Asymptotics of the single-source stochastic sandpile model

TL;DR

This paper investigates the asymptotic behavior of a stochastic sandpile model on a 2D grid, analyzing how the stabilization process scales with the number of grains and the distribution of toppling amounts, revealing phase transitions.

Contribution

It introduces a variant of the stochastic sandpile model, analyzes its asymptotics for large N, and identifies phase transitions in the case of binomial toppling distributions.

Findings

Both radius and avalanche numbers exhibit simple asymptotic behaviors.

A phase transition occurs at p ~ 1/N when the toppling distribution is binomial with parameter p.

Simulation results support the theoretical asymptotic analysis.

Abstract

In the single-source sandpile model, a number $N$ grains of sand are positioned at a central vertex on the 2-dimensional grid $\mathbb{Z}^2$. We study the stabilisation of this configuration for a stochastic sandpile model based on a parameter $M \in \mathbb{N}$. In this model, if a vertex has at least $4M$ grains of sand, it topples, sending $k$ grains of sand to each of its four neighbours, where $k$ is drawn according to some random distribution $\gamma$ with support $\{1,\cdots,M\}$. Topplings continue, a new random number $k$ being drawn each time, until we reach a stable configuration where all the vertices have less than $4M$ grains. This model is a slight variant on the one introduced by Kim and Wang. We analyse the stabilisation process described above as $N$ tends to infinity (for fixed $M$), for various probability distributions $\gamma$. We focus on two global parameters of the system, referred to as radius and avalanche numbers. The radius number is the greatest distance from the origin to which grains are sent during the stabilisation, while the avalanche number is the total number of topplings made. Our simulations suggest that both of these numbers have fairly simple asymptotic behaviours as functions of $\gamma$, $N$ and $M$ as $N$ tends to infinity. We also provide a more detailed analysis in the case where $\gamma$ is the binomial distribution with parameter $p$, in particular when $p$ tends to $1$. We exhibit a phase transition in that regime at the scale $p \sim 1/N$.