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October 1996 Random walks on the lamplighter group
Russell Lyons, Robin Pemantle, Yuval Peres
Ann. Probab. 24(4): 1993-2006 (October 1996). DOI: 10.1214/aop/1041903214

Abstract

Kaimanovich and Vershik described certain finitely generated groups of exponential growth such that simple random walk on their Cayley graph escapes from the identity at a sublinear rate, or equivalently, all bounded harmonic functions on the Cayley graph are constant. Here we focus on a key example, called $G_1$ by Kaimanovich and Vershik, and show that inward-biased random walks on $G_1$ move outward faster than simple random walk. Indeed, they escape from the identity at a linear rate provided that the bias parameter is smaller than the growth rate of $G_1$. These walks can be viewed as random walks interacting with a dynamical environment on $\mathbb{Z}$. The proof uses potential theory to analyze a stationary environment as seen from the moving particle.

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Russell Lyons. Robin Pemantle. Yuval Peres. "Random walks on the lamplighter group." Ann. Probab. 24 (4) 1993 - 2006, October 1996. https://doi.org/10.1214/aop/1041903214

Information

Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0879.60004
MathSciNet: MR1415237
Digital Object Identifier: 10.1214/aop/1041903214

Subjects:
Primary: 60B15
Secondary: 60J15

Keywords: bias , dynamical environment , Rate of escape , Speed

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 4 • October 1996
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