It is conjectured that many statistical physics models on planar lattices become invariant under rotations and even conformal maps in the scaling limit (i.e. when ``viewed from far away''). These include critical percolation, Ising model at critical temperature, self-avoiding polymers, FK and O(N) models. A classical example is the random walk (invariant only under rotations preserving the lattice) which in the scaling limit converges to conformally invariant Brownian motion.

Assuming the conformal invariance conjecture, physicists were able to make a number of striking but unrigorous predictions: e.g. dimension of a critical percolation cluster is almost surely 91/48; the number of simple length k trajectories of a Random Walk is about k^{11/32}·&mu^k, with &mu depending on the lattice, and so on.

We will discuss the recent progress in mathematical understanding of this area, based on combining ideas from probability, complex analysis, combinatorics.

Much of the course will be based on recent works of Rick Kenyon, Greg Lawler, Oded Schramm, and Wendelin Werner. Though some basic knowledge of probability or complex analysis is desirable, it is not necessary.

Zeit:       tba
Ort:        HG G 43 (Hermann-Weyl-Zimmer)
Beginn:   tba
 

M. Struwe