Random conformal weldings

Kari Astala Department of Mathematics and Statistics, University of Helsinki Antti Kupiainen Department of Mathematics and Statistics, University of Helsinki Eero Saksman Department of Mathematics and Statistics, University of Helsinki Peter Jones Department of Mathematics, Yale University

Analysis of PDEs Functional Analysis Probability mathscidoc: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.
No keywords uploaded!
[ Download ] [ 2017-01-08 20:33:57 uploaded by actaadmin ] [ 212 downloads ] [ 0 comments ] [ Cited by 16 ]
@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.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved