩ Z 4 , where a 1 , a 2 , a 3 , a 4 ∈ Z, and k ≥ 1. We say that such a cube is of scale k. If Q is dyadic cube, we denote by Q , respectively, Q , the cubes that are concentric with Q and have twice, respectively, four times, the side-length. Given v ∈ Z 4 , we denote by Q(v; k) the unique dyadic cube of scale k containing v.
The following inequality recasts the statement of Lemma 7.3. Let Q be a dyadic cube of scale k, p ∈ Q, q ∈ Q , such that dist ∞ (q, ∂Q ) = ε2 k . There is a constant C such that P (S q [0, σ Q ] ∩ S p [0, ∞) = ∅) ≤ C ε k .
(7.14) Fix x, y ∈ B z (u/2) and assume that I(x) ∩ J(x) ∩ I(y) ∩ J(y) occurs. We distinguish the following two cases: (I) the paths from x and y to π do not meet; (II) the paths from x and y to π meet at a vertex q(x, y) ∈ π .
In Case (I), using that the events J(x) and J(y) occur, let v, w ∈ π (u ), respectively, be the points where S p(x) and S p(y) , respectively, first hit π (u ). (Note that v and w may coincide.) It follows that P(Case (I)) ≤ P ∃v, w ∈ π (u ) : ξ Sx v < σ Sx Vz(u) , ξ Sy w < σ Sy Vz(u) .
We bound the probabilities of Cases (I) and (II) separately. The idea of the bound is not to sum over v and w, but rather, sum over the choice of suitable dyadic cubes that v and w fall into, and use the bound (7.14) for the probability of random walk intersections. Throughout, we write K for the integer such that 2 K−1 < u ≤ 2 K .

Case (I).
We may assume without loss of generality that the walk S z generating π hits v before w, as the other case follows by a symmetric argument. For convenience, we assume that z − v , v − w , x − v , y − w are all at least 32. At the end of the proof we comment on how to handle the remaining configurations of points. We define the following dyadic scales and cubes: Q(w) = Q(w; k w ).
We also let k z := max k ≥ 1 : 2 k+4 ≤ z − v ∞ k x := max k ≥ 1 : 2 k+4 ≤ x − v ∞ k y := max k ≥ 1 : 2 k+4 ≤ y − w ∞ . (7.15) A sketch of the argument is as follows: the walks S z and S x both have to hit Q (v), and then they intersect at a point of Q(v). Following the intersection, the walk S z has to hit Q (w), and so does the walk S y . These two walks then intersect at a point of Q(w).
Breaking up the paths into pieces, the various hitting and intersection events will give us the estimate:  We need to sum this estimate over the choices of x and y, and the choices of the boxes Q(v) and Q(w). In the summation we will need to distinguish a number of sub-cases according to the relative sizes of the scales k v , k w , k z , k x , k y .
We first establish the bound in (7.16). This is provided by the following lemma. Lemma 7.7. (Probability bound for Case (I)) Let R 1 and R 2 be dyadic boxes of scales k 1 and k 2 , and let x, y ∈ B z (u/2) be points such that: (i) R 1 and R 2 are disjoint; (ii) dist(z, R 1 ), dist(x, R 1 ) ≥ 2 k1 ; (iii) dist(y, R 2 ) ≥ 2 k2 . Define k x , k y , k z by the formulas (7.15) where Q(v) and Q(w) are replaced by R 1 and R 2 , respectively. Then P [∃v ∈ R 1 , ∃w ∈ R 2 s.t. Case (I)] Proof. We need to be careful about the event when the walk S z first hits R 1 , leaves R 1 and returns, before intersecting the path of S x . The following definitions take care of this possibility by introducing the variables 1 and 2 that count crossings from ∂R 1 to ∂R 1 and from ∂R 2 to ∂R 2 , respectively. The definitions are somewhat tedious to write down; however, estimating the resulting probabilities is then straightforward using the strong Markov property. Given 1 , 2 ≥ 0, let : S z [ξ R 1 , n] has made at least 1 crossings from ∂R 1 to R 1 } σ 1,R 1 = inf{n ≥ T 1 : S z (n) ∈ R 1 } ξ 1,R 2 = inf{n ≥ σ 1 ,R 1 : S z ∈ R 2 } T 1, 2 = inf{n ≥ ξ 1,R 2 : S z [ξ 1,R 2 , n] has made at least 2 crossings from ∂R 2 to R 2 } σ 1, 2,R 2 = inf{n ≥ T 1 , 2 : S z (n) ∈ R 2 }.
