RANDOM-WALKS ON THE LAMPLIGHTER GROUP

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 b ounded harmonic functions on the Cayley graph are constant. Here we fo cus 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 simp le 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 dyna mical environment on Z. The proof uses potential theory to analyze a s tationary environment as seen from the moving particle.