SLE et invariance conforme
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 925, pp. 15-28.

Les processus de Schramm-Loewner (SLE) induisent des courbes aléatoires du plan complexe, qui vérifient une propriété d'invariance conforme. Ce sont des outils fondamentaux pour la compréhension du comportement asymptotique en régime critique de certains modèles discrets intervenant en physique statistique ; ils ont permis notamment d'établir rigoureusement certaines conjectures importantes dans ce domaine.

The so-called Stochastic Loewner Evolutions form a family of random curves in the complex plane, which enjoy a (statistical) conformal invariance property. They have a crucial role in the analysis of the asymptotic behaviour of many discrete models in statistical physics. In particular, they have yielded rigorous proofs of several important conjectures in this field.

Classification : 60J65, 60K35, 82Bxx, 43xx, 30C35
Mot clés : Équation de Loewner stochastique, invariance conforme, mouvement brownien plan, percolation, marche aléatoire, changement d'échelle
Keywords: stochastic Loewner equation, conformal invariance, planar brownian motion, percolation, random walk, scaling limit 
@incollection{SB_2003-2004__46__15_0,
     author = {Bertoin, Jean},
     title = {SLE et invariance conforme},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     series = {Ast\'erisque},
     note = {talk:925},
     pages = {15--28},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     mrnumber = {2167200},
     zbl = {1083.60067},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__15_0/}
}
TY  - CHAP
AU  - Bertoin, Jean
TI  - SLE et invariance conforme
BT  - Séminaire Bourbaki : volume 2003/2004, exposés 924-937
AU  - Collectif
T3  - Astérisque
N1  - talk:925
PY  - 2005
SP  - 15
EP  - 28
IS  - 299
PB  - Association des amis de Nicolas Bourbaki, Société mathématique de France
PP  - Paris
UR  - http://www.numdam.org/item/SB_2003-2004__46__15_0/
LA  - fr
ID  - SB_2003-2004__46__15_0
ER  - 
%0 Book Section
%A Bertoin, Jean
%T SLE et invariance conforme
%B Séminaire Bourbaki : volume 2003/2004, exposés 924-937
%A Collectif
%S Astérisque
%Z talk:925
%D 2005
%P 15-28
%N 299
%I Association des amis de Nicolas Bourbaki, Société mathématique de France
%C Paris
%U http://www.numdam.org/item/SB_2003-2004__46__15_0/
%G fr
%F SB_2003-2004__46__15_0
Bertoin, Jean. SLE et invariance conforme, dans Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 925, pp. 15-28. http://www.numdam.org/item/SB_2003-2004__46__15_0/

[1] L. V. Ahlfors. Conformal Invariants, Topics in Geometric Function Theory. McGraw-Hill, 1973. | MR | Zbl

[2] V. Beffara. The dimension of the SLE curves. Prépublication. | Zbl

[3] J. Cardy. Critical percolation in finite geometries. J. Phys. A, pages L201-L206, 1992. | MR | Zbl

[4] J. B. Conway. Functions of one complex variable. II. Springer-Verlag, New York, 1995. | MR | Zbl

[5] B. Duplantier and K.-H. Kwon. Conformal invariance and intersections of random walks. Phys. Rev. Lett., 61 :2514-2517, 1988. | Zbl

[6] C. Itzykson and J.-M. Drouffe. Statistical field theory vol. 2. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1989. | MR | Zbl

[7] G. F. Lawler. An introduction to the Stochastic Loewner Evolution. In Random walks and geometry, pages 261-293. Walter de Gruyter, Berlin, 2004. | MR | Zbl

[8] G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents I : Half-plane exponents et II : Plane exponents. Acta Math., 187 :237-273 & 275-308, 2001. | MR | Zbl

[9] G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents III : Two-sided exponents. Ann. Inst. H. Poincaré. Probab. Statist., 38 :109-123, 2002. | Numdam | MR | Zbl

[10] G. F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32 :939-995, 2004. | MR | Zbl

[11] G. F. Lawler, O. Schramm, and W. Werner. On the scaling limit of self-avoiding walks. In Fractal geometry and application, A jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math. American Mathematical Society. à paraître. | Zbl

[12] G. F. Lawler, O. Schramm, and W. Werner. Conformal restriction properties. The chordal case. J. Amer. Math. Soc., 16 :917-955, 2003. | MR | Zbl

[13] D. Marshall and S. Rohde. The Loewner differential equation and slit mappings. Prépublication. | Zbl

[14] S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. à paraître. | Zbl

[15] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 :221-288, 2001. | MR | Zbl

[16] S. Smirnov. Critical percolation in the plane : conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333 :239-244, 2001. | MR | Zbl

[17] S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett., 8 :729-744, 2001. | MR | Zbl

[18] K. Symanzik. Euclidean Quantum Field Theory. In L. Jost, editor, Local Quantum Theory, Proc. International School of Physics “Enrico Fermi” XLV, page 152. Academic Press, New York, 1969.

[19] W. Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics, École d'été de probabilités de Saint-Flour 2002, volume 1840 of Lecture Notes in Mathematics, pages 107-195. Springer, 2004. | MR | Zbl