Sandpile Groups of Random Bipartite Graphs

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.