[Les processus SLE comme frontières d'amas de lacets browniens]
Nous étudions certaines propriétés de connectivité de la « soupe » de lacets browniens dans un domaine. On montre l'exitence d'une transition de phase : lorsque l'intensité c est petite, il y a un ensemble dénombrable d'amas disjoints alors que lorsque c est grand, il n'y a presque sûrement qu'un seul amas. Nous montrons que pour les petites valeurs de c, les frontières de ces amas sont des courbes simples de type SLE (Evolution de Loewner–Schramm) de paramètre κ∈(8/3,4] avec c=(6−κ)(3κ−8)/2κ. Ceci permet de construire une famille aléatoire de boucles de SLE disjointes sur toute surface de Riemann et est étroitement relié à la théorie conforme des champs.
In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLEκ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ−8)(6−κ)/2κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLEκ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory.
Accepté le :
Publié le :
Wendelin Werner 1
@article{CRMATH_2003__337_7_481_0, author = {Wendelin Werner}, title = {SLEs as boundaries of clusters of {Brownian} loops}, journal = {Comptes Rendus. Math\'ematique}, pages = {481--486}, publisher = {Elsevier}, volume = {337}, number = {7}, year = {2003}, doi = {10.1016/j.crma.2003.08.003}, language = {en}, }
Wendelin Werner. SLEs as boundaries of clusters of Brownian loops. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 481-486. doi : 10.1016/j.crma.2003.08.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.08.003/
[1] M. Bauer, D. Bernard, Conformal transformations and the SLE partition function martingales, Preprint, 2003
[2] V. Beffara, The dimension of the SLE curves, Preprint, 2002
[3] Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, Volume 241 (1984), pp. 333-380
[4] The random walk representation of classical spin systems and correlation inequalities, Comm. Math. Phys., Volume 83 (1982), pp. 123-150
[5] Conformal invariance and surface critical behavior, Nucl. Phys. B, Volume 240 (1984), pp. 514-532
[6] Connectivity properties of Mandelbrot's percolation process, Probab. Theory Related Fields, Volume 77 (1988), pp. 307-324
[7] J. Dubédat, SLE(κ,ρ) martingales and duality, Preprint, 2003
[8] R. Friedrich, J. Kalkkinen, On conformal field theory and stochastic Löwner evolutions, Preprint, 2003
[9] R. Friedrich, W. Werner, Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys. (2003), in press
[10] Fractal structure of 2-D quantum gravity, Mod. Phys. Lett. A, Volume 3 (1988), p. 819
[11] Values of Brownian intersection exponents I: Half-plane exponents, Acta Math., Volume 187 (2001), pp. 237-273
[12] Values of Brownian intersection exponents II: Plane exponents, Acta Math., Volume 187 (2001), pp. 275-308
[13] G.F. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. (2001), in press
[14] G.F. Lawler, O. Schramm, W. Werner, On the scaling limit of planar self-avoiding walks, in: M. Lapidus (Ed.), AMS Symp. Pure Math., Volume in honor of B.B. Mandelbrot, 2002, in press
[15] Conformal restriction properties. The chordal case, J. Amer. Math. Soc., Volume 16 (2003), pp. 915-955
[16] Universality for conformally invariant intersection exponents, J. Eur. Math. Soc., Volume 2 (2000), pp. 291-328
[17] G.F. Lawler, W. Werner, The Brownian loop soup, Probab. Theory Related Fields, in press
[18] The Fractal Geometry of Nature, Freeman, 1982
[19] Continuum Percolation, CUP, 1996
[20] S. Rohde, O. Schramm, Basic properties of SLE, Ann. Math. (2001), in press
[21] Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., Volume 118 (2000), pp. 221-288
[22] O. Schramm, S. Sheffield, (2003), in preparation
[23] W. Werner, Random planar curves and Schramm–Loewner evolutions, in: 2002 St-Flour Summer School, Lecture Notes in Math., Springer, 2002, in press
[24] W. Werner, Girsanov's theorem for SLE(κ,ρ) processes, intersection exponents and hiding exponents, Ann. Fac. Sci. Toulouse, in press
[25] W. Werner, Conformal restriction and related questions, Preprint, 2003
[26] W. Werner, (2003), in preparation
- On Clusters of Brownian Loops in d Dimensions, In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius, Volume 77 (2021), p. 797 | DOI:10.1007/978-3-030-60754-8_33
- The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model, Communications in Mathematical Physics, Volume 373 (2020) no. 1, p. 357 | DOI:10.1007/s00220-019-03620-3
- Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model, Communications on Pure and Applied Mathematics, Volume 73 (2020) no. 7, p. 1371 | DOI:10.1002/cpa.21884
- Conditioning a Brownian loop-soup cluster on a portion of its boundary, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55 (2019) no. 1 | DOI:10.1214/18-aihp883
- Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines, Communications in Mathematical Physics, Volume 361 (2018) no. 1, p. 53 | DOI:10.1007/s00220-018-3159-z
