Universality and conformal invariance in two dimensional statistical physics

There are a number of two dimensional lattice models in statistical physics (percolation, polymers or self-avoiding walks (SAW), nonintersecting random walks, uniform spanning trees or loop-erased walk (LEW), Ising model) which have been conjectured to have continuum limits at the critical temperature that are in some sense conformally invariant. Exact values of critical exponents have been predicted (nonrigorously) using conformal field theory.

This colloquium will discuss recent work with Oded Schramm and Wendelin Werner on these conjectures. The work focuses on those models which are expected to have a certain ``restriction property'' in the limit (this includes percolation, SAW, and nonintersecting random walks but not LEW or Ising model).

*It is shown that the exponents for all conformally invariant models with the restriction property can be given in terms of the intersection exponents for random walk/Brownian motion.

*Another process, the stochastic Loewner equation (SLE), introduced recently by Schramm, has the restriction property for a particular parameter value.

*The exponents for the SLE can be calculated exactly. By universality, we establish rigorously the exponents for Brownian motion, which in turn proves conjectures about the fractal properties of a Brownian path.

While we cannot prove anything about the continuum limit of percolation and SAW, the results do shed light about what kind of limits should be expected for these processes.