The conformally invariant measure on self-avoiding loops

Werner, Wendelin

doi:10.1090/S0894-0347-07-00557-7

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 that is contained in another Riemann surface is just the measure on restricted to those loops that stay in ). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.

1. M. Aizenman, B. Duplantier, A. Aharony (1999), Connectivity Exponents and External Perimeter in 2D Independent Percolation Models, Phys. Rev. Lett. 83, 1359.
2. Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
3. Michel Bauer and Denis Bernard, SLE martingales and the Virasoro algebra, Phys. Lett. B 557 (2003), no. 3-4, 309–316. MR 1972482, https://doi.org/10.1016/S0370-2693(03)00189-8
4. R. Bauer, R. Friedrich (2005), On chordal and bilateral SLE in multiply connected domains, preprint.
5. V. Beffara (2000), Mouvement Brownien plan, SLE, invariance conforme et dimensions fractales, Thèse de Doctorat, Université Paris-Sud.
6. Vincent Beffara, Hausdorff dimensions for 𝑆𝐿𝐸₆, Ann. Probab. 32 (2004), no. 3B, 2606–2629. MR 2078552, https://doi.org/10.1214/009117904000000072
7. Krzysztof Burdzy, Cut points on Brownian paths, Ann. Probab. 17 (1989), no. 3, 1012–1036. MR 1009442
8. 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
9. F. Camia, C. Newman (2005), The Full Scaling Limit of Two-Dimensional Critical Percolation, preprint.
10. J.L. Cardy (1984), Conformal invariance and surface critical behavior, Nucl. Phys. B 240, 514-532.
11. R. Friedrich, On connections of Conformal Field Theory and Stochastic Loewner Evolutions, math-ph/0410029.
12. R. Friedrich and J. Kalkkinen, On conformal field theory and stochastic Loewner evolution, Nuclear Phys. B 687 (2004), no. 3, 279–302. MR 2059141, https://doi.org/10.1016/j.nuclphysb.2004.03.025
13. Roland Friedrich and Wendelin Werner, Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys. 243 (2003), no. 1, 105–122. MR 2020222, https://doi.org/10.1007/s00220-003-0956-8
14. C. Garban, J.A. Trujillo-Ferreras (2005), The expected area of the Brownian loop is $\pi/5$ , Comm. Math. Phys., to appear.
15. M. Kontsevich (2003), CFT, SLE and phase boundaries, Preprint of the Max Planck Institute (Arbeitstagung 2003), 2003-60a.
16. Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR 2129588
17. 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, https://doi.org/10.1007/BF02392618
18. 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, https://doi.org/10.1007/BF02392618
19. 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, https://doi.org/10.4310/MRL.2001.v8.n1.a3
20. 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
21. Gregory Lawler, Oded Schramm, and Wendelin Werner, Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), no. 4, 917–955. MR 1992830, https://doi.org/10.1090/S0894-0347-03-00430-2
22. Gregory F. Lawler and José A. Trujillo Ferreras, Random walk loop soup, Trans. Amer. Math. Soc. 359 (2007), no. 2, 767–787. MR 2255196, https://doi.org/10.1090/S0002-9947-06-03916-X
23. 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, https://doi.org/10.1007/s100970000024
24. Gregory F. Lawler and Wendelin Werner, The Brownian loop soup, Probab. Theory Related Fields 128 (2004), no. 4, 565–588. MR 2045953, https://doi.org/10.1007/s00440-003-0319-6
25. 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
26. 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, https://doi.org/10.1007/BFb0084700
27. 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, https://doi.org/10.1016/S0764-4442(00)88575-4
28. 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
29. 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, https://doi.org/10.1080/17442508908833591
30. A. A. Kirillov and D. V. Yur′ev, Kähler geometry of the infinite-dimensional homogeneous space 𝑀=𝐷𝑖𝑓𝑓₊(𝑆¹)/𝑅𝑜𝑡(𝑆¹), Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35–46, 96 (Russian). MR 925071
31. Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883–924. MR 2153402, https://doi.org/10.4007/annals.2005.161.883
32. Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, https://doi.org/10.1007/BF02803524
33. O. Schramm, S. Sheffield (2005), in preparation.
34. S. Sheffield, W. Werner (2005), in preparation.
35. 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, https://doi.org/10.1016/S0764-4442(01)01991-7
36. 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, https://doi.org/10.1007/BF01192257
37. 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, https://doi.org/10.1007/978-3-540-39982-7_2
38. 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, https://doi.org/10.1016/j.crma.2003.08.003
39. Wendelin Werner, Girsanov’s transformation for 𝑆𝐿𝐸(𝜅,𝜌) 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
40. Wendelin Werner, Conformal restriction and related questions, Probab. Surv. 2 (2005), 145–190. MR 2178043, https://doi.org/10.1214/154957805100000113
41. W. Werner (2005), Some recent aspects of random conformally invariant systems, Lecture Notes from Les Houches summer school, July 2005.
42. W. Werner, in preparation.
43. 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, https://doi.org/10.1007/BF00531612
44. D. Zhan (2004), Random Loewner chains in Riemann surfaces, PhD dissertation, Caltech.