On a Cohen–Lenstra heuristic for Jacobians of random graphs
By J. Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, Melanie Matchett Wood. Journal of Algebraic Combinatorics, 2014
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In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen–Lenstra-type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p$$\end{document}, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b5f31bf47458f1f1fd9991f6576e1c3f177ef19a
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Citations in the corpus
- The distribution of sandpile groups of random regular graphs (2018, methodology): “In Lemma 57 above we concentrated only on the prime 2 for simplicity, but the using the same argument we can handle finitely many primes simultaneously by following the argument of Wood [20].”