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On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs

Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, Melanie Matchett Wood

TL;DR

This work extends the Cohen–Lenstra heuristic to Jacobians of random graphs by incorporating a canonical duality pairing, predicting a distribution on finite abelian p-groups with pairing that is inversely proportional to |Γ|·|Aut(Γ,δ)| up to a universal p-adic factor. It proves corresponding cokernel distributions for Haar-random symmetric matrices over Z_p and generalizes to Laplacian-like zero-sum matrices, then computes exact averages and moments for various statistics. Theoretical predictions are supported by substantial empirical data showing the Jacobian is cyclic with probability about 0.7935 and that p-parts behave independently across primes, with observed pairings matching the classified types. The results unify random graph Jacobians with p-adic random matrix theory and offer precise asymptotics for several natural statistics.

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.

On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs

TL;DR

This work extends the Cohen–Lenstra heuristic to Jacobians of random graphs by incorporating a canonical duality pairing, predicting a distribution on finite abelian p-groups with pairing that is inversely proportional to |Γ|·|Aut(Γ,δ)| up to a universal p-adic factor. It proves corresponding cokernel distributions for Haar-random symmetric matrices over Z_p and generalizes to Laplacian-like zero-sum matrices, then computes exact averages and moments for various statistics. Theoretical predictions are supported by substantial empirical data showing the Jacobian is cyclic with probability about 0.7935 and that p-parts behave independently across primes, with observed pairings matching the classified types. The results unify random graph Jacobians with p-adic random matrix theory and offer precise asymptotics for several natural statistics.

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.

Paper Structure

This paper contains 16 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. The pairing
  3. Conjectures
  4. Result for Haar random symmetric matrices
  5. Distribution of Cokernels of Haar Random Symmetric Matrices
  6. A generalization of Theorem \ref{['thm:pparts']} for matrices with row and column sums equal to zero.
  7. Computing Averages
  8. The normalizing constant
  9. The probability that $\Gamma$ has trivial or cyclic $p$-part
  10. Moments
  11. Empirical evidence
  12. The probability that a random graph is cyclic
  13. Relative frequencies of $2$-groups with duality pairings
  14. Relative frequencies of $3$-groups with duality pairings
  15. Relative frequencies at two places
  16. ...and 1 more sections

Key Result

Theorem 2

Let $\Gamma$ be a finite abelian $p$-group of rank $r$ with a duality pairing $\delta$, and let $A$ be a random $n \times n$ symmetric matrix with respect to additive Haar measure on ${\operatorname{Sym}}_n({\mathbb Z}_p)$. Then the probability that the cokernel of $A$ with its given duality pairing where ${\operatorname{Aut \;}}(\Gamma,\delta)$ is the set of automorphisms of $\Gamma$ that preserv

Theorems & Definitions (32)

  • Conjecture 1
  • Remark 1
  • Conjecture 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 22 more