Stochastic sandpile on a cycle

Abstract

Abstract In the stochastic sandpile (SS) 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 SS and the activated random walk models, and we use the coupling to show that for the SS with n particles on the cycle graph 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 → ∞.