Jonathan Tsai, CUHK, Hong Kong

Conformally Invariant Random Curves on Riemann Surfaces Satisfying the Restriction Property

 

Abstract

Let  be a family of random curves such that for each planar domain D with a and b on the boundary,  is a random curve in D from a to b. We say that the family of curves S satisfies the restriction property if for any compact set K in D avoiding a and b,  conditioned to avoid K is just . Lawler, Schramm and Werner discovered a way of constructing the family of curves S when D is a simply-connected domain. In particular, it is Stochastic Loewner evolution with parameter 8/3.

In this talk, we will outline an extension of this construction to non-simply connected finite Riemann surfaces with boundary. From this, we can deduce some asymptotic properties of the family of curves S.