Biased Random Walk on Lamplighter Groups and Graphs

Lyons, Pemantle, and Peres(4) found an example of a group on which a simple random walk has sublinear speed but an inward biased walk has linear speed. We examine a similar example by introducing a martingale that allows us to compute the exact speed of the inward biased walk. We apply the martingale techniques to the example of Lyons et al., and improve on their speed estimates for small amounts of bias. Finally, we examine the speed on a related family of graphs and construct an example on which the speed of the walk has a second phase change.