Permanental rank versus determinantal rank of random matrices over finite fields

TL;DR

This paper investigates the probability that random matrices over finite fields have zero permanent in all submatrices, revealing fundamental differences from the determinant and providing new insights into permanental rank.

Contribution

It establishes the asymptotic probability of zero permanental rank for random matrices over finite fields, highlighting key algebraic distinctions from the determinant.

Findings

Probability that a random matrix has zero permanental rank is approximately k/q^n for small k.

Contrast between permanental and determinantal rank probabilities over finite fields.

Main result applies when k is O(√n), revealing fundamental algebraic differences.

Abstract

This paper is motivated by basic complexity and probability questions about permanents of random matrices over finite fields, and in particular, about properties separating the permanent and the determinant. Fix $q = p^m$ some power of an odd prime, and let $k \leq n$ both be growing. For a uniformly random $n \times k$ matrix $A$ over $\mathbb{F}_q$, we study the probability that all $k \times k$ submatrices of $A$ have zero permanent; namely that $A$ does not have full "permanental rank". When $k = n$, this is simply the probability that a random square matrix over $\mathbb{F}_q$ has zero permanent, which we do not understand. We believe that the probability in this case is $\frac{1}{q} + o(1)$, which would be in contrast to the case of the determinant, where the answer is $\frac{1}{q} + \Omega_q(1)$. Our main result is that when $k$ is $O(\sqrt{n})$, the probability that a random $n \times k$ matrix does not have full permanental rank is essentially the same as the probability that the matrix has a $0$ column, namely $(1 +o(1)) \frac{k}{q^n}$. In contrast, for determinantal (standard) rank the analogous probability is $\Theta(\frac{q^k}{q^n})$. At the core of our result are some basic linear algebraic properties of the permanent that distinguish it from the determinant.