On drift and entropy growth for random walks on groups

Random Walks on Finite Groups

Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as t...

“ Isoperimetry on Solvable Groups and Random Walks ”

We exhibit a large class of locally compact amenable groups having the same isoperimetric profile and the same probability of return for random walks. This class of groups contains closed subgroups of triangular matrices over local fields and their quotients, but some of them are not linear. Abstract : In this talk, we introduce a variant of the isoperimetric profile on groups that we compute o...

Asymptotic Entropy and Green Speed for Random Walks on Countable Groups

We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjami...

Random walks on finite rank solvable groups

Abstract. We establish the lower bound p2t(e, e) exp(−t1/3), for the large times asymptotic behaviours of the probabilities p2t(e, e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r , such that any of its finitely generated subgroup admit...

An introduction to random walks on groups