Diffusing polygons and SLE(κ,ρ)

Abstract

We give a geometric derivation of $\text{SLE}(\kappa,\rho)$ in terms of conformally invariant random growing compact subsets of polygons. The parameters $\rho_j$ are related to the exterior angles of the polygons. We also show that $\text{SLE}(\kappa,\rho)$ can be generated by a metric Brownian motion, where metric and Brownian motion are coupled and the metric is a pull-back metric of the Euclidean metric of an evolving polygon.