On the distribution of the entries of a fixed-rank random matrix over a finite field

TL;DR

This paper proves that the distribution of entries from a fixed-rank random matrix over a finite field converges to a normal distribution as matrix dimensions grow large, revealing probabilistic structure in algebraic matrix models.

Contribution

It establishes a normal approximation for the distribution of entries in fixed-rank matrices over finite fields as dimensions tend to infinity.

Findings

Entries follow a normal distribution in large matrices

Distribution converges as matrix size increases

Results hold for fixed rank and subset of finite field

Abstract

Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$. Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over $\mathbb{F}_q$ taken with uniform distribution. We prove, in a precise sense, that, as $m, n \to +\infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ that belong to $\mathcal{A}$ approaches a normal distribution.