# A continuum of exponents for the rate of escape of random walks on groups.

A central question in the theory of random walks on groups is how symmetries

of the underlying space gives rise to structure and rigidity of the random

walks. For example, for nilpotent groups, it is known that random walks have

diffusive behavior, namely that the rate of escape, defined

as the expected distance of the walk from the identity satisfies

E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn|

~= n. (~= meaning upto (multiplicative) constants )-

In this work, for every 3/4 <= \beta< 1 we construct a finitely generated

group so that the expected distance of the simple random walk from its

starting point after n steps is n^\beta (up to constants). This answers a

question of Vershik, Naor and Peres. In other examples, the speed exponent can

fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1 , and were

based on lamplighter (wreath product) constructions.

(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups)

In this lecture we will describe how a variation of the lamplighter

construction, namely the permutational wreath product, can be used to get

precise bounds on the rate of escape in terms of return probabilities of the

random walk on some Schreier graphs. We will then show how groups of

automorphisms of rooted trees, related to automata groups , can then be

constructed and analyzed to get the desired rate of escape. This is joint

work with Balint Virag of the University of Toronto.

No previous knowledge of randopm walks, automaton groups or wreath products is

assumed.

- Last modified: 26/04/2012