Archive 2001 - 2002

Main speaker 2000-2001:

TOPICS IN PERCOLATION AND SPATIAL INTERACTION
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 solution.

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.