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

TL;DR

Under the assumption of the extended Riemann hypothesis, the paper proves that the characteristic polynomial of a random symmetric ±1-matrix is irreducible with high probability, advancing understanding of its algebraic properties.

Contribution

The paper introduces sharp estimates for the rank distribution of symmetric random ±1-matrices over finite fields, extending previous bounds to larger primes.

Findings

Characteristic polynomial is irreducible with high probability

Established new bounds for rank distribution over larger primes

Combined inverse Littlewood–Offord results for improved control

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 $2 < p \leq \exp(O(n^{1/4}))$. 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 $C > 1$.