Random walk on a group

Summary

A random walk on a group $G$ starts at the identity and multiplies by independent random elements of $G$. All these elements are drawn from one probability measure $\mu$. It is a Markov chain on $G$ whose transition probabilities are invariant under the group. When $\mu$ is uniform on a symmetric generating set, it is the simple random walk on the Cayley graph. The long-run behaviour of the walk reflects the algebra and geometry of the group. For example, let $G$ be finitely generated and let $\mu$ be symmetric with support generating $G$. By a theorem of Kesten, the return probabilities then decay exponentially exactly when $G$ is not amenable.