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

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 $p$.