The Rank of the Sandpile Group of Random Directed Bipartite Graphs

Abstract

We identify the asymptotic distribution of p-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös–Rényi random directed graph model, suggesting that the Sylow p-subgroups of this model may also be Cohen–Lenstra distributed. Our work builds on the results of Koplewitz who studied p-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of p-rank differs from the Cohen–Lenstra distribution. Koplewitz (sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017) conjectured that for random balanced bipartite graphs, the expected value of p-rank is O(1) for any p. This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.