On the permanent of random Bernoulli matrices

TL;DR

This paper proves that the permanent of large random Bernoulli matrices is almost surely non-zero and has a magnitude roughly exponential in n, with high probability.

Contribution

It establishes the asymptotic magnitude and non-vanishing property of the permanent for iid Bernoulli matrices, a significant advance in understanding their combinatorial properties.

Findings

Permanent magnitude is approximately n^{(1/2+o(1))n}

Permanent is almost surely non-zero for large matrices

High probability bounds on the permanent's size

Abstract

We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero.