Continuity of the drift for random walks on lamplighters
Given a discrete group $\Gamma$ and a random walk on it, one of the primary objectives is to study the asymptotic objects/quantities related to the walk, like the asymptotic drift/entropy, Poisson boundary, harmonic measure and its dimension for appropriate models for the boundary and so on. In particular, it is natural to ask how do these objects depend on the choice of the random walk, or, equivalently, the probability measure $\mu$ on $\Gamma$. Questions of this sort can be very far from trivial! In my talk I will present some methods which allow us to show continuity of the asymptotic drift for finitely supported random walks on lamplighter groups on trees, using a variety of tools from probability, metric geometry and dynamics. If time permits, I will mention possible ways to strengthen and generalize the results. Joint work (in progress) with Eduardo Silva.