Cutoff on Ramanujan complexes and classical groups

Michael Chapman; Ori Parzanchevski

doi:10.4171/CMH/537

Cutoff on Ramanujan complexes and classical groups

Michael Chapman, Ori Parzanchevski

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type A^z_d (d 1). As a result, we obtain explicit generators for the finite classical groups PGL_n.F_q/ for which the associated Cayley graphs exhibit total-variation cutoff.

Original language	American English
Pages (from-to)	431-456
Number of pages	26
Journal	Commentarii Mathematici Helvetici
Volume	97
Issue number	3
DOIs	https://doi.org/10.4171/CMH/537
State	Published - 2022

Bibliographical note

Funding Information:
Acknowledgements. The authors are grateful to Ori Gurel-Gurevich for his help with proving Proposition 4.2. They also thank Eyal Lubetzky, Alex Lubotzky and Nati Linial for helpful discussions and encouragement. M.C. was supported by ERC grant 339096 of Nati Linial and by ERC, BSF and NSF grants of Alex Lubotzky. O.P. was supported by ISF grant 2990/21.

Publisher Copyright:
© 2022 European Mathematical Society Publishing House. All rights reserved.

Keywords

Bruhat–Tits buildings
Cutoff
Ramanujan complexes
expanders
random walks

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Cite this

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abstract = "The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type Azd (d 1). As a result, we obtain explicit generators for the finite classical groups PGLn.Fq/ for which the associated Cayley graphs exhibit total-variation cutoff.",

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Cutoff on Ramanujan complexes and classical groups

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