On a CohenLenstra 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 CohenLenstra 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 padic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.5129
 Bibcode:
 2014arXiv1402.5129C
 Keywords:

 Mathematics  Combinatorics;
 05C80;
 15B52
 EPrint:
 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