# Some new results in random matrices over finite fields

## TL;DR

This paper investigates the distribution properties of random matrices over finite fields, providing new characterizations of random walks and analyzing eigenvalue and polynomial divisibility probabilities, revealing universal behaviors.

## Contribution

It introduces novel characterizations of random walks with large discrepancy and extends universality results for matrix eigenvalues and polynomial divisibility over finite fields.

## Key findings

- Distribution of ranks of random matrices over F_p analyzed
- Probability of eigenvalue-free matrices characterized
- Divisibility of characteristic polynomials by irreducible polynomials studied

## Abstract

In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.