[Champs conformes, propriété de restriction, représentations dégénérées et SLE]
Nous relions le processus de Schramm–Loewner (SLE) à certaines représentations de plus haut poids dégénérées de l'algèbre de Virasoro. Les propriétés de restriction du SLE étudiées dans [19] s'avèrent être importantes pour établir ce lien. Par ailleurs, diverses considérations et relations de la théorie conforme des champs peuvent ainsi être interprétées en termes du SLE. Ceci permet de faire le lien entre les modèles issus de la physique statistique et la théorie conforme des champs.
We relate the Schramm–Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The restriction properties of SLE that have been recently derived in [19] play a crucial role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables one to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory.
Accepté le :
Publié le :
@article{CRMATH_2002__335_11_947_0,
author = {Friedrich, Roland and Werner, Wendelin},
title = {Conformal fields, restriction properties, degenerate representations and {SLE}},
journal = {Comptes Rendus. Math\'ematique},
pages = {947--952},
publisher = {Elsevier},
volume = {335},
number = {11},
year = {2002},
doi = {10.1016/S1631-073X(02)02581-5},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02581-5/}
}
TY - JOUR AU - Friedrich, Roland AU - Werner, Wendelin TI - Conformal fields, restriction properties, degenerate representations and SLE JO - Comptes Rendus. Mathématique PY - 2002 SP - 947 EP - 952 VL - 335 IS - 11 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02581-5/ DO - 10.1016/S1631-073X(02)02581-5 LA - en ID - CRMATH_2002__335_11_947_0 ER -
%0 Journal Article %A Friedrich, Roland %A Werner, Wendelin %T Conformal fields, restriction properties, degenerate representations and SLE %J Comptes Rendus. Mathématique %D 2002 %P 947-952 %V 335 %N 11 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02581-5/ %R 10.1016/S1631-073X(02)02581-5 %G en %F CRMATH_2002__335_11_947_0
Friedrich, Roland; Werner, Wendelin. Conformal fields, restriction properties, degenerate representations and SLE. Comptes Rendus. Mathématique, Tome 335 (2002) no. 11, pp. 947-952. doi : 10.1016/S1631-073X(02)02581-5. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02581-5/
[1] M. Bauer, D. Bernard, SLEκ growth and conformal field theories, Preprint, 2002
[2] V. Beffara, The dimension of the SLE curves, Preprint, 2002
[3] Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B, Volume 241 (1984), pp. 333-380
[4] Conformal invariance and surface critical behavior, Nuclear Phys. B, Volume 240 (1984) no. FS12, pp. 514-532
[5] Exact surface and wedge exponents for polymers in two dimensions, Phys. Rev. Lett., Volume 57 (1986), pp. 3179-3182
[6] Kac–Moody and Virasoro algebras, A Reprint Volume for Physicists, Adv. Ser. Math. Phys., 3, World Scientific, 1988
[7] Conformal Invariance and Applications to Statistical Mechanics (Itzykson, C.; Saleur, H.; Zuber, J.-B., eds.), World Scientific, 1988
[8] Infinite-Dimensional Lie Algebras, CUP, 1990
[9] Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, Adv. Ser. Math. Phys., 2, World Scientific, 1987
[10] Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk, Phys. Rev. Lett., Volume 88 (2002), p. 130601
[11] Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc., Volume 30 (1994), pp. 1-61
[12] G.F. Lawler, An introduction to the stochastic Loewner evolution, 2001, to appear
[13] Values of Brownian intersection exponents I: Half-plane exponents, Acta Math., Volume 187 (2001), pp. 237-273
[14] Values of Brownian intersection exponents II: Plane exponents, Acta Math., Volume 187 (2001), pp. 275-308
[15] Values of Brownian intersection exponents III: Two sided exponents, Ann. Inst. H. Poincaré, Volume 38 (2002), pp. 109-123
[16] One-arm exponent for critical 2D percolation, Electron. J. Probab., Volume 7 (2002) no. 2
[17] G.F. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Preprint, 2001
[18] G.F. Lawler, O. Schramm, W. Werner, On the scaling limit of planar self-avoiding walks, Preprint, 2002
[19] G.F. Lawler, O. Schramm, W. Werner, Conformal restriction. The chordal case, Preprint, 2002
[20] G.F. Lawler, W. Werner, The loop-soup, 2002, in preparation
[21] A non-Hamiltonian approach to conformal field theory, Soviet Phys. JETP, Volume 39 (1974), pp. 10-18
[22] S. Rohde, O. Schramm, Basic properties of SLE, Preprint, 2001
[23] Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math., Volume 118 (2000), pp. 221-288
[24] Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris, Série I, Volume 333 (2001) no. 3, pp. 239-244
[25] Critical exponents for two-dimensional percolation, Math. Res. Lett., Volume 8 (2001), pp. 729-744
[26] W. Werner, Random planar curves and Schramm–Loewner evolutions, in: Lecture Notes of the 2002 St-Flour Summer School, 2002, to appear
Cité par Sources :
