Random Curves by Conformal Welding

TL;DR

This paper constructs a conformally invariant family of random closed curves in the plane using Gaussian Free Field-based homeomorphisms, conjecturing their relation to SLE curves for certain parameters.

Contribution

It introduces a new method of generating random conformally invariant curves via welding of Gaussian Free Field-based homeomorphisms, linking to SLE theory.

Findings

Constructed a conformally invariant family of random curves.

Proposed a conjectural relation to SLE$()$ curves.

Established a novel connection between Gaussian Free Field and conformal welding.

Abstract

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.