Random curves by conformal welding

Kari Astala; Peter Jones; Antti Kupiainen; Eero Saksman

doi:10.1016/j.crma.2009.12.014

Complex Analysis/Mathematical Physics

Random curves by conformal welding

Presented by: Paul Malliavin

Kari Astala ¹; Peter Jones ²; Antti Kupiainen ¹; Eero Saksman ¹

¹ University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finland
² Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06510, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 257-262.

Abstract
Résumé

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to $SLE (κ)$ for $κ < 4$ .

On construit une famille aléatoire conformément invariante de courbes fermées dans le plan par soudure d'un cercle unité donné en terme d'exponentielle d'un champ libre gaussien. On conjecture que nos courbes sont localement reliées à $SLE (κ)$ pour $κ < 4$ .

Received: 2009-12-11
Published online: 2010-02-24

DOI: 10.1016/j.crma.2009.12.014

Author's affiliations:

Kari Astala ¹; Peter Jones ²; Antti Kupiainen ¹; Eero Saksman ¹

¹ University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finland
² Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06510, USA

@article{CRMATH_2010__348_5-6_257_0,
     author = {Kari Astala and Peter Jones and Antti Kupiainen and Eero Saksman},
     title = {Random curves by conformal welding},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {257--262},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2009.12.014},
     language = {en},
}

TY  - JOUR
AU  - Kari Astala
AU  - Peter Jones
AU  - Antti Kupiainen
AU  - Eero Saksman
TI  - Random curves by conformal welding
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 257
EP  - 262
VL  - 348
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2009.12.014
LA  - en
ID  - CRMATH_2010__348_5-6_257_0
ER  -

%0 Journal Article
%A Kari Astala
%A Peter Jones
%A Antti Kupiainen
%A Eero Saksman
%T Random curves by conformal welding
%J Comptes Rendus. Mathématique
%D 2010
%P 257-262
%V 348
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2009.12.014
%G en
%F CRMATH_2010__348_5-6_257_0

Kari Astala; Peter Jones; Antti Kupiainen; Eero Saksman. Random curves by conformal welding. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 257-262. doi : 10.1016/j.crma.2009.12.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.014/

References
Cited by

[1] H. Airault; P. Malliavin; A. Thalmaier Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows, J. Math. Pures Appl., Volume 83 (2004), pp. 955-1018

[2] K. Astala; T. Iwaniec; G. Martin Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, vol. 47, Princeton University Press, 2009

[3] K. Astala; P. Jones; A. Kupiainen; E. Saksman Random conformal weldings, 2009 | arXiv

[4] E. Bacry; J.F. Muzy Log-infinitely divisible multifractal processes, Comm. Math. Phys., Volume 236 (2003), pp. 449-475

[5] J. Barral; B. Mandelbrot Random multiplicative multifractal measures. I–III, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 17-52

[6] A. Beurling; L.V. Ahlfors The boundary correspondence under quasiconformal mappings, Acta Math., Volume 96 (1956), pp. 125-142

[7] B. Duplantier; S. Sheffield Liouville quantum gravity and KPZ, 2008 | arXiv

[8] B. Duplantier; S. Sheffield Duality and the Knizhnik–Polyakov–Zamolodchikov relation in Liouville quantum gravity, Phys. Rev. Lett., Volume 102 (2009) no. 150603 (4 pp)

[9] Y.V. Fyodorov; J.P. Bouchaud Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential, J. Phys. A: Math. Theor., Volume 41 (2008), p. 372001

[10] P. Jones; S. Smirnov Removability theorems for Sobolev functions and quasiconformal maps, Arkiv för Matematik, Volume 38 (2000), pp. 263-279

[11] J.-P. Kahane Sur le chaos multiplicatif, Ann. Sci. Math. Québec, Volume 9 (1985), pp. 435-444

[12] O. Lehto Homeomorphisms with a given dilatation, Oslo, 1968 (Lecture Notes in Mathematics), Volume vol. 118, Springer (1970), pp. 58-73

[13] T. Reed On the boundary correspondence of quasiconformal mappings of domains bounded by quasicircles, Pacific J. Math., Volume 28 (1969), pp. 653-661

[14] O. Schramm Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., Volume 118 (2000), pp. 221-288

Cited by Sources:

^☆ We thank M. Bauer, D. Bernard, S. Rohde and S. Smirnov for discussions and L. Dubois for help in French.

Comments - Policy