Acta Mathematica

Random conformal weldings

Kari Astala, Antti Kupiainen, Eero Saksman, and Peter Jones

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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.

Article information

Source
Acta Math., Volume 207, Number 2 (2011), 203-254.

Dates
Received: 8 September 2009
Revised: 3 August 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892580

Digital Object Identifier
doi:10.1007/s11511-012-0069-3

Mathematical Reviews number (MathSciNet)
MR2892610

Zentralblatt MATH identifier
1253.30032

Rights
2011 © Institut Mittag-Leffler

Citation

Astala, Kari; Kupiainen, Antti; Saksman, Eero; Jones, Peter. Random conformal weldings. Acta Math. 207 (2011), no. 2, 203--254. doi:10.1007/s11511-012-0069-3. https://projecteuclid.org/euclid.acta/1485892580


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