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Journal of Algebraic Combinatorics

, Volume 42, Issue 3, pp 701–723 | Cite as

On a Cohen–Lenstra heuristic for Jacobians of random graphs

  • Julien Clancy
  • Nathan Kaplan
  • Timothy Leake
  • Sam PayneEmail author
  • Melanie Matchett Wood
Article

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 \({\mathbb {Z}}_p\), distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.

Keywords

Random graphs Cohen–Lenstra heuristics Cokernels of random matrices Sandpile groups Graph Jacobians 

Notes

Acknowledgments

The authors thank Matt Baker, Wei Ho, Matt Kahle, and the referees. The fourth author was supported in part by NSF grant DMS-1068689 and NSF CAREER grant DMS-1149054. The fifth author was supported by an American Institute of Mathematics Five-Year Fellowship and National Science Foundation grants DMS-1147782 and DMS-1301690.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Julien Clancy
    • 1
  • Nathan Kaplan
    • 1
  • Timothy Leake
    • 1
  • Sam Payne
    • 1
    Email author
  • Melanie Matchett Wood
    • 2
    • 3
  1. 1.Yale UniversityNew HavenUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.American Institute of MathematicsSan JoseUSA

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