Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908718102216704 |
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| author | Becker, Oren Breuillard, Emmanuel |
| author_facet | Becker, Oren Breuillard, Emmanuel |
| contents | 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 $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks Becker, Oren Breuillard, Emmanuel Group Theory Combinatorics Operator Algebras Probability 22D40 (Primary) 05C81, 11G50 (Secondary) 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 $p$. |
| title | Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks |
| topic | Group Theory Combinatorics Operator Algebras Probability 22D40 (Primary) 05C81, 11G50 (Secondary) |
| url | https://arxiv.org/abs/2512.15364 |