On the permanent of a random symmetric matrix

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. In our proof, we build on and extend some techniques introduced by Tao and Vu, studying the evolution of permanents of submatrices in a random matrix process.