Universality for cokernels of random matrix products

TL;DR

This paper proves that the distribution of cokernels of products of large random integer matrices converges to a universal distribution, extending the Cohen-Lenstra measure, with implications for matrix cokernels over finite fields.

Contribution

It establishes universality results for cokernel distributions of random matrix products and characterizes the limiting distributions via a generalized Cohen-Lenstra measure.

Findings

Distribution converges to a universal limit as matrix size grows.

Universal limits are characterized by a generalization of Cohen-Lenstra measure.

Explicit universal distribution for coranks over finite fields is derived.

Abstract

For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution converges to a universal one as $n \to \infty$ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of $\operatorname{cok}(M_1),\operatorname{cok}(M_1M_2),\ldots,\operatorname{cok}(M_1 \cdots M_k)$. Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over $\mathbb{F}_p$ as the matrix size tends to infinity.