Sandpile Groups of Random Bipartite Graphs

TL;DR

This paper analyzes the asymptotic distribution of the p-rank of sandpile groups in random bipartite graphs, revealing a dependence on the ratio of vertices on each side and contrasting with Erdős-Rényi graphs.

Contribution

It introduces a novel analysis of sandpile groups in bipartite graphs, showing their distribution depends on graph properties rather than a universal random group model.

Findings

Distribution depends on the ratio of vertices on each side.

A threshold at ratio = 1/p determines the p-rank behavior.

Contrasts with Erdős-Rényi graph sandpile groups.

Abstract

We determine the asymptotic distribution of the p-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to 1/p. We follow the approach of Melanie Wood and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples, in order to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of Erdos-Renyi random graphs, the distribution of the sandpile groups of random bipartite graphs depends on the properties of the graph, rather than coming from some more general random group model.