Asymptotic Height Distribution in High-Dimensional Sandpiles

We give an asymptotic formula for the single-site height distribution of Abelian sandpiles on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^d$$\end{document} as d→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \rightarrow \infty $$\end{document}, in terms of Poisson(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {Poisson}(1)$$\end{document} probabilities. We provide error estimates.


Introduction
We consider the Abelian sandpile model on the nearest neighbour lattice Z d ; see Sect. 1.1 for definitions and background. Let P denote the weak limit of the stationary distributions P L in finite boxes [−L, L] d ∩ Z d . Let η denote a sample configuration from the measure P. Let p d (i) = P[η(o) = i], i = 0, . . . , 2d − 1, denote the height probabilities at the origin in d dimensions. The following theorem is our main result that states the asymptotic form of these probabilities as d → ∞.
In particular, The appearance of the Poisson(1) distribution in the above formula is closely related to the result of Aldous [1] that the degree distribution of the origin in the uniform spanning forest in Z d tends to 1 plus a Poisson(1) random variable as d → ∞. Indeed our proof of (1.1) is achieved by showing that in the uniform spanning forest of Z d , the number of neighbours w of the origin o, such that the unique path from w to infinity passes through o is asymptotically the same as the degree of o minus 1, that is, Poisson (1).
In [11], we compared the formula (1.1) to numerical simulations in d = 32 on a finite box with L = 128, and there is excellent agreement with the asymptotics already for these values.
Other graphs where information on the height distribution is available are as follows. Dhar and Majumdar [7] studied the Abelian sandpile model on the Bethe lattice, and the exact expressions for various distribution functions including the height distribution at a vertex were obtained using combinatorial methods. For the single-site height distribution they obtained (see [7,Eqn. (8.2) If one lets the degree d → ∞ in this formula, one obtains the form in the right hand side of (1.1) for any fixed i (with 2d replaced by d).
Exact expressions for the distribution of height probabilities were derived by Papoyan and Shcherbakov [20] on the Husimi lattice of triangles with an arbitrary coordination number q. However, on d-dimensional cubic lattices of d ≥ 2, exact results for the height probability are only known for d = 2; see [13,14,18,21,22].

Definitions and Background
Sandpiles are a lattice model of self-organized criticality, introduced by Bak, Tang and Wiesenfeld [3] and have been studied in both physics and mathematics. See the surveys [6,9,10,15,23]. Although the model can easily be defined on an arbitrary finite connected graph, in this paper we will restrict to subsets of Z d . Let For simplicity, we suppress the d-dependence in our notation. We let G L = (V L ∪ {s}, E L ) denote the graph obtained from Z d by identifying all vertices in Z d \V L that becomes s, and removing loop-edges at s. We call s the sink. A sandpile η is a collection of indistinguishable particles on V L , specified by a map η : V L → {0, 1, 2, . . . }.
We say that η is stable at x ∈ V L , if η(x) < 2d. We say that η is stable, if η(x) < 2d, for all x ∈ V L . If η is unstable (i.e. η(x) ≥ 2d for some x ∈ V L ), x is allowed to topple which means that x passes one particle along each edge to its neighbours. When the vertex x topples, the particles are re-distributed as follows: Particles arriving at s are lost, so we do not keep track of them. Toppling a vertex may generate further unstable vertices. Given a sandpile ξ on V L , we define its stabilization by carrying out all possible topplings, in any order, until a stable sandpile is reached. It was shown by Dhar [5] that the map ξ → ξ • is well-defined, that is, the order of topplings does not matter.
We now define the sandpile Markov chain. The state space is the set of stable sandpiles L . Fix a positive probability distribution p on V L , i.e. x∈V L p(x) = 1 and p(x) > 0 for all x ∈ V L . Given the current state η ∈ L , choose a random vertex X ∈ V according to p, add one particle at X and stabilize. The one-step transition of the Markov chain moves from η to (η + 1 X ) • . Considering the sandpile Markov chain on G L , there is only one recurrent class [5]. We denote the set of recurrent sandpiles by R L . It is known [5] that the invariant distribution P L of the Markov chain is uniformly distributed on R L .
Majumdar and Dhar [19] gave a bijection between R L and spanning trees of G L . This maps the uniform measure P L on R L to the uniform spanning tree measure UST L . A variant of this bijection was introduced by Priezzhev [22] and is described in more generality in [8,12]. The latter bijection enjoys the following property that we will exploit in this paper. Orient the spanning tree towards s, and let π L (x) denote the oriented path from a vertex x to s. Let Then, we have that This has the following consequence for the height probabilities. Let The measures P L have a weak limit P = lim L→∞ P L [2], and hence, p(i) = lim L→∞ p L (i) exist, i = 0, . . . , 2d − 1. Although the q L (i) depend on the non-local variable W L , one also has that q(i) = lim L→∞ q L (i) exist, i = 0, . . . , 2d −1; see [12]. In fact, q(i) is given by the following natural analogue of its finite volume definition. Consider the uniform spanning forest measure USF on Z d ; defined as the weak limit of UST L ; see [16,Chapter 10]. Let π(x) denote the unique infinite self-avoiding path in the spanning forest starting at x, and let Therefore, we have

