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 Zn, the system stabilizes in O(n 3 ) time for all initial particle configurations, provided that p(n) tends to 1 sufficiently rapidly as n → ∞.