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The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

Gilyoung Cheong, Yifeng Huang

TL;DR

The paper studies the cokernel distribution of random $\mathbb{Z}_p$-matrices under a concentrated-residue model $X= A_n+pB_n$, extending Wood’s universality to a setting where the residue is fixed. It derives an explicit distribution for $\mathrm{cok}(P(X))$ with a non-constant monic $P(t)$ when $B_n$ is Haar-random, and shows that for special $A_n$ a broad class of $B_n$ yields the same distribution, revealing a universality phenomenon in this concentrated-residue regime. The authors implement a linearization approach to reduce the problem to equidistribution over the ring $R=\mathbb{Z}_p[t]/(P(t))$, and develop a noncommutative Weierstrass preparation theorem to construct measure-preserving maps that underpin the equidistribution results. The analysis combines module-theoretic invariants (Betti numbers, Hom/Ext dimensions) with a moment-method framework in a diamond-category setting (Sawin–Wood) to uniquely determine the distribution from moments. Overall, the work advances understanding of Cohen–Lenstra-type distributions for random integral matrices under fixed-residue constraints and introduces new noncommutative algebra tools that may apply to broader matrix-model universality questions.

Abstract

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of $p$-adic integers whose entries are independent. Previously, Wood showed that regardless of the distribution of $X_n$, as long as each entry of $X_n$ is not too concentrated on a single residue modulo $p$, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen--Lenstra distribution, as $n \rightarrow \infty$. In this paper, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$, where $B_n$ is a random $n \times n$ matrix over $\mathbb{Z}_p$. We show that for every fixed $n$ and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$, we can explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ when $B_n$ is a Haar-random matrix. Using this, we also show that for specific choices of $A_n$ a much wider class of random matrices $B_n$ gives the same distribution of $\mathrm{cok}(P(X_n))$. For the Haar-random $B_n$, we deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$.

The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

TL;DR

The paper studies the cokernel distribution of random -matrices under a concentrated-residue model , extending Wood’s universality to a setting where the residue is fixed. It derives an explicit distribution for with a non-constant monic when is Haar-random, and shows that for special a broad class of yields the same distribution, revealing a universality phenomenon in this concentrated-residue regime. The authors implement a linearization approach to reduce the problem to equidistribution over the ring , and develop a noncommutative Weierstrass preparation theorem to construct measure-preserving maps that underpin the equidistribution results. The analysis combines module-theoretic invariants (Betti numbers, Hom/Ext dimensions) with a moment-method framework in a diamond-category setting (Sawin–Wood) to uniquely determine the distribution from moments. Overall, the work advances understanding of Cohen–Lenstra-type distributions for random integral matrices under fixed-residue constraints and introduces new noncommutative algebra tools that may apply to broader matrix-model universality questions.

Abstract

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime and a positive integer , consider a random matrix over the ring of -adic integers whose entries are independent. Previously, Wood showed that regardless of the distribution of , as long as each entry of is not too concentrated on a single residue modulo , the distribution of the cokernel of , up to isomorphism, weakly converges to the Cohen--Lenstra distribution, as . In this paper, we consider the case when has a concentrated residue so that , where is a random matrix over . We show that for every fixed and a non-constant monic polynomial , we can explicitly compute the distribution of when is a Haar-random matrix. Using this, we also show that for specific choices of a much wider class of random matrices gives the same distribution of . For the Haar-random , we deduce our result from an interesting equidistribution result for matrices over , which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring of matrices over .
Paper Structure (27 sections, 32 theorems, 118 equations)

This paper contains 27 sections, 32 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Relevance to past and future works
  3. Methodology and brief outline of the paper
  4. Proof of Theorem \ref{['main']} from an equidistribution result
  5. Linearization and equidistribution
  6. Proofs of Proposition \ref{["prep'"]} and Theorem \ref{["Ha'"]}
  7. Minimal resolutions
  8. Fixing a residue class
  9. Some specifics about $\mathbb{Z}_p[t]/(P(t))$
  10. Proofs of Proposition \ref{["prep'"]} and Theorem \ref{["Ha'"]} assuming Theorems \ref{['thm:fw_general']} and \ref{['thm:fw_fiberwise']}
  11. Preliminaries in commutative algebra for proofs of Theorem \ref{['thm:fw_general']} and Lemma \ref{['lem:equidistribution_residue']}
  12. Proof of Theorem \ref{['thm:fw_general']}
  13. Proof of Lemma \ref{['lem:equidistribution_residue']}
  14. Reduction of proof of Theorem \ref{['equi']}
  15. A noncommutative Weierstrass preparation theorem and proof of Lemma \ref{['lem:equidist_map']}
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $0 < \epsilon < 1$ be a real number, and fix a finite abelian $p$-group $G$. For each $n \in \mathbb{Z}_{\geq 1}$, suppose that $\mathrm{M}_n(\mathbb{Z}_p) = \mathbb{Z}_{p}^{n^2}$ is equipped with a probability measure, where each random $X \in \mathrm{M}_n(\mathbb{Z}_p)$ has $n^2$ independent e in terms of the notation digits. Then

Theorems & Definitions (72)

  • Remark 1.1
  • Theorem 1.2: Wood
  • Remark 1.3
  • Theorem 1.4: Friedman and Washington
  • Conjecture 1.5: Cheong and Kaplan
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 62 more