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Abstract : |
Abstract. Kaimanovich and Vershik (1983) 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 G1 by Kaimanovich and Vershik, and show that inward-biased random walks on G1 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 G1. These walks can be viewed as random walks interacting with a dynamical environment on Z. The proof uses potential theory to analyze a stationary environment as seen from the moving particle. x1. Introduction. The study of random walks and harmonic functions on finitely generated groups has a long history. For a random walk supported by a finite generating set, Avez (1974) showed that a necessary condition for the existence of non-constant bounded harmonic functions, |
