The Rank of the Sandpile Group of Random Directed Bipartite Graphs

TL;DR

This paper determines the asymptotic distribution of the p-rank of the sandpile group in random directed bipartite graphs, showing it aligns with Erdős-Rényi models and confirming a conjecture about balanced graphs.

Contribution

It proves Koplewitz's conjecture that the expected p-rank is bounded for balanced bipartite graphs and provides the exact p-rank distribution for directed cases.

Findings

p-rank distribution matches Erdős-Rényi model

Confirms bounded p-rank expectation for balanced graphs

Provides exact distribution for directed bipartite graphs

Abstract

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 Erd{\"o}s-R{\'e}nyi 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 \cite{K} 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.