# Asymptotic Height Distribution in High-Dimensional Sandpiles

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## Abstract

We give an asymptotic formula for the single-site height distribution of Abelian sandpiles on \(\mathbb {Z}^d\) as \(d \rightarrow \infty \), in terms of \(\mathsf {Poisson}(1)\) probabilities. We provide error estimates.

## Keywords

Abelian sandpile Uniform spanning forest Wilson’s method Loop-erased random walk## Mathematics Subject Classification (2010)

Primary 60K35 Secondary 82C20## 1 Introduction

We consider the Abelian sandpile model on the nearest neighbour lattice \(\mathbb {Z}^d\); see Sect. 1.1 for definitions and background. Let \(\mathbf {P}\) denote the weak limit of the stationary distributions \(\mathbf {P}_L\) in finite boxes \([-L,L]^d \cap \mathbb {Z}^d\). Let \(\eta \) denote a sample configuration from the measure \(\mathbf {P}\). Let \(p_d(i) = \mathbf {P}[\eta (o) = i]\), \(i = 0, \dots , 2d-1\), 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 \(d \rightarrow \infty \).

### Theorem 1.1

- (i)For \(0 \le i \le d^{1/2}\), we have$$\begin{aligned} p_d(i) = \sum _{j=0}^i \frac{e^{-1} \frac{1}{j!}}{2d-j} + O\Big (\frac{i}{d^2}\Big ) = \frac{1}{2d} \sum _{j=0}^i e^{-1} \frac{1}{j!} + O\Big (\frac{i}{d^2}\Big ). \end{aligned}$$(1.1)
- (ii)If \(d^{1/2} < i \le 2d-1\), we haveIn particular, \(p_d(i) \sim (2d)^{-1}\), if \(i,d \rightarrow \infty \).$$\begin{aligned} p_d(i) = p_d(d^{1/2}) + O(d^{-3/2}). \end{aligned}$$

The appearance of the \(\mathsf {Poisson}(1)\) 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 \(\mathbb {Z}^d\) tends to 1 plus a \(\mathsf {Poisson}(1)\) random variable as \(d \rightarrow \infty \). Indeed our proof of (1.1) is achieved by showing that in the uniform spanning forest of \(\mathbb {Z}^d\), 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, \(\mathsf {Poisson}(1)\).

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.

*i*(with 2

*d*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 \(d\ge 2\), exact results for the height probability are only known for \(d = 2\); see [13, 14, 18, 21, 22].

### 1.1 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 \(\mathbb {Z}^d\).

Let \(V_L = [-L,L]^d \cap \mathbb {Z}^d\) be a box of radius *L*, where \(L \ge 1\). For simplicity, we suppress the *d*-dependence in our notation. We let \(G_L = (V_L \cup \{s \},E_L)\) denote the graph obtained from \(\mathbb {Z}^d\) by identifying all vertices in \(\mathbb {Z}^d {\setminus } V_L\) that becomes *s*, and removing loop-edges at *s*. We call *s* the *sink*. A *sandpile*\(\eta \) is a collection of indistinguishable particles on \(V_L\), specified by a map \(\eta : V_L \rightarrow \{ 0, 1, 2, \dots \}\).

*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:

*s*are lost, so we do not keep track of them. Toppling a vertex may generate further unstable vertices. Given a sandpile \(\xi \) on \(V_L\), we define its stabilization

We now define the sandpile Markov chain. The state space is the set of stable sandpiles \(\Omega _L\). Fix a positive probability distribution *p* on \(V_L\), i.e. \(\sum _{x\in V_L} p(x) = 1 \) and \(p(x) > 0\) for all \(x\in V_L\). Given the current state \(\eta \in \Omega _L\), choose a random vertex \(X \in V\) according to *p*, add one particle at *X* and stabilize. The one-step transition of the Markov chain moves from \(\eta \) to \((\eta + \mathbf {1}_X)^{\circ }\). Considering the sandpile Markov chain on \(G_L\), there is only one recurrent class [5]. We denote the set of recurrent sandpiles by \(\mathcal {R}_L\). It is known [5] that the invariant distribution \(\mathbf {P}_{L}\) of the Markov chain is uniformly distributed on \(\mathcal {R}_L\).

