The distribution of sandpile groups of random graphs

Abstract

We determine the distribution of the sandpile group (or Jacobian) of the Erdős-Rényi random graph G ( n , q ) G(n,q) as n n goes to infinity. We prove the distribution converges to a specific distribution conjectured by Clancy, Leake, and Payne. This distribution is related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the “moments” of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. The methods developed to prove this result include inverse Littlewood-Offord theorems over finite rings and new techniques for studying homomorphisms of finite abelian groups with not only precise structure but also approximate versions of that structure. We then show these moments determine a unique distribution despite their p k 2 p^{k^2} -size growth. In particular, our theorems imply universality of singularity probability and ranks mod p p for symmetric integral matrices.