April 12 -  2002:

Abstracts:

Olle Häggström

TOPICS IN PERCOLATION AND SPATIAL INTERACTION

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.
 

Bernard Nienhuis

Exact correlations in 2D critical percolation and in a 1D sandpile model

Percolation has been studied by physicists and mathematicians for many decades, and in many forms. It describes a phase transition in porous matter between macroscopically permeable (or conducting) and impermeable (or isolating) phases.
The archetypical model is simply a completely uncorrelated mixture of permeable and impermeable elements. In physical systems these elements are pores or grains, and in models they are typically lattice sites (site percolation) or edges (bond percolation).

In this lecture I will consider the bond percolation model on a square lattice, on an L by infinity strip with a variety of boundary conditions, periodic and otherwise. When the strip is cut into two half-infinite strips, any two sites on the cut may or may not be connected to each other via the pores in one of the half strips. These connections collectively for all sites on the cut may be
called a connectivity configuration (CC). Each of these CC occurs with a characteristic probability or weight. For instance in the two
most probable CC all sites are mutually connected, or none of them are.

It turns out that these probabilities have surprising properties. For instance, each one is an integer multiple of the weight of the
least probable CC. A very detailed but ill-understood connection has emerged of these probabilities with the number of Alternating Sign Matrices, a problem in combinatorics which has received much attention recently. In the lecture this connection will be presented and many other observations on the critical percolation probabilities will be given. The results are all in the form of conjectures, as none of these observations have been proved to date.

The critical state of the two dimensional critical percolation will be shown to be equivalent to the time evolution of a stochastic process. This process is very similar to sandpile models that describe the distribution of avalanches in flowing sand.



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