Random conformal welding for finitely connected regions

TL;DR

This paper develops a method for random conformal welding of finitely connected regions in the Riemann sphere using Gaussian free fields, extending previous results to multiply connected domains.

Contribution

It introduces a new construction of random homeomorphisms on boundary components and proves their existence and uniqueness for multiply connected regions.

Findings

Established existence and uniqueness of random conformal welding for finitely connected regions.

Generalized previous results from simply connected to multiply connected domains.

Connected the problem to a non-uniformly elliptic Beltrami equation with random complex dilatation.

Abstract

Given a finitely connected region $\Omega$ of the Riemann sphere whose complement consists of $m$ mutually disjoint closed disks $\bar{U}_j$, the random homeomorphism $h_j$ on the boundary component $\partial U_j$ is constructed using the exponential Gaussian free field. The existence and uniqueness of random conformal welding of $\Omega$ with $h_j$ is established by investigating a non-uniformly elliptic Betrami equation with a random complex dilatation. This generalizes the result of Astala, Jones, Kupiainen and Saksman to multiply connected domains.