MARK KAC SEMINAR

introduction	talks	archive	contact	location

Main speaker 2000-2001:

Wendelin Werner:  Critical exponents, conformal invariance and planar Brownian motion

Understanding and proving mathematically the striking predictions made by theoretical physics (Nienhuis, Cardy, Duplantier etc.) concerning the existence and value of critical exponents for some two-dimensional systems in statistical physics (such as self-avoiding walks, critical percoaltion, intersections of simple random walk) is a tantalising challenge for mathematicians that seems to be related to several braches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras).

The aim of these three lectures is to give an introduction to the subject and to present some mathematical progress obtained in recent joint work with Greg Lawler and Oded Schramm. In particular, we will describe the construction and properties of a conformally invariant random curve discovered by Schramm that turns out to be the only possible conformally invariant scaling limit of critical percolation cluster interfaces. For this random curve, one can recover most conjectures made by physicists concerning critical percolation and self-avoiding walks, and obtain a complete mathematical proof of the conjectures for simple random walks and Brownian motions.

We shall also try to explain why the geometry of critical percolation interfaces in their scaling limit should be identical to that of planar Brownian motions.

Other related papers and preprints can also be downloaded from Werners website