Random conformal weldings

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 βX, where X is the restriction of the 2-dimensional free field on the circle and the parameter β 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 dilatation restricted to dyadic cells of various scales. For the uniqueness we invoke a result by Jones and Smirnov on conformal removability of Hölder curves. Our curves are closely related to SLE(ϰ) for ϰ<4.