The conformally invariant measure on self-avoiding loops
Author:
Wendelin Werner
Journal:
J. Amer. Math. Soc. 21 (2008), 137-169
MSC (2000):
Primary 60D05; Secondary 82B41, 82B43, 30C99, 60J65
DOI:
https://doi.org/10.1090/S0894-0347-07-00557-7
Published electronically:
February 20, 2007
MathSciNet review:
2350053
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show that there exists a unique (up to multiplication by constants) and natural measure on simple loops in the plane and on each Riemann surface, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surface $S’$ that is contained in another Riemann surface $S$ is just the measure on $S$ restricted to those loops that stay in $S’$). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.
-
ADA M. Aizenman, B. Duplantier, A. Aharony (1999), Connectivity Exponents and External Perimeter in 2D Independent Percolation Models, Phys. Rev. Lett. 83, 1359.
- Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
- Michel Bauer and Denis Bernard, SLE martingales and the Virasoro algebra, Phys. Lett. B 557 (2003), no. 3-4, 309–316. MR 1972482, DOI https://doi.org/10.1016/S0370-2693%2803%2900189-8 BF R. Bauer, R. Friedrich (2005), On chordal and bilateral SLE in multiply connected domains, preprint. Bephd V. Beffara (2000), Mouvement Brownien plan, SLE, invariance conforme et dimensions fractales, Thèse de Doctorat, Université Paris-Sud.
- Vincent Beffara, Hausdorff dimensions for $\rm SLE_6$, Ann. Probab. 32 (2004), no. 3B, 2606–2629. MR 2078552, DOI https://doi.org/10.1214/009117904000000072
- Krzysztof Burdzy, Cut points on Brownian paths, Ann. Probab. 17 (1989), no. 3, 1012–1036. MR 1009442
- Krzysztof Burdzy and Gregory F. Lawler, Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal, Ann. Probab. 18 (1990), no. 3, 981–1009. MR 1062056 CN F. Camia, C. Newman (2005), The Full Scaling Limit of Two-Dimensional Critical Percolation, preprint. Ca J.L. Cardy (1984), Conformal invariance and surface critical behavior, Nucl. Phys. B 240, 514-532. F R. Friedrich, On connections of Conformal Field Theory and Stochastic Loewner Evolutions, math-ph/0410029.
- R. Friedrich and J. Kalkkinen, On conformal field theory and stochastic Loewner evolution, Nuclear Phys. B 687 (2004), no. 3, 279–302. MR 2059141, DOI https://doi.org/10.1016/j.nuclphysb.2004.03.025
- Roland Friedrich and Wendelin Werner, Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys. 243 (2003), no. 1, 105–122. MR 2020222, DOI https://doi.org/10.1007/s00220-003-0956-8 GT C. Garban, J.A. Trujillo-Ferreras (2005), The expected area of the Brownian loop is $\pi /5$, Comm. Math. Phys., to appear. Ko M. Kontsevich (2003), CFT, SLE and phase boundaries, Preprint of the Max Planck Institute (Arbeitstagung 2003), 2003-60a.
- Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR 2129588
- Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273. MR 1879850, DOI https://doi.org/10.1007/BF02392618
- Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273. MR 1879850, DOI https://doi.org/10.1007/BF02392618
- Gregory F. Lawler, Oded Schramm, and Wendelin Werner, The dimension of the planar Brownian frontier is $4/3$, Math. Res. Lett. 8 (2001), no. 1-2, 13–23. MR 1825256, DOI https://doi.org/10.4310/MRL.2001.v8.n1.a3
- Gregory F. Lawler, Oded Schramm, and Wendelin Werner, On the scaling limit of planar self-avoiding walk, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 339–364. MR 2112127, DOI https://doi.org/10.1214/aop/1079021469
- Gregory Lawler, Oded Schramm, and Wendelin Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), no. 4, 917–955. MR 1992830, DOI https://doi.org/10.1090/S0894-0347-03-00430-2
- Gregory F. Lawler and José A. Trujillo Ferreras, Random walk loop soup, Trans. Amer. Math. Soc. 359 (2007), no. 2, 767–787. MR 2255196, DOI https://doi.org/10.1090/S0002-9947-06-03916-X
- Gregory F. Lawler and Wendelin Werner, Universality for conformally invariant intersection exponents, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 4, 291–328. MR 1796962, DOI https://doi.org/10.1007/s100970000024
- Gregory F. Lawler and Wendelin Werner, The Brownian loop soup, Probab. Theory Related Fields 128 (2004), no. 4, 565–588. MR 2045953, DOI https://doi.org/10.1007/s00440-003-0319-6
- Jean-François Le Gall, On the connected components of the complement of a two-dimensional Brownian path, Random walks, Brownian motion, and interacting particle systems, Progr. Probab., vol. 28, Birkhäuser Boston, Boston, MA, 1991, pp. 323–338. MR 1146456
- Jean-François Le Gall, Some properties of planar Brownian motion, École d’Été de Probabilités de Saint-Flour XX—1990, Lecture Notes in Math., vol. 1527, Springer, Berlin, 1992, pp. 111–235. MR 1229519, DOI https://doi.org/10.1007/BFb0084700
- Paul Malliavin, The canonic diffusion above the diffeomorphism group of the circle, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 4, 325–329 (English, with English and French summaries). MR 1713340, DOI https://doi.org/10.1016/S0764-4442%2800%2988575-4
- Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [Series for the Referee]. MR 665254
- T. S. Mountford, On the asymptotic number of small components created by planar Brownian motion, Stochastics Stochastics Rep. 28 (1989), no. 3, 177–188. MR 1020270, DOI https://doi.org/10.1080/17442508908833591
- A. A. Kirillov and D. V. Yur′ev, Kähler geometry of the infinite-dimensional homogeneous space $M={\rm Diff}_+(S^1)/{\rm Rot}(S^1)$, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35–46, 96 (Russian). MR 925071
- Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883–924. MR 2153402, DOI https://doi.org/10.4007/annals.2005.161.883
- Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, DOI https://doi.org/10.1007/BF02803524 ScSh O. Schramm, S. Sheffield (2005), in preparation. ShW S. Sheffield, W. Werner (2005), in preparation.
- Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244 (English, with English and French summaries). MR 1851632, DOI https://doi.org/10.1016/S0764-4442%2801%2901991-7
- Wendelin Werner, Sur la forme des composantes connexes du complémentaire de la courbe brownienne plane, Probab. Theory Related Fields 98 (1994), no. 3, 307–337 (French, with English and French summaries). MR 1262969, DOI https://doi.org/10.1007/BF01192257
- Wendelin Werner, Random planar curves and Schramm-Loewner evolutions, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, Springer, Berlin, 2004, pp. 107–195. MR 2079672, DOI https://doi.org/10.1007/978-3-540-39982-7_2
- Wendelin Werner, SLEs as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337 (2003), no. 7, 481–486 (English, with English and French summaries). MR 2023758, DOI https://doi.org/10.1016/j.crma.2003.08.003
- Wendelin Werner, Girsanov’s transformation for ${\rm SLE}(\kappa ,\rho )$ processes, intersection exponents and hiding exponents, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 121–147 (English, with English and French summaries). MR 2060031
- Wendelin Werner, Conformal restriction and related questions, Probab. Surv. 2 (2005), 145–190. MR 2178043, DOI https://doi.org/10.1214/154957805100000113 Wln3 W. Werner (2005), Some recent aspects of random conformally invariant systems, Lecture Notes from Les Houches summer school, July 2005. Wip W. Werner, in preparation.
- Marc Yor, Loi de l’indice du lacet brownien, et distribution de Hartman-Watson, Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 71–95 (French). MR 576898, DOI https://doi.org/10.1007/BF00531612 Zhan D. Zhan (2004), Random Loewner chains in Riemann surfaces, PhD dissertation, Caltech.
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 60D05, 82B41, 82B43, 30C99, 60J65
Retrieve articles in all journals with MSC (2000): 60D05, 82B41, 82B43, 30C99, 60J65
Additional Information
Wendelin Werner
Affiliation:
Université Paris-Sud, Laboratoire de Mathématiques, Université Paris-Sud, Bât. 425, 91405 Orsay cedex, France and DMA, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex, France
Email:
wendelin.werner@math.u-psud.fr
Received by editor(s):
December 17, 2005
Published electronically:
February 20, 2007
Additional Notes:
This work was supported by the Institut Universitaire de France
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.