The Annals of Probability

Random walks and isoperimetric profiles under moment conditions

Laurent Saloff-Coste and Tianyi Zheng

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Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $\vert \cdot\vert $. We study the behavior of the probability of return for random walks driven by symmetric measures $\mu$ that are such that $\sum\rho(\vert x\vert )\mu(x)<\infty$ for increasing regularly varying or slowly varying functions $\rho$, for instance, $s\mapsto(1+s)^{\alpha}$, $\alpha\in(0,2]$, or $s\mapsto(1+\log(1+s))^{\varepsilon}$, $\varepsilon>0$. For this purpose, we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp $L^{2}$-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.

Article information

Ann. Probab., Volume 44, Number 6 (2016), 4133-4183.

Received: January 2015
Revised: October 2015
First available in Project Euclid: 14 November 2016

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Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Random walk isoperimetric profile wreath product moment


Saloff-Coste, Laurent; Zheng, Tianyi. Random walks and isoperimetric profiles under moment conditions. Ann. Probab. 44 (2016), no. 6, 4133--4183. doi:10.1214/15-AOP1070.

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