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 $\det(A)$.
Categories: math.CO cs.CC math.PR