SLE and conformal invariance

Jean Bertoin

Séminaire Bourbaki (2003-2004)

  • Volume: 46, page 15-28
  • ISSN: 0303-1179

Abstract

top
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.

How to cite

top

Bertoin, Jean. "SLE et invariance conforme." Séminaire Bourbaki 46 (2003-2004): 15-28. <http://eudml.org/doc/252142>.

@article{Bertoin2003-2004,
abstract = {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.},
author = {Bertoin, Jean},
journal = {Séminaire Bourbaki},
keywords = {stochastic Loewner equation; conformal invariance; planar brownian motion; percolation; random walk; scaling limit},
language = {fre},
pages = {15-28},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {SLE et invariance conforme},
url = {http://eudml.org/doc/252142},
volume = {46},
year = {2003-2004},
}

TY - JOUR
AU - Bertoin, Jean
TI - SLE et invariance conforme
JO - Séminaire Bourbaki
PY - 2003-2004
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 46
SP - 15
EP - 28
AB - 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.
LA - fre
KW - stochastic Loewner equation; conformal invariance; planar brownian motion; percolation; random walk; scaling limit
UR - http://eudml.org/doc/252142
ER -

References

top
  1. [1] L.V. Ahlfors. Conformal Invariants, Topics in Geometric Function Theory. McGraw-Hill, 1973. Zbl0272.30012MR357743
  2. [2] V. Beffara. The dimension of the SLE curves. Prépublication. Zbl1165.60007
  3. [3] J. Cardy. Critical percolation in finite geometries. J. Phys. A, pages L201–L206, 1992. Zbl0965.82501MR1151081
  4. [4] J.B. Conway. Functions of one complex variable. II. Springer-Verlag, New York, 1995. Zbl0277.30001MR1344449
  5. [5] B. Duplantier and K.-H. Kwon. Conformal invariance and intersections of random walks. Phys. Rev. Lett., 61 :2514–2517, 1988. Zbl0652.60073
  6. [6] C. Itzykson and J.-M. Drouffe. Statistical field theory vol. 2. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1989. Zbl0825.81002MR1175177
  7. [7] G.F. Lawler. An introduction to the Stochastic Loewner Evolution. In Random walks and geometry, pages 261–293. Walter de Gruyter, Berlin, 2004. Zbl1061.60107MR2087784
  8. [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. Zbl1005.60097MR1879850
  9. [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. Zbl1006.60075MR1899232
  10. [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. Zbl1126.82011MR2044671
  11. [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. Zbl1069.60089
  12. [12] G.F. Lawler, O. Schramm, and W. Werner. Conformal restriction properties. The chordal case. J. Amer. Math. Soc., 16 :917–955, 2003. Zbl1030.60096MR1992830
  13. [13] D. Marshall and S. Rohde. The Loewner differential equation and slit mappings. Prépublication. Zbl1078.30005
  14. [14] S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. à paraître. Zbl1081.60069
  15. [15] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 :221–288, 2001. Zbl0968.60093MR1776084
  16. [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. Zbl0985.60090MR1851632
  17. [17] S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett., 8 :729–744, 2001. Zbl1009.60087MR1879816
  18. [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. [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. Zbl1057.60078MR2079672

NotesEmbed ?

top

You must be logged in to post comments.