2009 Probability Seminar - 07

The Schramm-Loewner evolution (SLE) is a one-parameter family of random growth processes that has been successfully used to analyze a number of models from two-dimensional statistical mechanics. Currently there is interest in trying to formalize our understanding of conformal field theory using SLE. Smirnov recently showed that the scaling limit of interfaces of the 2d critical Ising model can be described by SLE(3). The primary goal of this talk is to explain how a certain non-local observable of the 2d critical Ising model studied by Arguin and Saint-Aubin can be rigorously described using multiple SLE(3) and Smirnov's result. As an extension of this result, we explain how to compute the probability that a Brownian excursion and an SLE(k) curve, 0<4, do not intersect.