The Annals of Probability

Random walks on the lamplighter group

Russell Lyons, Robin Pemantle, and Yuval Peres

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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.

Article information

Ann. Probab., Volume 24, Number 4 (1996), 1993-2006.

First available in Project Euclid: 6 January 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J15

Bias speed rate of escape dynamical environment


Lyons, Russell; Pemantle, Robin; Peres, Yuval. Random walks on the lamplighter group. Ann. Probab. 24 (1996), no. 4, 1993--2006. doi:10.1214/aop/1041903214.

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