Cohen-Lenstra flag universality for random matrix products

TL;DR

This paper demonstrates that the distribution of cokernels of products of random integer matrices converges to a universal Cohen-Lenstra type measure, revealing deep connections between random matrix products and algebraic structures.

Contribution

It establishes the universality of cokernel distributions for products of random matrices over integers, extending Cohen-Lenstra heuristics to matrix product settings.

Findings

Convergence of cokernel flags to Cohen-Lenstra measure as matrix size grows

Universality holds for any nondegenerate entry distribution

Conditional distributions relate to Hall-Littlewood structure constants

Abstract

For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this flag converges universally, for any nondegenerate entry distribution, to the Cohen-Lenstra type measure which weights each flag inversely proportional to the size of its automorphism group. As a corollary, we prove universality of certain formulas for the limiting conditional distribution of $\mathrm{cok}(M_1M_2)$ given $\mathrm{cok}(M_1),\mathrm{cok}(M_2)$ in terms of Hall-Littlewood structure constants, which were previously obtained only for Haar matrices over $\mathbb{Z}_p$. Our proofs combine the general technology of Sawin-Wood, matrix product moment computations following those of Nguyen-Van Peski, and the computation done previously for Haar $p$-adic matrices by Huang.