Acceleration of Lamplighter Random Walks

Details Acceleration of Lamplighter Random Walks Acceleration of Lamplighter Random Walks

2007-08-28

arxiv.org/abs/0708.3767v2

Mathematics

Probability

20 pages, accepted for publication in Markov Processes and Related Fields

Scientific paper

Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(\mathbb{Z}/ 2\mathbb{Z})\wr G$, such that its projection on $G$ is transient and has finite first moment. This random walk can be interpreted as a lamplighter random walk on $G$. Our aim is to show that the random walk on the wreath product escapes to infinity with respect to a suitable (pseudo-)metric faster than its projection onto $G$. We also address the case where the pseudo-metric is the length of a shortest ``travelling salesman tour''. In this context, and excluding some degenerate cases if $G=\mathbb{Z}$, the linear rate of escape is strictly bigger than the rate of escape of the lamplighter random walk's projection on $G$.

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