2021/06/11 by Bhargava, Atal, DePascale, Jack, Koenig, Jake
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2106.06352
We identify the asymptotic distribution of p-rank of the sandpile group of a random directed bipartite graphs which are not too imbalanced. We show this matches exactly that of the Erdös-Rényi random directed graph model, suggesting the Sylow p-subgroups of this model may also be Cohen-Lenstra distributed. Our work builds on 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 \citeK 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.