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Nearly-symmetric matrices in the Cohen-Lenstra universality class

Elia Gorokhovsky

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.

Nearly-symmetric matrices in the Cohen-Lenstra universality class

Abstract

In this paper, we study cokernels of random matrices over with symmetry conditions determined by fixed alternating bilinear forms on . These include perturbations of random symmetric matrices at a very small (but unbounded with ) 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.
Paper Structure (7 sections, 23 theorems, 86 equations)

This paper contains 7 sections, 23 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Background on Cokernel Distributions and the Moment Method
  4. Reduction to Symmetric Case
  5. Isotropy over a Finite Field and Very Small Perturbations
  6. Computing Isotropy Probabilities
  7. Examples

Key Result

Theorem 1.2

Let $\varepsilon > 0$. For each $n$, let $C_n$ be a deterministic $n\times n$ alternating matrix over $\mathbb Z_p$. Let $X_n$ be a random $C_n$-symmetric matrix over $\mathbb Z_p$ with independent $\varepsilon$-balanced entries on and above the diagonal. Assume that we have Then for any finite abelian $p$-group $G$, we have In other words, the distribution of $\operatorname{coker}(X_n)$ converg

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1: Consequence of wood2019matrices, see also sawin2024momentproblemrandomobjects
  • Remark 3.2
  • Lemma 3.3: wood2019matrices, wood2017sandpile, see also clancyCohenLenstraHeuristic2015
  • Definition 3.4
  • Theorem 3.5: wood2019matrices
  • Theorem 3.6: Consequence of wood2017sandpile
  • Remark 3.7
  • ...and 34 more