Acta Mathematica

Random conformal weldings

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

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2011 © Institut Mittag-Leffler


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.

Export citation


  • Adler RJ, Taylor JE (2007) Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York
  • Airault H, Malliavin P, Thalmaier A (2004) 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., 83, 955–1018.
  • — Brownian measures on Jordan–Virasoro curves associated to theWeil–Petersson metric. J. Funct. Anal., 259 (2010), 3037–3079.
  • Anderson GD, Vamanamurthy MK, Vuorinen MK (1997) Conformal Invariants, Inequalities, and Quasiconformal Maps. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York
  • Astala, K., Iwaniec, T. & Martin, G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009.
  • Bacry, E. & Muzy, J. F., Log-infinitely divisible multifractal processes. Comm. Math. Phys., 236 (2003), 449–475.
  • Barral, J., Techniques for the study of infinite products of independent random functions (Random multiplicative multifractal measures III), in Fractal Geometry and Applications: a Jubilee of Benoˆıt Mandelbrot, Proc. Sympos. Pure Math., 72, Part 2, pp. 53–90. Amer. Math. Soc., Providence, RI, 2004.
  • Barral, J. & Mandelbrot, B. B., Introduction to infinite products of independent random functions (Random multiplicative multifractal measures I), in Fractal Geometry and Applications: a Jubilee of Benoˆıt Mandelbrot, Proc. Sympos. Pure Math., 72, Part 2, pp. 3–16. Amer. Math. Soc., Providence, RI, 2004.
  • — Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures II), in Fractal Geometry and Applications: a Jubilee of Benoˆıt Mandelbrot, Proc. Sympos. Pure Math., 72, Part 2, pp. 17–52. Amer. Math. Soc., Providence, RI, 2004.
  • Beurling, A. & Ahlfors, L., The boundary correspondence under quasiconformal mappings. Acta Math., 96 (1956), 125–142.
  • Binder, I. & Smirnov, S., Personal communication, 2010.
  • Bouchaud, J. P. & Fyodorov, Y. V., Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A, 41 (2008), 372001, 12 pp.
  • Cardy, J., Scaling and Renormalization in Statistical Physics. Cambridge Lecture Notes in Physics, 5. Cambridge University Press, Cambridge, 1996.
  • Carpentier, D. & Le Doussal, P., Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models. Phys. Rev. E, 63 (2001), 026110, 33 pp.
  • Duplantier, B. & Sheffield, S., Liouville quantum gravity and KPZ. Invent. Math., 185 (2011), 333–393.
  • Fyodorov, Y. V., Le Doussal, P. & Rosso, A., Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields. J. Stat. Mech. Theory Exp., 2009 (2009), P10005, 32 pp.
  • Jerison, D. S. & Kenig, C.E., Hardy spaces, A1, and singular integrals on chord-arc domains. Math. Scand., 50 (1982), 221–247.
  • Jones, P.W. & Smirnov, S. K., Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat., 38 (2000), 263–279.
  • Kahane, J.-P., Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985.
  • — Sur le chaos multiplicatif. Ann. Sci. Math. Qu´ebec, 9 (1985), 105–150.
  • — Positive martingales and random measures. Chinese Ann. Math. Ser. B, 8 (1987), 1–12.
  • Kahane, J.-P. & Peyrière, J., Sur certaines martingales de Benoit Mandelbrot. Adv. Math., 22 (1976), 131–145.
  • Lehto, O., Homeomorphisms with a given dilatation, in Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968), Lecture Notes in Mathematics, 118, pp. 58–73. Springer, Berlin–Heidelberg, 1970.
  • Mandelbrot, B., Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech., 62 (1974), 331–358.
  • Molchan, G. M., Scaling exponents and multifractal dimensions for independent random cascades. Comm. Math. Phys., 179 (1996), 681–702.
  • Oikawa, K., Welding of polygons and the type of Riemann surfaces. Kōdai Math. Sem. Rep., 13 (1961), 37–52.
  • Reed, T. J., On the boundary correspondence of quasiconformal mappings of domains bounded by quasicircles. Pacific J. Math., 28 (1969), 653–661.
  • Samorodnitsky, G., Probability tails of Gaussian extrema. Stochastic Process. Appl., 38 (1991), 55–84.
  • Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221–288.
  • — Conformally invariant scaling limits: an overview and a collection of problems, in International Congress of Mathematicians (Madrid, 2006). Vol. I, pp. 513–543. Eur. Math. Soc., Zurich, 2007.
  • Sheffield, S., Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Preprint, 2010. arXiv:1012.4797 [math.PR].
  • Smirnov, S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math., 172 (2010), 1435–1467.
  • Talagrand, M., Sharper bounds for Gaussian and empirical processes. Ann. Probab., 22 (1994), 28–76.
  • Vainio, J. V., Conditions for the possibility of conformal sewing. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 53 (1985), 43 pp.
  • Vuorinen, M., Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, 1319. Springer, Berlin–Heidelberg, 1988.