Main speaker 2000-2001:
from Chalmers University, Goteborg, Sweden,
The main speaker in spring will be professor
on: Topics in Percolation and Spatial Interaction.
TOPICS IN PERCOLATION AND SPATIAL
In these lectures, I intend to discuss recent progress made in some selected
areas of percolation theory and the stochastic modelling of
spatially interacting systems.
LECTURE 1: Uniqueness (and non-uniqueness) of infinite clusters
One of the most important and celebrated theorems in percolation theory is the
uniqueness of the infinite cluster for i.i.d. percolation on Z^d.
Although this result dates back to the late 1980's, the issue of uniqueness of
infinite clusters is still very much alive as a research
area. For instance, what happens when we move to various types of dependent
models, or to other lattices? And what happens if we focus
not on connected components, but instead on entangled components, or on rigid
components? Some answers to these questions will be given.
LECTURE 2: The phase transition behavior of the Widom-Rowlinson model
The Widom-Rowlinson lattice gas model is a Gibbs system where each node can be
in state -1, 0 or +1, and the interaction is a hard-core repulsion
between -1's and +1's. This model admits, to some extent, a
percolation-theoretic analysis similar to the (Fortuin-Kasteleyn) random-cluster
analysis of Ising and Potts models. However, there are subtleties in the
Widom-Rowlinson case that make certain monotonicity issues less tractable. For
instance, it is still not known whether the phase transition in the
Widom-Rowlinson model on Z^d is monotonic, in the sense that nonuniqueness of
Gibbs measures at some value of the activity parameter implies the same thing at
higher values. This lecture will (after reviewing some basics) center around
this open problem, and report on some side-results obtained in the search for a
LECTURE 3: Models for spatial growth and competition
Consider a stochastic particle system on Z^d where the sites are in state 0
(healthy) or 1 (infected), and infected sites infect each neighbor at unit rate.
There is no recovery, and initially there is just one infected site. This is the
so-called Richardson model, which can also be formulated as a first-passage
percolation problem with exponential passage times on the edges. The central
result says, roughly, that the set of infected sites grows linearly in each
direction, and has an asymptotic shape. The shape, however, is not known. I will
discuss this model and also some more recent generalizations, to continuous
space and to models with several types of infection that compete against each
other for space.