The distribution of sandpile groups of random regular graphs

TL;DR

This paper investigates the distribution of sandpile groups in random regular graphs, confirming Cohen-Lenstra heuristics for directed graphs and related distributions for undirected graphs, with implications for graph invertibility.

Contribution

It proves the distribution of sandpile groups follows Cohen-Lenstra heuristics for directed graphs and extends results to undirected graphs, answering open questions.

Findings

Sandpile groups follow Cohen-Lenstra heuristics for directed graphs.

For finitely many primes, events become independent in the limit.

Results imply high probability of adjacency matrix invertibility in random regular graphs.

Abstract

We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is invertible with high probability. Note that for directed graphs this was recently proved by Huang. It also gives an alternate proof of a theorem of Backhausz and Szegedy.