Sharp threshold for universality of cokernels of random matrices over finite fields

TL;DR

This paper establishes the precise threshold at which the cokernels of certain random matrices over finite fields exhibit universal distributional behavior, resolving an open problem in the field.

Contribution

It proves the sharp threshold for universality of cokernels of random matrices over finite fields with entries having limited probability bounds, extending previous results.

Findings

Cokernels converge to a universal distribution beyond the threshold

The threshold is sharp and depends on the parameter c

Answers an open problem by Wood (2022)

Abstract

In this paper, we determine the sharp threshold for universality of cokernels of random matrices over finite fields. More precisely, we prove the following: given any constant $c>1$, let $A(n)$ be a random $n \times n$ matrix over $\mathbb{F}_p$ whose entries are independent and take any given value of $\mathbb{F}_p$ with probability at most $1 - \frac{c \log n}{n}$. Then the cokernels of $A(n)$ converge in distribution, as $n \to \infty$, to the same limiting law as the cokernels of uniform random $n \times n$ matrices over $\mathbb{F}_p$. This answers an open problem posed by Wood (2022).