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November 2016 Random walks and isoperimetric profiles under moment conditions
Laurent Saloff-Coste, Tianyi Zheng
Ann. Probab. 44(6): 4133-4183 (November 2016). DOI: 10.1214/15-AOP1070

Abstract

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.

Citation

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Laurent Saloff-Coste. Tianyi Zheng. "Random walks and isoperimetric profiles under moment conditions." Ann. Probab. 44 (6) 4133 - 4183, November 2016. https://doi.org/10.1214/15-AOP1070

Information

Received: 1 January 2015; Revised: 1 October 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1378.60019
MathSciNet: MR3572333
Digital Object Identifier: 10.1214/15-AOP1070

Subjects:
Primary: 20F65, 60B05, 60J10

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • November 2016
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