Jeudi 09 janvier 2003
Polymers are often modeled by self-avoiding walks (SAW) on the integer lattice. While this is a simple model to define, it is a difficult problem to determine what a typical "long" SAW looks like, or, similarly, to determine the continuum limit of SAWs. However, it is widely believed that in two dimensions that the limit should have certain "conformal invariance" properties. I will describe recent work with Oded Schramm (Microsoft) and Wendelin Werner (Orsay) that shows that there is essentially only one possibility for the limit that is consisitent with the conformal invariance hypothesis. This continuous process is a particular case of the stochastic Loewner evolution (SLE) first introduced by Schramm. Critical exponents for SAWs can be interpreted as scaling exponents for SLE, and the values of these exponents for SLE can be computed rigorously. It remains an open mathematical problem to prove that the lattice SAW approaches this limit.
As time allows, I will also describe the relationship between this measure and two other important measures in two-dimensions that are "conformally invariant" --- planar Brownian motion and the scaling limit of critical percolation.