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.