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November 1- 2002:

11:15-13:00: Harry Kesten
        How far apart are the trees in the uniform spanning forest?


14:15-16:00: Antal Jarai
         Incipient infinite clusters in 2D percolation and prospects for high dimensions
 

How far apart are the trees in the uniform spanning forest?
(joint with I. Benjamini, Y. Peres and O. Schramm)

A spanning tree of a connected graph G is a subgraph of G which is a tree and which contains all vertices of G. A uniform spanning tree of a finite G is a random spanning tree of G, chosen uniformly from among all spanning trees of G. Such
spanning trees arise in some computer science problems. They also have close relations to random walks on G via Wilson's algorithm for choosing a uniform spanning tree (we shall explain this algorithm).

In response to a question of Russell Lyons, Pemantle considered the weak limit (as n tends to infinity) of a uniform spanning tree on the cube \{-n, \dots,n\}^d. This limit is a probability measure on subgraphs of Z^d without loops and with vertex set equal to all vertices of Z^d. Pemantle showed that it is connected, i.e., a tree, if and only if the dimension d is less than or equal to 4. The limit is called the uniform spanning forest (USF).

We shall consider the following random variables on the UFS:

N(x,y) := minimum number of edges outside the USF in a path joining x and y 
(where x,y \in Z^d).

It turns out that the maximum over all x and y in Z^d of N(x,y) is almost surely constant and is a curious function of the dimension d. A principal tool in our proof is the concept of the stochastic dimension of a relation on Z^d (i.e., of a subset of Z^d times Z^d), and the behavior of stochastic dimension under the composition of two
relations.

Antal Jarai

Incipient infinite clusters in 2D percolation and prospects for high dimensions

Abstract: This talk will review some results about the `incipient infinite cluster' defined by Kesten (1986). It is widely believed that in all common percolation models there is no infinite cluster at the critical point a.s., and this is a theorem for two- and high dimensional models. The IIC captures the infinite structure that is
emerging, but is not quite present at the critical point. It is obtained by conditioning the cluster of the origin, at criticality, to be connected to the boundary of a large box, and letting the size of the box go to infinity. In the limit an infinite object is obtained, which describes the lattice-scale view (local properties) of large critical clusters. The IIC shows up in some models of self-organized criticality, for example invasion percolation. I will mainly discuss 2D results with some outlook on high-dimensional
analogues which includes joint work with R. van der Hofstad.