February 7-2003

11:15-13:00: Akira Sakai Eurandom, The Netherlands
Critical behavior for the contact process.

14:15-16:00: Kurt Johansson from the KTH, Stockholm, Sweden.

Talk 1: Probability measures from random matrix theory

Abstracts:

The contact process models an infection in a society in which every individual
does not move around, such as an orchard. It is known that this model exhibits
a phase transition between the healthy phase (the process almost surely dies out
in a finite time) and the infection phase (the process continues forever with
positive probability), as the infection rate crosses its critical point.

We are interested in the relations among critical exponents that represent
singular behavior of observables around the critical point. We prove that

1) if the spatial dimension is above four and the infection range is finite
but widely spread out, then the critical exponents exist and take on the values
of the corresponding critical exponents for the branching random walk;

2) if we assume existence of the critical exponents, then they satisfy
so-called hyperscaling inequalities that imply existence of at least one
non-branching random walk exponent when the spatial dimension is less than four.

The canonical measure of super-Brownian motion is defined by the scaling limit
of the critical branching random walk emerging from one individual and
conditioned to survive for some time. It was recently proved by van der Hofstad
and Slade that, for every positive integer n, the scaling limit of the n+1 point
connectivity function for spread-out oriented percolation (discrete-time contact
process) converges the n-th moment measure of the canonical measure of
super-Brownian motion, if the spatial dimension is above four. We will also
discuss the same results for the spread-out contact process and difficulty
arising in the continuous-time model. This is ongoing work with van der Hofstad.

Kurt Johansson
Talk 1: Probability measures from random matrix theory

Talk 2: Random growth and random matrices
Talk 3: Random permutations and random tilings

Abstract: The eigenvalue measures from random matrix theory give
rise to probability measures which arise not only within random
matrix theory itself but also in other contexts. I will review
the basic facts from random matrix theory and discuss in
particular the occurrence of the largest eigenvalue distribution
in last-passage percolation, random permutations, certain two-dimensional
growth models and in random tilings.


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