Wilson's Method
Given a finite path γ = [s 0 , s 1 , . . . , s k ] in Z d , we erase loops from γ chronologically, as they are created. We trace γ until the first time t, if any, when Then, we continue tracing γ and follow the same procedure to remove loops until there are no more loops to remove. This gives the loop-erasure π = L E(γ ) of γ , which is a self-avoiding path [17]. If γ is generated from a random walk process, the loop-erasure of γ is called the loop-erased random walk (LERW). When d ≥ 3, the USF on Z d can be sampled via Wilson's method rooted at infinity [4], [16,Section 10], that is described as follows. Let s 1 , s 2 , . . . be an arbitrary enumeration of the vertices and let T 0 be the empty forest with no vertices. We start a simple random walk γ n at s n and γ n stops when T n−1 is hit, otherwise we let it run indefinitely. L E(γ n ) is attached to T n−1 , and the resulting forest is denoted by T n . We continue the same procedure until all the vertices are visited. The above gives a random sequence of forests T 1 ⊂ T 2 ⊂ . . . , where T = ∪ n T n is a spanning forest of Z d . The extension of Wilson's theorem [24] to transient infinite graphs proved in [4] implies that T is distributed as the USF.

Proof of the Main Theorem
Let (S x n ) n≥0 be a simple random walk started at x (independent between x's on Z d ) and let π(x) be the path in the USF from x to infinity. We introduce the events: Then, recall that (2.1) [17] states that for all non-negative integers n and all d ≥ 1, we have

Preliminary
Based on above, we have Since D 4 (k)dk and D 6 (k)dk state the probability that S o returns to o in 4 and 6 steps each, by counting the number of ways to return, they are bounded by dimensionindependent multiples of 1/d 2 and 1/d 3 , respectively. We have D n (k)dk = 0 with odd n, and for 6 < n ≤ d − 1 and n even, we have D n (k)dk ≤ D 6 (k)dk. Hence, The last sum in (2.2) can be bounded as: since we can take d > 4 and π 4 < 1.
Hence, we have the required results

Lower Bounds
Let us fix the vertices x 1 , . . . , where N = {y ∈ Z d : |y| ≤ 1}.  o) for all x, y. Therefore, combining above results together, we get P[S o n / ∈ N for n ≥ ., x i and then started at x i+1 , . . . , x 2d−1 . We obtain the following:

Lemma 2.3 P[B
If the first step is not to o, the first step could be in one of the e 1 , . . . , e i directions, say e j , with probability i/2d. Then, the probability to hit x j is 1/2d+O(1/d 2 ). Hence, the probability that S x k hits {x 1 , . . . , Then, by (2.1), Then, the result follows The above lemma gives a lower bound for q d , and we now prove an upper bound.

Upper Bounds
Recall that π(o) denotes the unique infinite self-avoiding path in the spanning forest starting at o and letĀ o = {π(o) visits only one neighbour of o}.

Lemma 2.5 P[π(o) visits more than one neighbour of o] = P[Ā c o ] = O(1/d).
Proof LetĀ all ={∀w ∼ o: either π(w) does not visit o or π(w) visits o at the first step}.

Lemma 2.6 P[∃w ∼ o : π(w) visits o but not at the first step] = P[Ā c all ] = O(1/d).
Proof For a given w, w ∼ o, use Wilson's algorithm with a walk started at w. Consider that if S w 1 = o, or S w 1 = o, but S w returns to w subsequently and then this loop starting from w in S w is erased, π(w) does not visit o at the first step. Hence, we have the inequality: We bound the two terms as follows. For the first term, let us append a step from o to w at the beginning of the walk and analyse it as if the walk started at o. Since S o 1 ∈ N \{o}, by symmetry, we may assume S o 1 = w. Then, if S o 2 = o, S o will need at least 2 more steps to return to o.
For the second term in the right hand side of (2.4), we first note that we have P[S w 1 = o, S w 2 = w] = 1/(2d) 2 . If S w does not return to w in the first two steps, S w will need at least 4 steps to return to w. Then, we have that the right hand side of (2.4) is Due to Lemmas 2.5 and 2.6, we have Here, where The right hand side of (2.5) is contained in the event Therefore, we have  Due to Lemma 2.9, we have that the right hand side of (2.7) is (2.9) . Proof Due to Lemma 2.7, (2.6) and (2.9), we have

Lemma 2.11
For k = 1, . . . , 3 and distinct w 1 , . . . , w k ∼ o, we have This lemma can be proved using ideas used to prove Lemma 2.7. We are left to prove statement (ii). The uniform distribution for d 1/2 ≤ i ≤ 2d − 1 can be obtained from the monotonicity: