Stein's method and the rank distribution of random matrices over finite fields

TL;DR

This paper uses Stein's method to establish precise bounds on the total variation distance between the finite-sample rank distribution of random matrices over finite fields and its limit distribution, with extensions to various matrix classes.

Contribution

It introduces Stein's method to derive sharp bounds for the rank distribution convergence of random matrices over finite fields, including several matrix subclasses.

Findings

Established bounds for the total variation distance between finite and limiting rank distributions.

Extended results to symmetric, skew-symmetric, Hermitian, and other matrix classes.

Provided explicit decay rates of the distributional difference as matrix size grows.

Abstract

With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_q$ of size $q\ge2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\rightarrow\infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\|{\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.