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Kesten’s theorem for uniformly recurrent subgroups

Part of: Graph theory

Published online by Cambridge University Press:  13 March 2019

MIKOLAJ FRACZYK Open the ORCID record for MIKOLAJ FRACZYK [Opens in a new window]
MIKOLAJ FRACZYK*
Affiliation:
Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053, Budapest, Hungary email fraczyk@renyi.hu
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Abstract

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application, we extend Kesten’s theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$.

Keywords

group actionstopological dynamicsrandom walks

MSC classification

Primary: 05C81: Random walks on graphs
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2778 - 2787
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Copyright
© Cambridge University Press, 2019

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