January 31, 2015

Fall 2005 Renormalization and Universality in Mathematics and Mathematical Physics

Coxeter Lecture Series

September 13-15, 2005 --3:30 p.m.
Oded Schramm (Microsoft)
Scaling limits of two dimensional random systems

To be held at the Fields Institute

The simple random walk on the square grid in the plane converges to Brownian motion under appropriate scaling. Planar Brownian motion is rotationally invariant, while the random walk is not. In fact, Brownian motion enjoys conformal invariance, which is far richer. Many other random systems, such as critical percolation and the critical Ising model of magnetism, also seem to exhibit conformal invariance in the scaling limit. This conformal invariance implies that certain paths arising naturally from these processes fall into a particular one-parameter family of random fractal paths called Stochastic Loewner Evolution, or SLE.

Audio and slides of the talks are available here.

The first talk will be a survey talk about SLE, its definition and properties, and the random processes converging to SLE.

The second talk will demonstrate how the SLE framework facilitates the calculation of the large-scale properties of these processes.

The last talk will be about the Gaussian free field and its level sets, which are SLE with parameter 4.

The Fields Institute Coxeter Lecture Series (CLS) brings a leading mathematician to the Institute to give a series of three lectures in the field of the current thematic program. The first talk is an overview for a general mathematical audience, postdoctoral fellows and graduate students. The other two talks are chosen, in collaboration with the organizers of the thematic program, to target specialists in the field.

Index of Fields Distinguished and Coxeter Lectures.