May 3 -  2002:

Abstracts:

Neil O'Connell

A path-transformation for random walks and the Robinson-Schensted correspondence

In [O'Connell and Yor (2002)] a path-transformation $G^{(k)}$
was introduced with the property that, for $X$ belonging to a certain
class of random walks on $\Z_+^k$, the transformed walk $G^{(k)}(X)$
has the same law as that of the original walk conditioned never
to exit the Weyl chamber $\{x:\ x_1\le\cdots\le x_k\}$.  The proof
of this representation theorem is based on symmetry and reversibility
properties of queues in series (or, equivalently, the aysymmetric
exclusion process).  I will recall the main ideas of the proof.

It turns out that $G^{(k)}$ is closely related to the Robinson-Schensted
algorithm, and this connection leads to a new proof of the above
representation theorem.  The new proof is valid for a larger class
of random walks and yields additional information about the joint
law of $X$ and $G^{(k)}(X)$.  The corresponding results for the
Brownian model are recovered by Donsker's theorem.  These are
connected with Hermitian Brownian motion and the Gaussian Unitary
Ensemble of random matrix theory.  The connection we make between
the path-transformation $G^{(k)}$ and the RS algorithm also
provides a new formula and interpretation for the latter.
This can be used to study properties of the RS algorithm and,
moreover, extends easily to a continuous setting.

 

Martin Barlow

Anomalous diffusion, transition probabilities and stability
 

Anomalous diffusion typically arises in media in which obstacles of
all sizes are present. In the case of random walks on graphs, which
is the context of this talk, this means that the mean square
displacement $E(d(x,X_n)^2$ grows more slowly than $n$.

In these talks I will discuss examples of anomalous diffusion,
and give the form the transition probabilities take. I will also
present some recent work with R. Bass on the stability of these
estimates under small perturbations.

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