The central limit theorem for entries of random matrices with specific rank over finite fields

TL;DR

This paper proves a central limit theorem for the entries of large-rank random matrices over finite fields, extending previous results to cases where the rank grows with matrix size.

Contribution

It establishes a new CLT for entries of random matrices with increasing rank over finite fields, generalizing prior fixed-rank results.

Findings

Entries of large-rank matrices are normally distributed as size grows.

The CLT holds when the rank increases with matrix dimensions.

Extends previous fixed-rank CLT to variable-rank scenarios.

Abstract

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathcal{A}$ a non-empty proper subset of $\mathbb{F}_q$. Let $\mathbf{M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \to \infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ in $\mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively.