On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs
Abstract
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 the p-adic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.
Keywords
Cite
@article{arxiv.1402.5129,
title = {On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs},
author = {Julien Clancy and Nathan Kaplan and Timothy Leake and Sam Payne and Melanie Matchett Wood},
journal= {arXiv preprint arXiv:1402.5129},
year = {2016}
}
Comments
20 pages. v2: Improved exposition and appended code used to generate experimental evidence after the \end{document} line in the source file. To appear in J. Algebraic Combin