*s*, and let \(\pi _L(x)\) denote the oriented path from a vertex

*x*to

*s*. Let

*q*(

*i*) is given by the following natural analogue of its finite volume definition. Consider the uniform spanning forest measure \(\mathsf {USF}\) on \(\mathbb {Z}^d\); defined as the weak limit of \(\mathsf {UST}_L\); see [16, Chapter 10]. Let \(\pi (x)\) denote the unique infinite self-avoiding path in the spanning forest starting at

*x*, and let

### 1.2 Wilson’s Method

Given a finite path \(\gamma = [s_0, s_1, \ldots , s_k] \) in \(\mathbb {Z}^d\), we erase loops from \(\gamma \) chronologically, as they are created. We trace \(\gamma \) until the first time *t*, if any, when \(s_t \in \{s_0, s_1, \ldots , s_{t-1}\}\), i.e. there is a loop. We suppose \(s_t = s_i\), for some \(i \in \{0,1,\ldots ,t-1\}\) and remove the loop \([s_i,s_{i+1},\ldots ,s_t=s_i]\). Then, we continue tracing \(\gamma \) and follow the same procedure to remove loops until there are no more loops to remove. This gives the loop-erasure \(\pi = LE(\gamma )\) of \(\gamma \), which is a self-avoiding path [17]. If \(\gamma \) is generated from a random walk process, the loop-erasure of \(\gamma \) is called the loop-erased random walk (LERW).

When \(d \ge 3\), the \(\mathsf {USF}\) on \(\mathbb {Z}^d\) can be sampled via Wilson’s method rooted at infinity [4, 16, Section 10], that is described as follows. Let \(s_1, s_2,\dots \) be an arbitrary enumeration of the vertices and let \(\mathcal {T}_0\) be the empty forest with no vertices. We start a simple random walk \(\gamma _n\) at \(s_n\) and \(\gamma _n\) stops when \(\mathcal {T}_{n-1}\) is hit, otherwise we let it run indefinitely. \(LE(\gamma _n)\) is attached to \(\mathcal {T}_{n-1}\), and the resulting forest is denoted by \(\mathcal {T}_n\). We continue the same procedure until all the vertices are visited. The above gives a random sequence of forests \(\mathcal {T}_1 \subset \mathcal {T}_2 \subset \dots \), where \(\mathcal {T}= \cup _{n} \mathcal {T}_n \) is a spanning forest of \(\mathbb {Z}^d\). The extension of Wilson’s theorem [24] to transient infinite graphs proved in [4] implies that \(\mathcal {T}\) is distributed as the \(\mathsf {USF}\).

## 2 Proof of the Main Theorem

*x*(independent between

*x*’s on \(\mathbb {Z}^d\)) and let \(\pi (x)\) be the path in the USF from

*x*to infinity. We introduce the events:

### 2.1 Preliminary

### Lemma 2.1

We have \(\mathbf {P}[S^o_n = o\text { for some } n \ge 2] = O(1/d)\) and \(\mathbf {P}[S^o_n = o\text { for some } n \ge 4] = O(1/d^2)\), as \(d \rightarrow \infty \).

### Proof

*d*dimensions of the one-step distribution of RW. Lemma A.3 in [17] states that for all non-negative integers

*n*and all \(d\ge 1\), we have

*o*in 4 and 6 steps each, by counting the number of ways to return, they are bounded by dimension-independent multiples of \(1/d^2\) and \(1/d^3\), respectively. We have \(\int {\hat{D}}^n(k) dk = 0\) with odd

*n*, and for \(6 < n \le d-1\) and

*n*even, we have \(\int {\hat{D}}^n(k)dk \le \int {\hat{D}}^6(k)dk\). Hence,

### 2.2 Lower Bounds

### Lemma 2.2

We have \(\mathbf {P}[A_0] \ge 1- O(i/d).\)

### Proof

*o*in two further steps, the remaining neighbours will need at least 4 steps to hit, so, by Lemma 2.1, we have

*x*,

*y*. Therefore, combining above results together, we get \(\mathbf {P}[S_n^o \not \in \mathcal {N}\text { for } n \ge 2 | S_1^o \ne x_1,\dots ,x_i] \ge 1 - O(1/d)\) as required. \(\square \)

Let us label the neighbours of *o* different from \(x_1, \dots , x_i\) as \(x_{i+1}, \dots , x_{2d}\), in any order. On the event \(A_0\), the first step of \(\pi (o)\) is to a neighbour of *o* in \(\{x_{i+1},\dots ,x_{2d}\}\) and we could assume \(x_{2d}\) to be the first step of \(\pi (o)\). Then, \(\pi (o)\) does not visit other vertices in \(\mathcal {N} \backslash \{o\}\). Define \(A_j =\{S_1^{x_j} = o\}\) for \(j = 1,2,\dots ,i\) and then \(\mathbf {P}[A_j] = 1/2d\).

### Lemma 2.3

\(\mathbf {P}[B_k] \ge 1 - 1/2d - O(i/d^2)\), where \(i+1 \le k \le 2d-1\).

