Permanents of random matrices over finite fields

Abstract

Fix a finite field ${\mathbb{F}}_q$ and let $A\in {\mathbb{F}}_q^{n\times n}$ be a uniformly random $n\times n$ matrix over ${\mathbb{F}}_q$. The asymptotic distribution of the determinant $\det (A)$ is well-understood, but the asymptotic distribution of the permanent $\mathrm{per}(A)$ is still something of a mystery. In this paper we make a first step in this direction, proving that $\mathrm{per}(A)$ is significantly more uniform than $\det (A)$.