Integer invariants of abelian Cayley graphs

Ducey, Joshua E.; Jalil, Deelan M.

doi:10.48550/arxiv.1308.2335

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2016

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“…The cokernel of the adjacency matrix is called the Smith group, and has been studied e.g. in [CSX14,DJ13].…”

Section: Introductionmentioning

confidence: 99%

The distribution of sandpile groups of random graphs

Wood

2016

J. Amer. Math. Soc.

127

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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.

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“…The cokernel of the adjacency matrix is called the Smith group, and has been studied e.g. in [CSX14,DJ13].…”

Section: Introductionmentioning

confidence: 99%