Title: Convergence of percolation interface to SLE(6)
Abstract:
Schramm-Loewner evolutions (SLE) are random curves in the plane satisfying conformal invariance and a certain Markov property. Such random curves arise naturally as interfaces in the 2D lattice models of statistical physics. The theory of SLEs has been a very active field in mathematics over the past decade. In this talk I'll illustrate the convergence of a discrete random curve to one of the SLEs by giving details in a concrete example: the percolation model could be seen as a model for a porous media, say porous rock. In two dimensions the interface, which the external boundary of this piece of matter, is a random curve. It is desired to understand the scaling limit of this random curve as the size of microscopic scale is taken to zero. It turns out that at the criticality the scaling limit will be a SLE curve with variance parameter 6. The talk will include the following things:
* definition of percolation
* Russo-Seymour-Welsh theory
* holomorphic observables and Cardy-Smirnov formula
* short review of SLE
* precompactness methods for random curves
* uniqueness of the scaling limit using holomorphic observables