Rank of Near Uniform Matrices

TL;DR

This paper investigates the rank distribution of near uniform random matrices over finite fields, establishing tighter convergence bounds and extending results to symmetric, alternating, and perturbed matrices.

Contribution

It introduces a new near uniform condition that yields sharper bounds on rank statistics and extends universality results to broader classes of matrices with elementary methods.

Findings

Achieves bounds of q^{-cn} for rank statistic convergence.

Extends results to symmetric and alternating matrices.

Applicable to matrices with deterministic entries or non-identical distributions.

Abstract

A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the behavior occurs for a larger class of random variables. The rank statistics of random matrices chosen uniformly from $\operatorname{Mat}(\mathbf{F}_q)$ over a finite field are well understood. The universality properties of these statistics are not yet fully understood however. Recently Wood [39] and Maples [26] considered a natural requirement where the random variables are not allowed to be too close to constant and they showed that the rank statistics match with the uniform model up to an error of type $e^{-cn}$. In this paper we explore a condition called near uniform, under which we are able to prove tighter bounds $q^{-cn}$ on the asymptotic convergence of the rank statistics. Our method is completely elementary, and allows for a small number of the entries to be deterministic, and for the entries to not be identically distributed so long as they are independent. More importantly, the method also extends to near uniform symmetric, alternating matrices. Our method also applies to two models of perturbations of random matrices sampled uniformly over $\operatorname{GL}_n(\mathbf{F}_q)$: subtracting the identity or taking a minor of a uniformly sampled invertible matrix.