ansobol's Werner:W [4 articles]

Abstract

This is the preliminary version of the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007. ...

Abstract

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees. ...

Abstract

We show that there exists (up to multiplicative constants) a unique and natural measure on simple loops on Riemann surfaces, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surface S' that is contained in another Riemann surface S, is just the measure on S restricted to those loops that stay in S'). We then study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops. ...