The distribution of sandpile groups of random graphs

Abstract

Abstract. We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erdős-Rényi random graph G(n, q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups. We prove the distributions of Sylow p-subgroups converge to specific distributions conjectured by Clancy, Leake, and Payne. These distributions are 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. We then show these moments determine a unique distribution despite their p k 2 -size growth.

The distribution of sandpile groups of random regular graphs

“…Theorem 1 follows from Theorem 2 using the results of Wood [22] on the moment problem. The general question is the following.…”

“…This is exactly the distribution appearing in Theorem 1. Note that this is not the original formula given in [5], but it can be easily deduced from it, see [22]. Here, a map φ :…”

“…Instead of abelian p-groups, one can consider groups which are direct sums of finite abelian p i -groups for a fixed finite set of primes. See [22] for details. Note that for even d the moments of the sandpile groups of H n are larger than the bounds above.…”

“…We can give a more explicit formula for the limiting probability above by using the following fact from [22]. If G = i Z/p λ i Z with λ 1 ≥ λ 2 ≥ · · · and µ is the transpose of the partition λ, then…”

Graphs with few trivial critical ideals

“…Based on this, D. Wagner conjectured [35] that almost every connected simple graph has a cyclic critical group. A recent study [37] concluded that the probability that the critical group of a random graph is cyclic is asymptotically at most…”