Permanents of random matrices over finite fields
This addresses a theoretical gap in random matrix theory for mathematicians and computer scientists, though it is incremental as a first step.
The paper tackled the problem of understanding the asymptotic distribution of the permanent of random matrices over finite fields, showing that the permanent is significantly more uniform than the determinant.
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 $\det(A)$.