A countdown process, with application to the rank of random matrices over $\mathbb F_q(n)$

TL;DR

This paper introduces a countdown process linked to random integer partitions to analyze the distribution of the corank of random matrices over finite fields, providing sharper bounds on their convergence to the limit distribution.

Contribution

It defines a novel countdown process driven by geometric variables that improves bounds on the total variation distance for matrix corank distributions.

Findings

Sharper bounds on total variation distance established

Countdown process effectively models matrix corank distribution

Connections made between partitions and matrix properties

Abstract

Motivated by the work of Fulman and Goldstein, comparing the distribution of the corank of random matrices in $\mathbb F_q[n]$ with the limit distribution as $n \to \infty$, we define a countdown process, driven by independent geometric random variables related to random integer partitions. Analysis of this process leads to sharper bounds on the total variation distance.