The conformally invariant measure on self-avoiding loops

Werner, Wendelin¹

¹ 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

Journal of the American Mathematical Society, Tome 21 (2008) no. 1, pp. 137-169

Résumé

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.

Détail

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

Bibliographie

[1] Bass, Richard F. Probabilistic techniques in analysis 1995

[2] Bauer, Michel, Bernard, Denis SLE martingales and the Virasoro algebra Phys. Lett. B 2003 309 316

[3] Beffara, Vincent Hausdorff dimensions for 𝑆𝐿𝐸₆ Ann. Probab. 2004 2606 2629

[4] Burdzy, Krzysztof Cut points on Brownian paths Ann. Probab. 1989 1012 1036

[5] Burdzy, Krzysztof, Lawler, Gregory F. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal Ann. Probab. 1990 981 1009

[6] Friedrich, R., Kalkkinen, J. On conformal field theory and stochastic Loewner evolution Nuclear Phys. B 2004 279 302

[7] Friedrich, Roland, Werner, Wendelin Conformal restriction, highest-weight representations and SLE Comm. Math. Phys. 2003 105 122

[8] Lawler, Gregory F. Conformally invariant processes in the plane 2005

[9] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Values of Brownian intersection exponents. I. Half-plane exponents Acta Math. 2001 237 273

[10] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin Values of Brownian intersection exponents. I. Half-plane exponents Acta Math. 2001 237 273

[11] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin The dimension of the planar Brownian frontier is 4/3 Math. Res. Lett. 2001 13 23

[12] Lawler, Gregory F., Schramm, Oded, Werner, Wendelin On the scaling limit of planar self-avoiding walk 2004 339 364

[13] Lawler, Gregory, Schramm, Oded, Werner, Wendelin Conformal restriction: the chordal case J. Amer. Math. Soc. 2003 917 955

[14] Lawler, Gregory F., Trujillo Ferreras, Josã© A. Random walk loop soup Trans. Amer. Math. Soc. 2007 767 787

[15] Lawler, Gregory F., Werner, Wendelin Universality for conformally invariant intersection exponents J. Eur. Math. Soc. (JEMS) 2000 291 328

[16] Lawler, Gregory F., Werner, Wendelin The Brownian loop soup Probab. Theory Related Fields 2004 565 588

[17] Le Gall, Jean-Franã§Ois On the connected components of the complement of a two-dimensional Brownian path 1991 323 338

[18] Le Gall, Jean-Franã§Ois Some properties of planar Brownian motion 1992 111 235

[19] Malliavin, Paul The canonic diffusion above the diffeomorphism group of the circle C. R. Acad. Sci. Paris Sér. I Math. 1999 325 329

[20] Mandelbrot, Benoit B. The fractal geometry of nature 1982

[21] Mountford, T. S. On the asymptotic number of small components created by planar Brownian motion Stochastics Stochastics Rep. 1989 177 188

[22] Kirillov, A. A., Yur′Ev, D. V. Kähler geometry of the infinite-dimensional homogeneous space 𝑀 Funktsional. Anal. i Prilozhen. 1987

[23] Rohde, Steffen, Schramm, Oded Basic properties of SLE Ann. of Math. (2) 2005 883 924

[24] Schramm, Oded Scaling limits of loop-erased random walks and uniform spanning trees Israel J. Math. 2000 221 288

[25] Smirnov, Stanislav Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits C. R. Acad. Sci. Paris Sér. I Math. 2001 239 244

[26] Werner, Wendelin Sur la forme des composantes connexes du complémentaire de la courbe brownienne plane Probab. Theory Related Fields 1994 307 337

[27] Werner, Wendelin Random planar curves and Schramm-Loewner evolutions 2004 107 195

[28] Werner, Wendelin SLEs as boundaries of clusters of Brownian loops C. R. Math. Acad. Sci. Paris 2003 481 486

[29] Werner, Wendelin Girsanov’s transformation for 𝑆𝐿𝐸(𝜅,𝜌) processes, intersection exponents and hiding exponents Ann. Fac. Sci. Toulouse Math. (6) 2004 121 147

[30] Werner, Wendelin Conformal restriction and related questions Probab. Surv. 2005 145 190

[31] Yor, Marc Loi de l’indice du lacet brownien, et distribution de Hartman-Watson Z. Wahrsch. Verw. Gebiete 1980 71 95