Nearly-symmetric matrices in the Cohen-Lenstra universality class

TL;DR

This paper investigates how small perturbations in symmetry conditions of random integer matrices affect the distribution of their cokernels, showing convergence to the Cohen-Lenstra distribution under weak assumptions.

Contribution

It demonstrates that cokernel distributions of nearly-symmetric matrices converge to the Cohen-Lenstra distribution, revealing sensitivity to minor symmetry perturbations.

Findings

Cokernel distributions of nearly-symmetric matrices converge to Cohen-Lenstra distribution

Weak conditions on entry distributions suffice for convergence

Small symmetry perturbations significantly influence cokernel distribution

Abstract

In this paper, we study cokernels of random $n\times n$ matrices over $\mathbb Z$ with symmetry conditions determined by fixed alternating bilinear forms on $\mathbb Z^n$. These include perturbations of random symmetric matrices at a very small (but unbounded with $n$) number of entries. We show that, subject to fairly weak conditions on the distributions of the entries, the distribution of these cokernels converges weakly to the Cohen-Lenstra distribution, which is the limiting distribution of cokernels of random matrices with no symmetry constraints. This result demonstrates that the cokernel distributions of symmetric matrices are quite sensitive to small perturbations of the symmetry conditions.