On the permanent of a random symmetric matrix

TL;DR

This paper proves that the permanent of a random symmetric matrix with Rademacher entries typically has magnitude around $n^{n/2}$, confirming a conjecture and extending to broader models.

Contribution

It resolves Vu's conjecture by establishing the typical magnitude of the permanent for a class of random symmetric matrices.

Findings

Permanent magnitude is $n^{n/2+o(n)}$ with high probability.

Results extend to more general random matrix models.

Confirms a longstanding conjecture in random matrix theory.

Abstract

Let $M_{n}$ denote a random symmetric $n\times n$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values $\pm 1$ with probability $1/2$ each). Resolving a conjecture of Vu, we prove that the permanent of $M_{n}$ has magnitude $n^{n/2+o(n)}$ with probability $1-o(1)$. Our result can also be extended to more general models of random matrices.