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2015 Non-Liouville groups with return probability exponent at most 1/2
Michał Kotowski, Bálint Virág
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Electron. Commun. Probab. 20: 1-12 (2015). DOI: 10.1214/ECP.v20-3774


We construct a finitely generated group $G$ without the Liouville property such that the return probability of a random walk satisfies $p_{2n}(e,e) \gtrsim e^{-n^{1/2+ o(1)}}$. This shows that the constant $1/2$ in a recent theorem by Saloff-Coste and Zheng, saying that return probability exponent less than $1/2$ implies the Liouville property, cannot be improved. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs.


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Michał Kotowski. Bálint Virág. "Non-Liouville groups with return probability exponent at most 1/2." Electron. Commun. Probab. 20 1 - 12, 2015.


Accepted: 14 February 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1318.60007
MathSciNet: MR3314647
Digital Object Identifier: 10.1214/ECP.v20-3774

Primary: 60B15
Secondary: 20F65

Keywords: permutational wreath products , Random walks , return probabilities

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