We continue with the bound for Case (I). We break up Case (I) into the following sub-cases (that partially overlap, but together cover all possibilities): (I-1) k v < k z , k x and k w < k y ; (I- 4) k v = k z ≤ k x and k w = k y ; (I-2) k v < k z , k x and k w = k y ; (I- 5) k v = k x ≤ k z and k w < k y ; (I- 3) k v = k z ≤ k x and k w < k y ; (I- 6) k v = k x ≤ k z and k w = k y .
Fixing the scales k v , k w , k z , k x , k y , we bound the number of choices of x, y and the dyadic boxes containing v and w in each case separately, and apply Lemma 7.7. Then we sum over the scales allowed in each sub-case. A depiction of case (I-1) may be found in Figure 2.  x and y are dashed. S x and S z intersect at v, which is surrounded by boxes Q(v), Q (v), and Q (v). Walks S z and S y intersect at w (boxes not shown). Note that in this case, the factor limiting the size of Q(v) is the proximity of the vertex w. After intersection, S z terminates upon intersecting the sink s (i.e., the boundary of the large square).
Sub-case (I-1). The number of choices for Q(v) is of order 2 4kz /2 4kv . The number of choices for x is of order 2 4kx . Given Q(v), the number of choices for Q(w) is O(1) (note that k w = k v ), and the number of choices for y is of order 2 4ky . Mutiplying these bound ≤ y∈Bo(4r) u∈Z d G(o, u) G(u, x) G(y, u) + G(o, x) G(x, u) G(y, u) ≤ Cr 6−d . (7.21) Write future x = {y ∈ Z d : x ∈ past y }. Using Cauchy-Schwarz we have This yields WSF(| T o (r/2)| > t) ≥ c 1 > 0 for some c 1 > 0 and sufficiently small δ. Fix such δ. We get a lower bound on |future o ∩ A o (r, (3/2)r)| by considering the number of loopfree points of the random walk S o generating the path from o to ∞. A result of Lawler says that with probability 1, the fraction of loop-free points is asymptotically a positive constant in d ≥ 5; see [23,Section 7.7]. Therefore, we can find ε > 0 small enough, such that WSF(|future o ∩ A o (r, (3/2)r)| ≥ εr 2 ) ≥ 1 − c 1 /2. With these choices of δ and ε, the right hand side of (7.22) is at least r −d ((c 1 /2) ε r 2 ) 2 = c 2 r 4−d . Substituting this and (7.21) into the bound given by Theorem 1.6 we obtain WSF |past o | ≥ t ≥ c r −2 = c t −1/2 . As in Section 6.2, we have ν(|Av| > t) ≥ 1 2d WSF(|past o | > t), and the claim of the theorem follows.

Upper bounds: d ≥ 5
Unlike the case of the radius, an upper bound on the size of waves does not yield directly an upper bound on the size of the avalanche. However, we can still get a power law upper bound, that we prove in this section.
Proof of theorem 1.5(iv); upper bound. Fix some k ≥ 1. Recalling that N denotes the number of waves, we have ν(S > t) = ν(S > t, N > k) + ν(S > t, N ≤ k). (7.23) From Lemma 2.3 we have N ≤ R, and we can upper bound the first term in the expression above by ν(R > k). Next, writing S j for the size of the j-th wave, if S > t and N ≤ k, then we have S j > t/k for some 1 ≤ j ≤ N . Hence, ν(S > t) ≤ ν(R > k) + ν(S j > t/k, for some j ≤ N ) (7.24) where we used Theorem 1.2(iv) for the first term. We now control the second term on the right in equation (7.24).  Proof. We note that an application of Theorem 1.2(iv) yields ≤ (t 1/4 ) −2+o(1) + WSF o (|B(t 1/4 ) ∩ T o | > t). (7.25) Due to Wilson's algorithm, we have WSF o (x ∈ T o ) ≤ C |x| 2−d , and hence we have Using the above lemma and optimizing the number of waves k in (7.24), we get ν(S > t) ≤ t −2/5+o(1) , thereby completing the proof of the upper bound in Theorem 1.5(iv).