Random Conformal Weldings at criticality

TL;DR

This paper constructs a family of random Jordan curves in the plane via conformal welding using a boundary homeomorphism derived from a critical Gaussian Free Field, extending previous results to the critical case.

Contribution

It introduces a novel construction of random Jordan curves at criticality using Gaussian Free Field-based measures, extending existing theorems to this critical regime.

Findings

Constructed random Jordan curves via conformal welding at criticality.

Represented Gaussian Free Field using vaguelets, offering a new analytical tool.

Extended prior theorems to the critical case of Gaussian Free Field.

Abstract

We construct a family of random Jordan curves in the plane by welding together two disks on their boundaries using a random homeomorphism. This homeomorphism arises from a random measure whose density, in a generalized sense, is the exponentiated Gaussian Free Field at criticality. We also introduce a representation of the Gaussian Free Field in terms of vaguelets, which may be of separate interest. The result extends a theorem of Astala, Jones, Kupiainen and Saksman (\cite{AJKS09}) to criticality.