The stochastic sandpile model on complete bipartite graphs

The stochastic sandpile model (SSM) generalises the standard Abelian sandpile model (ASM) by making topplings of unstable vertices random. When unstable, a vertex sends one grain to each of its neighbours independently with probability p ∈ (0, 1). We study the SSM on complete bipartite graphs. We characterise recurrent configurations of the model in terms of a simple series of inequalities. This allows us to exhibit a bijection between sorted recurrent configurations and pairs of compatible Ferrers diagrams. We also provide a stochastic version of Dhar’s burning algorithm to check if a given (stable) configuration is recurrent or not, with linear complexity on sorted configurations.