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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The distribution of sandpile groups of random regular graphs

Author: András Mészáros
Journal: Trans. Amer. Math. Soc. 373 (2020), 6529-6594
MSC (2010): Primary 05C80, 15B52, 60B20
Published electronically: July 3, 2020
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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 $ \vert\mathrm {Aut}(P)\vert^{-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.

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András Mészáros
Affiliation: Department of Mathematics and its Applications, Central European University, Budapest; and Department of Combinatorics and Discrete Mathematics, Alfréd Rényi Institute of Mathematics, Budapest

Received by editor(s): October 17, 2018
Received by editor(s) in revised form: December 16, 2019, and January 27, 2020
Published electronically: July 3, 2020
Additional Notes: The author was partially supported by the ERC Consolidator Grant 648017 and the Hungarian National Research, Development and Innovation Office, NKFIH grant K109684.
Article copyright: © Copyright 2020 American Mathematical Society