Abelian and stochastic sandpile models on complete bipartite graphs

TL;DR

This paper investigates Abelian and stochastic sandpile models on complete bipartite graphs, introducing algorithms and combinatorial bijections to analyze recurrent configurations efficiently.

Contribution

It develops a linear-time stochastic version of Dhar's burning algorithm and establishes new combinatorial bijections for recurrent configurations in both models.

Findings

Linear complexity stochastic burning algorithm

Bijection between recurrent configurations and Ferrers diagrams

Interpretation of ASM configurations via Motzkin paths

Abstract

In the sandpile model, vertices of a graph are allocated grains of sand. At each unit of time, a grain is added to a randomly chosen vertex. If that causes its number of grains to exceed its degree, that vertex is called unstable, and topples. In the Abelian sandpile model (ASM), topplings are deterministic, whereas in the stochastic sandpile model (SSM) they are random. We study the ASM and SSM on complete bipartite graphs. For the SSM, we provide a stochastic version of Dhar's burning algorithm to check if a given (stable) configuration is recurrent or not, with linear complexity. We also exhibit a bijection between sorted recurrent configurations and pairs of compatible Ferrers diagrams. We then provide a similar bijection for the ASM, and also interpret its recurrent configurations in terms of labelled Motzkin paths.