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Cokernels of random matrices satisfy the Cohen-Lenstra heuristics

Kenneth Maples

TL;DR

The paper proves that the cokernel of a random $n\times n$ matrix with iid entries over $\mathbb{Z}_p$ (and more generally over $\mathbb{Z}/N\mathbb{Z}$) follows the Cohen-Lenstra distribution up to exponentially small error, under a mild non-degeneracy (min-entropy) condition. It introduces a column-exposure framework, uses enlarged submodules and Young-diagram encodings to track the evolving cokernel, and employs a suite of analytic-combinatorial tools, including a Swapping Lemma, to control all submodule regimes. The main contribution is a robust universality principle that binds a wide class of non-uniform random matrices to the Cohen-Lenstra measure, alongside a composite-modulus generalization. This strengthens the heuristic view that Cohen-Lenstra statistics arise from general random-cokernel behavior rather than a specific random-model construction, with potential extensions to broader rings and distributions.

Abstract

Let A be an n by n random matrix with iid entries taken from the p-adic integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A has a universal probability distribution. In particular, the p-part of an iid random matrix over the integers has cokernel distributed according to the Cohen-Lenstra measure up to an exponentially small error.

Cokernels of random matrices satisfy the Cohen-Lenstra heuristics

TL;DR

The paper proves that the cokernel of a random matrix with iid entries over (and more generally over ) follows the Cohen-Lenstra distribution up to exponentially small error, under a mild non-degeneracy (min-entropy) condition. It introduces a column-exposure framework, uses enlarged submodules and Young-diagram encodings to track the evolving cokernel, and employs a suite of analytic-combinatorial tools, including a Swapping Lemma, to control all submodule regimes. The main contribution is a robust universality principle that binds a wide class of non-uniform random matrices to the Cohen-Lenstra measure, alongside a composite-modulus generalization. This strengthens the heuristic view that Cohen-Lenstra statistics arise from general random-cokernel behavior rather than a specific random-model construction, with potential extensions to broader rings and distributions.

Abstract

Let A be an n by n random matrix with iid entries taken from the p-adic integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A has a universal probability distribution. In particular, the p-part of an iid random matrix over the integers has cokernel distributed according to the Cohen-Lenstra measure up to an exponentially small error.

Paper Structure

This paper contains 16 sections, 28 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The column exposure process
  4. Statement of the universality proposition
  5. The sequence of partial cokernels
  6. Young diagrams
  7. Enlarged submodules
  8. Entropy bounds
  9. Universality for enlarged submodules
  10. Types of enlarged submodules
  11. Analytic and Combinatorial Tools
  12. Sparse submodules
  13. Unsaturated submodules
  14. Swapping Lemma
  15. Semi-saturated submodules
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $A \in M(n,\mathbb{Z}_p)$ be an iid random matrix with min-entropy $\alpha$. Then for all finite abelian $p$-groups $G$, where the constants are absolute.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 37 more