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#### Analysis of PDEsFunctional AnalysisProbabilitymathscidoc:1701.03004

Acta Mathematica, 207, (2), 203-254, 2009.9
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.
@inproceedings{kari2009random,
title={Random conformal weldings},
author={Kari Astala, Antti Kupiainen, Eero Saksman, and Peter Jones},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357287856745},
booktitle={Acta Mathematica},
volume={207},
number={2},
pages={203-254},
year={2009},
}

Kari Astala, Antti Kupiainen, Eero Saksman, and Peter Jones. Random conformal weldings. 2009. Vol. 207. In Acta Mathematica. pp.203-254. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357287856745.