Stochastic Sandpile on a Cycle

Abstract

In the stochastic sandpile model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability $0<p<1$ of not moving. These interactions continue until each site has no more than one particle on it. We provide a formal coupling between the stochastic sandpile and the activated random walk models, and we use the coupling to show that for the stochastic sandpile with $n$ particles on the cycle graph $\mathbb{Z}_n,$ the system stabilizes in $O(n^3)$ time for all initial particle configurations, provided that $p(n)$ tends to $1$ sufficiently rapidly as $n \rightarrow \infty$.