Limit shape of single-source stochastic sandpiles with $p$-topplings on $\mathbb{Z}$

TL;DR

This paper studies the asymptotic shape and boundary fluctuations of a stochastic sandpile model on the integer line, revealing a symmetric limit shape and Gaussian fluctuations in the large particle limit.

Contribution

It introduces a new analysis of the limit shape and boundary fluctuations for the p-toppling stochastic sandpile model on or the first time.

Findings

The limit shape is a symmetric interval around the origin.

Boundary fluctuations converge to a Gaussian distribution after rescaling.

Abstract

We investigate the limit shape of the single-source model for stochastic sandpiles on the integer line subject to $p$--topplings. In this model, an initial configuration of $n\in\mathbb{N}$ particles is placed at the origin and stabilized according to a random toppling rule depending on $p\in (0,1)$: an unstable vertex sends exactly one particle to its left neighbor with probability $p$, and independently sends exactly one particle to its right neighbor with probability $p$. We prove that as $n \to \infty$, the macroscopic limit shape of the final stable configuration is a symmetric interval around the origin. Furthermore, by analyzing the center of mass martingale, we establish a central limit theorem for the boundary fluctuations, showing that after proper rescaling, they converge to a Gaussian distribution.