Random walks have been objects of intensive study by
mathematicians over many
decades with a well developed classical theory now in place. In
recent years there have been a number of advances in this area which
have spurred on research and have seen exciting progress on some novel
problems. This workshop will cover recent work on random walk across a
broad spectrum of activity but will have some major themes of focus.
The key areas that we wish to focus on are specifically random walks
in random media, random walks on groups and random walks with
self-interaction. Our aim is to ensure that all are aware of new
techniques and the challenges that are arising for our areas of
focus. By bringing together a broad range of random walk experts we
have the potential to provide significant impact on the development of
these rich areas of research.
Our aim is to have speakers from a wide range of topics within the
area of random walk. However we will provide more focus on three
topics in which there has been exciting progress in the last few
Random walks in random environment:
The field of random media has been the object of intensive
mathematical research over the last thirty years. It collects together
a variety of models, mainly originating from physical sciences such as
condensed matter physics, physical chemistry, and geology, where one
is interested in materials which have defects or inhomogeneities. Many
of these features can be modelled by taking a medium given by a
suitable random process. The study of `transport phenomena' (heat
conduction, fluid flow) in random media leads to study of the
properties of the Laplacian and also of random motion or diffusion in
such media. The study of random media is hard, and quite frequently
what was initially thought of as a simple toy model has ended up as a
major mathematical challenge.
Self-interacting random walk:
Self-interacting random walks (SIRW) are random processes evolving in an
environment constantly modified by their own behaviour. The processes
can be self-repelling or self-attracting, in other words more likely
to stay away from or to come back to the places already visited
before. These non-Markovian random walks ``learn'' from their past
behaviour, either localizing on particular subsets or scattering on
the graph, as a consequence of the interaction feature.
Random walks on graphs and groups:
Aside from these two areas of focus we will consider more classical
problems of relevance such as the behaviour of random walks on groups and graphs.
As a result we have invited experts in analysis and probability for
random walks in this setting.