June 7  2002:

Abstracts:

Rob van den Berg

Box-crossings in two-dimensional percolation

Crossing arguments play an important role in 2-dimensional percolation.

After an introduction and discussion of some classical  results (including the RSW theorem) I will present a result with Antal Jarai about the distance between the lowest crossing and the boundary of the domain, in ordinary percolation at the critical point.

I will also mention some problems, conjectures and partial results concerning crossing probabilities in a {\em dependent} percolation model studied with Rachel Brouwer, and discuss their background and possible implications for a peculiar infinite-volume epidemics (or forest-fire) model.

 

Silke Rolles

Reinforced random walks

Let G be a locally finite graph. All edges are given non-negative numbers as weights. Reinforced random walk on G is defined as follows: In each step, the random walker jumps to a neighboring vertex with a probability proportional to the weight of the traversed
edge. Each time an edge is traversed, its weight is increased by 1.

Although the model may seem somewhat artificial, I will show in the talk that reinforced random walk arises quite naturally: Its distribution can be characterized by a few natural properties. Furthermore, I will present a limit theorem for reinforced random walk on a finite graph. Connections with random walk in random environment will be shown. Finally, I will discuss recurrence questions and open problems.

The talk is based on joint work with Mike Keane. References can be found on my homepage.

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