Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Abstract

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$-matrices over ${\mathbb{F}}_p$ for primes $2<p\le \exp (O(n^{1/4}))$. Previously, such estimates were available only for $p=o(n^{1/8})$. At the heart of our proof is a way to combine multiple inverse Littlewood--Offord-type results to control the contribution to singularity-type events of vectors in ${\mathbb{F}}_p^n$ with anticoncentration at least $1/p+\Omega (1/p^2)$. Previously, inverse Littlewood--Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C>1$.

12 pages; comments welcome!