### Proof

We have \(\mathbf {P}[S_1^{x_k} \ne o] = 1 - 1/2d\). If the first step is not to *o*, the first step could be in one of the \(e_1,\dots ,e_i\) directions, say \(e_j\), with probability *i* / 2*d*. Then, the probability to hit \(x_j\) is \(1/2d + O(1/d^2)\). Hence, the probability that \(S^{x_k}\) hits \(\{x_1,\dots ,x_i\}\) is \(O(i/d^2)\). \(\square \)

### Lemma 2.4

\(q_d(i) \ge e^{-1} \frac{1}{i!}\left( 1+O\left( \frac{i^2}{d}\right) \right) .\)

### Proof

The above lemma gives a lower bound for \(q_d\), and we now prove an upper bound.

### 2.3 Upper Bounds

Recall that \(\pi (o)\) denotes the unique infinite self-avoiding path in the spanning forest starting at *o* and let \({\bar{A}}_o = \{\pi (o)\text { visits only one neighbour of } o\}\).

### Lemma 2.5

\(\mathbf {P}[\pi (o) \text { visits more than one neighbour of } o] = P[{\bar{A}}_o^c] = O(1/d).\)

### Proof

*o*, denoted by

*w*, then \(P[{\bar{A}}_o^c]\)

Let \({\bar{A}}_\mathrm{all} = \{ \forall w \sim o{:} \text { either } \pi (w) \text { does not visit } o \text { or } \pi (w) \text { visits } o~\mathrm{at~the~first~step} \}\).

### Lemma 2.6

\(\mathbf {P}[\exists w\sim o: \pi (w)\text { visits } o \text { but not at the first step}] = \mathbf {P}[{\bar{A}}_\mathrm{all}^c] = O(1/d).\)

### Proof

*w*, \(w \sim o\), use Wilson’s algorithm with a walk started at

*w*. Consider that if \(S_1^w \ne o\), or \(S_1^w = o\), but \(S^w\) returns to

*w*subsequently and then this loop starting from

*w*in \(S^w\) is erased, \(\pi (w)\) does not visit

*o*at the first step. Hence, we have the inequality:

*o*to

*w*at the beginning of the walk and analyse it as if the walk started at

*o*. Since \(S_1^o \in \mathcal {N} \backslash \{o\}\), by symmetry, we may assume \(S_1^o = w\). Then, if \(S_2^o \ne o\), \(S^o\) will need at least 2 more steps to return to

*o*.

*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

### Lemma 2.7

\(\mathbf {P}[H_j | {\bar{A}}_{o,x_1,\dots ,x_i} \cap \bigcap _{1\le j' < j}H_{j'}] = 1/2d + O(1/d^2)\), where \(j = 1,\dots ,i\).

### Proof

Given that \(\pi (o)\) visits only one neighbour of *o* which is not in \(\{x_1,\dots ,x_i\}\) and the first steps of \(\pi (x_1),\dots ,\pi (x_{j-1})\) are all to *o*, the probability that \(H_j\) happens is \(\mathbf {P}[S_1^{x_j} = o] = 1/2d\) with the error term of \(O(1/d^2)\) due to the loop-erasure. \(\square \)

### Lemma 2.8

### Proof

Consider Wilson’s algorithm with random walks started at the remaining neighbours \(x_{i+1}, \dots ,x_{2d}\). Assume \(x_{2d}\) to be the neighbour of *o* that \(\pi (o)\) goes through. The probability that \(\pi (x_k)\) does not go through *o* is \(1 - 1/2d + O(1/d^2)\) for \(k\in \{i+1,\dots ,2d-1\}\).

If \(\pi (x_k)\) visits \(x_{k'}\), where \(k < k' \le 2d-1\), the probability that \(\pi (x_{k'})\) does not go through *o* is 1 instead of \(1-1/2d+O(1/d^2)\), since the LERW from \(x_{k'}\) stops immediately and \(\pi (x_{k'}) \subset \pi (x_k)\), which does not go through *o*. \(\square \)

### Lemma 2.9

On the event \({\bar{A}}_\mathrm{rest}\), \(N \le B\), where \(B \sim \mathsf {Binom}(2d-i-1, p)\), \(p = 1/2d + O(1/d^2)\).

### Proof

Since we have \((2d-i-1)\) trials with probability at most \(1/2d+O(1/d^2)\). \(\square \)

### Lemma 2.10

\(q_d(i) \le O(\frac{1}{d}) + e^{-1}\frac{1}{i!}(1+O(\frac{i}{d})).\)

### Proof

### Lemma 2.11

This lemma can be proved using ideas used to prove Lemma 2.7.

### 2.4 Proof of the Asymptotic Formula

### Proof of Theorem 1.1

## Notes

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