Random Walks on countable groups

Michael Björklund

Published 2015-02-13Version 1

We begin by giving a short and essentially self-contained proof of the equivalence between the vanishing of the drift of a finitely generated symmetric measured group with finite first moment and the absence of bounded harmonic functions; a result due to Kaimanovich-Vershik and Karlsson-Ledrappier. Given a measured group $(G,\mu)$, we introduce the new notion of weak $(G,\mu)$-mixing and show that the Poisson boundary is weakly $(G,\mu)$-mixing. In particular, this gives a new proof of the fact that the "double" Poisson boundary is weakly mixing in the non-singular sense, which was first observed by Kaimanovich. Finally, we show that (non-singular) weak mixing for ergodic $(G,\mu)$-spaces is equivalent to the absence of a probability measure preserving factor with discrete spectrum.