Tuesday, April 1, 2014
13th Annual Thomas Wolff Memorial Lectures in Mathematics
Random Walks and their Scaling Limits
Gregory F. Lawler, Professor of Mathematics, Mathematics, University of Chicago
April 1, 2014
The simple random walk on the integer lattice is well understood as well as its scaling limit, Brownian motion. However, there are a number of models of random walks with strong interactions for which we are still trying to determine the behavior. My first talk will be a survey of the state of knowledge for three problems: intersections of random walk, loop-erased random walk, and self-avoiding random walk.
April 3, 2014
There has been considerable work in the last twenty years on the planar case and I will spend the second two lectures discussing aspect of this. In the the second lecture, I will consider a particular case, the loop-erased random walk, and describe its relationship to other models in particular spanning trees, determinants of the Laplacian and the random walk loop measures. If time allows. I will discuss a recent result with C. Benes and F. Viklund about the probability that a loop-erased walk goes though a point.
April 7, 2014
The third talk will focus on the continuum limit of many of these walks, the Schramm-Loewner evolution (SLE). Many of the properties SLE can be seen as continuous analogues of properties of the loop-erased walk and I will discuss some of them including recent work on fractal properties with a number of coauthors.