# Random Conformal Weldings

## TL;DR

This paper constructs a conformally invariant random family of closed curves in the plane using welding of random homeomorphisms derived from the exponential of a free field, employing probabilistic estimates and Beltrami equations.

## Contribution

It introduces a novel method to generate conformally invariant random curves via welding of homeomorphisms based on free fields, extending the understanding of random conformal geometry.

## Key findings

- Constructed a new family of random conformal curves.
- Proved existence using Lehto's method and probabilistic estimates.
- Suggested a connection to SLE curves for future research.

## Abstract

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of $\beta X$ where $X$ is the restriction of the two dimensional free field on the circle and the parameter $\beta$ is in the "high temperature" regime $\beta<\sqrt 2$. The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilation restricted to dyadic cells of various scales. For uniqueness we invoke a result by Jones and Smirnov on conformal removability of H\"older curves. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.