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The distribution of sandpile groups of random graphs

Author: Melanie Matchett Wood
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 05C80, 15B52, 60B20
Published electronically: August 19, 2016
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Abstract: We determine the distribution of the sandpile group (or Jacobian) of the Erdős-Rényi random graph $ G(n,q)$ as $ 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}$-size growth. In particular, our theorems imply universality of singularity probability and ranks mod $ p$ for symmetric integral matrices.

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Additional Information

Melanie Matchett Wood
Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53705, and American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306-2244

Received by editor(s): October 21, 2014
Received by editor(s) in revised form: April 21, 2016, July 6, 2016, and July 17, 2016
Published electronically: August 19, 2016
Additional Notes: This work was done with the support of an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, and National Science Foundation grants DMS-1147782 and DMS-1301690.
Article copyright: © Copyright 2016 American Mathematical Society