Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks

TL;DR

This paper establishes uniform spectral gaps for random walks on semisimple algebraic groups, leading to anti-concentration results, non-abelian Littlewood-Offord inequalities, and bounds on escape from subvarieties, with applications to expansion in finite simple groups.

Contribution

It introduces a uniform spectral gap for quasi-regular representations of linear groups, enabling new anti-concentration and expansion results in algebraic and finite group settings.

Findings

Proves uniform exponential anti-concentration for random walks on algebraic groups.

Derives a non-abelian Littlewood-Offord inequality.

Establishes logarithmic bounds for escape from algebraic subvarieties.

Abstract

We show that random walks on semisimple algebraic groups do not concentrate on proper algebraic subvarieties with uniform exponential rate of anti-concentration. This is achieved by proving a uniform spectral gap for quasi-regular representations of countable linear groups. The method makes key use of Diophantine heights and the Height Gap theorem. We also deduce a non-abelian version of the Littlewood--Offord inequalities and prove logarithmic bounds for escape from subvarieties. In a sequel to this paper, we will show how to transform this uniform gap into uniform expansion for Cayley graphs of finite simple groups of bounded rank $G(p)$ over almost all primes $p$.