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

Oren Becker, Emmanuel Breuillard

TL;DR

The paper proves a uniform spectral gap for quasi-regular representations of countable linear groups arising from Zariski-dense actions on semisimple algebraic groups, with constants depending only on the ambient dimension. It then translates this gap into strong, uniform anti-concentration results for random walks on linear groups and a non-abelian Littlewood–Offord inequality, together with a logarithmic bound for escaping subvarieties. The core methodology blends a ping-pong with overlaps technique, adelic heights and the Height Gap theorem, and local-field dynamics to achieve global, uniform control across all fields and characteristics. A forthcoming sequel promises to leverage these results to uniform expansion in finite simple groups of bounded rank, via Bourgain–Gamburd-type machinery.

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 $p$.

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

TL;DR

The paper proves a uniform spectral gap for quasi-regular representations of countable linear groups arising from Zariski-dense actions on semisimple algebraic groups, with constants depending only on the ambient dimension. It then translates this gap into strong, uniform anti-concentration results for random walks on linear groups and a non-abelian Littlewood–Offord inequality, together with a logarithmic bound for escaping subvarieties. The core methodology blends a ping-pong with overlaps technique, adelic heights and the Height Gap theorem, and local-field dynamics to achieve global, uniform control across all fields and characteristics. A forthcoming sequel promises to leverage these results to uniform expansion in finite simple groups of bounded rank, via Bourgain–Gamburd-type machinery.

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 over almost all primes .

Paper Structure

This paper contains 37 sections, 34 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Uniform gap for quasi-regular representations
  3. Anti-concentration estimates for random walks
  4. Non-commutative Littlewood-Offord estimates
  5. Escape from subvarieties
  6. Some further remarks.
  7. Weak containment
  8. The spectral gap property and invariant means
  9. Proving the spectral gap property without uniformity
  10. A reformulation of Theorem \ref{['main1bis']}
  11. Other sufficient geometric conditions on the isotropy group
  12. Anti-concentration on other sets and neighborhoods
  13. One-percent versus ninety-nine percents
  14. An overview of the proof of Theorem \ref{['main1bis']}.
  15. Organisation of the paper.
  16. ...and 22 more sections

Key Result

Theorem 1.1

There is $\varepsilon_d>0$ depending on $d$ only, such that the following holds. Let $K$ be a field and $\mathbb{G}\leq \mathop{\mathrm{GL}}\nolimits_d$ a connected semisimple algebraic $K$-group. Suppose $H\leq \Gamma$ are countable subgroups of $\mathbb{G}(K)$, with $H$ not Zariski-dense in $\math provided $S$ is finite and generates a Zariski-dense subgroup of $\mathbb{G}$.

Theorems & Definitions (65)

  • Theorem 1.1: uniform spectral gap for quasi-regular representations
  • Corollary 1.2
  • Theorem 1.3: anti-concentration on algebraic subvarieties
  • Theorem 1.4: non-abelian Littlewood--Offord inequality
  • Corollary 1.5: Logarithmic escape
  • Lemma 2.1: ping-pong with overlaps
  • proof : Proof of Lemma \ref{['ping-pong with overlaps']}
  • Lemma 2.2
  • proof
  • Lemma 3.1: Escape lemma
  • ...and 55 more