Rate of Escape on the Lamplighter Tree

Details Rate of Escape on the Lamplighter Tree Rate of Escape on the Lamplighter Tree

2007-08-28

arxiv.org/abs/0708.3766v1

Mathematics

Probability

16 pages; accepted for publication in Journal of Mathematical Sciences (N.Y.)

Scientific paper

Suppose we are given a homogeneous tree $\mathcal{T}_q$ of degree $q\geq 3$, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product $(\mathbb{Z}/2)\wr \Gamma$, where $\Gamma=\ast_{i=1}^q \mathbb{Z}/2$ is the free product group of $q$ factors $\mathbb{Z}/2$. We consider a transient random walk on a Cayley graph of $(\mathbb{Z}/2)\wr \Gamma$, for which we want to compute lower and upper bounds for the rate of escape, that is, the speed at which the random walk flees to infinity.

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