ABSTRACTS:
Christina Goldschmidt, Cambridge:
Coagulationfragmentation duality, PoissonDirichlet distributions
and random recursive trees
We give a new example of duality between certain coagulation
and fragmentation operators. More specifically, if we start with
a random variable with PoissonDirichlet PD(alpha, theta) distribution
then, after application of our fragmentation operator, we obtain
a random variable with PD(alpha, theta + 1) distribution. The
coagulation operator goes back the other way. These relations
provide a counterpart to Pitman's relations between PD(alpha,
theta) and PD(alpha x beta, theta). Repeated application of our
fragmentation operator gives rise to a Markov fragmentation chain,
which can be encoded naturally by certain random recursive trees.
[Based on joint work with Rui Dong (Berkeley) and James Martin
(Oxford).]
Russell Lyons, Indiana:
Unimodularity and Stochastic Processes
Stochastic processes on vertextransitive graphs, especially
Cayley graphs of groups, have been studied for 50 years (not counting
the special case of integer lattices, which goes back hundreds
of years). The assumption of invariance under graph automorphisms
plays a key role, but investigations of the last 15 years have
shown that an additional assumption is also extremely useful.
This newer assumption is the property of unimodularity, which
is equivalen to the MassTransport Principle. We shall review
some wellknown applications and also discuss recent work with
David Aldous. This includes three theorems on RWRE.
Robin Pemantle, Dept. of Math,
Univ. of Pennsylvania:
Multivariate generating function techniques and an application
to quantum random walks
A number of problems in combinatorics and probability may be
encoded into a generating function, F, and limit theorems extracted
analytically. The extraction depends on details of the function
F. I will discuss the case where F = G/H is a rational 3variable
function
and H(1+x,1+y,1+z) is a conic.
Why should you care about this case? It turns out that this encompasses
many cases of the "Arctic Circle" phenomena:i in tiling
examples, randomness is confined to a linearly growing disk; the
location of a quantum random walk is uniformly spread over such
a disk.
This work is in progress and is joint with Yuliy Baryshnikov.
Yuval Peres, Berkeley:
The Unreasonable Effectiveness of Martingales
Martingales are often used by combinatorialists for their concentration
properties. The goal of this survey talk is to illustrate the
usefullness of optional stopping arguments for natural problems
on critical random graphs, random walks, and embeddings of graphs
in Euclidean space. In particular the same martingale lemma can
be used to prove
(1) the twothirds power law for the largest component of the
critical random graph G(n,1/n),
(2) the same law for critical percolation on a random 3regular
graph, and
(3) the 1/2 power law bound for rate of decay of the transition
probabilities p^k(x,y) of random walk on any graph of bounded
degree. (Talk based on joint works with Asaf Nachmias and with
Ben Morris)
For further information please contact gensci@fields.utoronto.ca
