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.
Citation
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
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