- Connection Probabilities for Conformal Loop Ensembles, Communications in Mathematical Physics, Volume 362 (2018) no. 2, p. 415 | DOI:10.1007/s00220-018-3207-8
- The law of a point process of Brownian excursions in a domain is determined by the law of its trace, Electronic Journal of Probability, Volume 23 (2018) no. none | DOI:10.1214/18-ejp258
- The nested simple conformal loop ensembles in the Riemann sphere, Probability Theory and Related Fields, Volume 165 (2016) no. 3-4, p. 835 | DOI:10.1007/s00440-015-0647-3
- Random walk loop soups and conformal loop ensembles, Probability Theory and Related Fields, Volume 166 (2016) no. 1-2, p. 553 | DOI:10.1007/s00440-015-0666-0
- On the Spatial Markov Property of Soups of Unoriented and Oriented Loops, Séminaire de Probabilités XLVIII, Volume 2168 (2016), p. 481 | DOI:10.1007/978-3-319-44465-9_19
- Planar Ising magnetization field I. Uniqueness of the critical scaling limit, The Annals of Probability, Volume 43 (2015) no. 2 | DOI:10.1214/13-aop881
- The Appropriate Use of Employment At-Will in County Sheriffs’ Departments, Review of Public Personnel Administration, Volume 34 (2014) no. 3, p. 199 | DOI:10.1177/0734371x13500319
- The Hausdorff dimension of the CLE gasket, The Annals of Probability, Volume 42 (2014) no. 4 | DOI:10.1214/12-aop820
- From CLE(
) to SLE( )'s, Electronic Journal of Probability, Volume 18 (2013) no. none | DOI:10.1214/ejp.v18-2376 - Convex hull ofnplanar Brownian paths: an exact formula for the average number of edges, Journal of Physics A: Mathematical and Theoretical, Volume 46 (2013) no. 1, p. 015004 | DOI:10.1088/1751-8113/46/1/015004
- Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Annals of Mathematics, Volume 176 (2012) no. 3, p. 1827 | DOI:10.4007/annals.2012.176.3.8
- Critical Ising on the Square Lattice Mixes in Polynomial Time, Communications in Mathematical Physics, Volume 313 (2012) no. 3, p. 815 | DOI:10.1007/s00220-012-1460-9
- Connection probabilities and RSW‐type bounds for the two‐dimensional FK Ising model, Communications on Pure and Applied Mathematics, Volume 64 (2011) no. 9, p. 1165 | DOI:10.1002/cpa.20370
- Conformally invariant scaling limits: an overview and a collection of problems, Selected Works of Oded Schramm (2011), p. 1161 | DOI:10.1007/978-1-4419-9675-6_34
- Universal Behavior of Connectivity Properties in Fractal Percolation Models, Electronic Journal of Probability, Volume 15 (2010) no. none | DOI:10.1214/ejp.v15-805
- Exploration trees and conformal loop ensembles, Duke Mathematical Journal, Volume 147 (2009) no. 1 | DOI:10.1215/00127094-2009-007
- Equilibrium Stranski-Krastanow and Volmer-Weber models, EPL (Europhysics Letters), Volume 86 (2009) no. 1, p. 16002 | DOI:10.1209/0295-5075/86/16002
- The Scaling Limit of (Near-)Critical 2D Percolation, New Trends in Mathematical Physics (2009), p. 117 | DOI:10.1007/978-90-481-2810-5_9
- Ising (conformal) fields and cluster area measures, Proceedings of the National Academy of Sciences, Volume 106 (2009) no. 14, p. 5457 | DOI:10.1073/pnas.0900700106
- Scaling limits of two‐dimensional percolation: an overview, Statistica Neerlandica, Volume 62 (2008) no. 3, p. 314 | DOI:10.1111/j.1467-9574.2008.00400.x
- The conformally invariant measure on self-avoiding loops, Journal of the American Mathematical Society, Volume 21 (2007) no. 1, p. 137 | DOI:10.1090/s0894-0347-07-00557-7
- On Malliavin measures, SLE, and CFT, Proceedings of the Steklov Institute of Mathematics, Volume 258 (2007) no. 1, p. 100 | DOI:10.1134/s0081543807030108
- The Expected Area of the Filled Planar Brownian Loop is π/5, Communications in Mathematical Physics, Volume 264 (2006) no. 3, p. 797 | DOI:10.1007/s00220-006-1555-2
- Two-Dimensional Critical Percolation: The Full Scaling Limit, Communications in Mathematical Physics, Volume 268 (2006) no. 1, p. 1 | DOI:10.1007/s00220-006-0086-1
- Stochastic Loewner Evolutions, Encyclopedia of Mathematical Physics (2006), p. 80 | DOI:10.1016/b0-12-512666-2/00401-6
- The Scaling Limit Geometry of Near-Critical 2D Percolation, Journal of Statistical Physics, Volume 125 (2006) no. 5-6, p. 1155 | DOI:10.1007/s10955-005-9014-6
- Course 3 Conformal random geometry, Mathematical statistical physics, École d'ÉtÉ de physique des houches session LXXXIII, Volume 83 (2006), p. 101 | DOI:10.1016/s0924-8099(06)80040-5
- Course 2 Some recent aspects of random conformally invariant systems, Mathematical statistical physics, École d'ÉtÉ de physique des houches session LXXXIII, Volume 83 (2006), p. 57 | DOI:10.1016/s0924-8099(06)80039-9
- 2D growth processes: SLE and Loewner chains, Physics Reports, Volume 432 (2006) no. 3-4, p. 115 | DOI:10.1016/j.physrep.2006.06.002
- Some mathematical aspects of the scaling limit of critical two-dimensional systems, Pramana, Volume 64 (2005) no. 5, p. 757 | DOI:10.1007/bf02704581
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