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November 2013 Behaviors of entropy on finitely generated groups
Jérémie Brieussel
Ann. Probab. 41(6): 4116-4161 (November 2013). DOI: 10.1214/12-AOP761

Abstract

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq\alpha\leq\beta\leq1$, there is a group $\Gamma $ with measure $\mu $ equidistributed on a finite generating set such that

\[\liminf\frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\alpha ,\qquad\limsup \frac{\log H_{\Gamma ,\mu }(n)}{\log n}=\beta .\]

The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_{n}$ on such groups are also evaluated, providing an example of group with return probability satisfying

\[\liminf\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=\frac{1}{3},\qquad\limsup\frac{{\log}|{\log P}(Y_{n}=_{\Gamma }1)|}{\log n}=1\]

and drift satisfying

\[\liminf\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=\frac{1}{2},\qquad\limsup\frac{\log{\mathbb{E}}\|Y_{n}\|}{\log n}=1.\]

Citation

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Jérémie Brieussel. "Behaviors of entropy on finitely generated groups." Ann. Probab. 41 (6) 4116 - 4161, November 2013. https://doi.org/10.1214/12-AOP761

Information

Published: November 2013
First available in Project Euclid: 20 November 2013

zbMATH: 1280.05123
MathSciNet: MR3161471
Digital Object Identifier: 10.1214/12-AOP761

Subjects:
Primary: 05C81 , 20E08 , 60B15

Keywords: automorphisms of rooted tree , Entropy , Random walk on groups , return probability

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • November 2013
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