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 ${\hat{I}}^{2}X$ where $X$ is the restriction of the two dimensional free field on the circle and the parameter ${\hat{I}}^2$ is in the "high temperature" regime ${\hat{I}}^2<\sqrt{2}$. 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 Hölder curves. We conjecture that our curves are locally related to SLE$(\hat{I}º)$ for $\hat{I}º<4$.

36 pages, 2 figures