Greg Lawler

Abstract:   It has been predicted by theoretical physicists that the continuum limit of a number of two dimensional models from lattice statistical physics "at criticality" are in some sense conformally invariant. Among these models are: nonintersecting random walks, self-avoiding random walks, percolation, loop-erased random walks, uniform spanning trees, Ising model, Potts models. With this assumption, they used techniques from conformal field theory to give (nonrigorous) predictions for critical exponents.

Recently, a number of these predictions have been proved and there is much active research in doing more. The main new ingredient is the stochastic Loewner evolution (SLE) first introduced by Oded Schramm a few years ago. SLE gives an important one parameter family of conformally invariant measures. There is another important family that we call restriction measures, which are related to SLE but are not the same. These tools allow the continuum limit of many models to be understood and in some cases we can rigorously show that that lattice models approach these limits. We now can understand limits in terms of measures on paths and clusters rather than just in terms of the "fields" that these measures give.

I will give a survey of work in this area from the last couple of years. The majority of the talk will deal with research I have done in collaboration with Oded Schramm (Microsoft) and Wendelin Werner (Orsay).