Permanents of random matrices over finite fields

TL;DR

This paper investigates the distribution of the permanent of random matrices over finite fields, revealing that it is more uniformly distributed than the determinant, thus advancing understanding of matrix invariants in finite field settings.

Contribution

It provides the first analysis showing the permanent's distribution is more uniform than the determinant's for random matrices over finite fields.

Findings

Permanent distribution is more uniform than determinant

First step towards understanding permanent distribution over finite fields

Highlights differences between permanent and determinant distributions

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 $\operatorname{per}(A)$ is still something of a mystery. In this paper we make a first step in this direction, proving that $\operatorname{per}(A)$ is significantly more uniform than $\det(A)$.