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Cours spécialisés - Parutions - 16 (2009)

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Percolation et modèle d'Ising
Wendelin Werner
Cours spécialisés 16 (2009), 161 pages

Résumé :
Ces notes de cours constituent une introduction mathématique à l'étude de modèles probabilistes sur réseau, issus de la physique statistique. À travers les exemples de la percolation et du modèle d'Ising, nous abordons les phénomènes de changements de phases et nous introduisons un certain nombre de techniques classiques. Nous présentons également – c'est l'un des buts principaux de ce cours – des résultats récents, dus à Stanislas Smirnov concernant l'invariance conforme de ces deux modèles en dimension deux.

Mots-clefs : Physique statistique, mécanique statistique, probabilités, transitions de phases, percolation, modèle d'Ising, modèle de Potts, invariance conforme

Abstract:
Ising model and percolation
These lecture notes provide a mathematical introduction to the study of random lattice-based models from statistical physics. Via the study of percolation and of the Ising model, we introduce the notion of phase transitions and we describe some classical techniques. One of the main goals of these notes is also to present recent results of Stanislav Smirnov concerning the conformal invariance of these models in two-dimensional space.

Keywords: Statistical physics, statistical mecanics, probability theory, phase transitions, percolation, Ising model, Potts model, conformal invariance

Class. math. : 60-01, 82-01, 82B05, 82B20, 82B26, 82B27, 82B43


ISBN : 978-2-85629-276-1
ISSN : 1284-6090

Bibliographie:

1
Ahlfors, Lars V.
Complex analysis
McGraw-Hill Book Co., 1978
Math Reviews MR510197
2
Aizenman, Michael and Barsky, David J.
Sharpness of the phase transition in percolation models
Comm. Math. Phys. 108 (1987) 489–526
Math Reviews MR874906
Zentralblatt 618.60098
3
Aizenman, Michael and Kesten, Harry and Newman, Charles M.
Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation
Comm. Math. Phys. 111 (1987) 505–531
Math Reviews MR901151
Zentralblatt 642.60102
4
Aizenman, Michael and Newman, Charles M.
Tree graph inequalities and critical behavior in percolation models
J. Statist. Phys. 36 (1984) 107–143
Math Reviews MR762034
Zentralblatt 586.60096
5
Berg, J. (van den) and Keane, M.
On the continuity of the percolation probability function
in Conference in modern analysis and probability (New Haven, Conn., 1982)
Contemp. Math. 26 (1984) 61–65
Math Reviews MR737388
Zentralblatt 541.60099
6
Berg, J. (van den) and Kesten, Harry
Inequalities with applications to percolation and reliability
J. Appl. Probab. 22 (1985) 556–569
Math Reviews MR799280
Zentralblatt 571.60019
7
Bollobás, Béla and Riordan, Oliver
Percolation
Cambridge Univ. Press, 2006
Math Reviews MR2283880
Zentralblatt 1118.60001
8
Burton, R. M. and Keane, M.
Density and uniqueness in percolation
Comm. Math. Phys. 121 (1989) 501–505
Math Reviews MR990777
Zentralblatt 662.60113
9
Cardy, J.L.
Conformal invariance and surface critical behavior
Nucl. Phys. B 240 (1984) 514–532
10
Cardy, J.L. and Riva, V.
Holomorphic parafermions in the Potts model and SLE
J. Stat. Mech. (2006) english0612:P001
11
Fortuin, C. M.
On the random-cluster model. II. The percolation model
Physica 58 (1972) 393–418
Math Reviews MR0378660
12
Fortuin, C. M.
On the random-cluster model. III. The simple random-cluster model
Physica 59 (1972) 545–570
Math Reviews MR0432137
13
Fortuin, C. M. and Kasteleyn, P. W.
On the random-cluster model. I. Introduction and relation to other models
Physica 57 (1972) 536–564
Math Reviews MR0359655
14
Fortuin, C. M. and Kasteleyn, P. W. and Ginibre, J.
Correlation inequalities on some partially ordered sets
Comm. Math. Phys. 22 (1971) 89–103
Math Reviews MR0309498
Zentralblatt 346.06011
15
Georgii, Hans-Otto and Häggström, Olle and Maes, Christian
The random geometry of equilibrium phases
in Phase transitions and critical phenomena
Phase Transit. Crit. Phenom. 18 (2001) 1–142
Math Reviews MR2014387
16
Grimmett, Geoffrey
The stochastic random-cluster process and the uniqueness of random-cluster measures
Ann. Probab. 23 (1995) 1461–1510
Math Reviews MR1379156
Zentralblatt 852.60105
17
Grimmett, Geoffrey
Percolation and disordered systems
in Lectures on probability theory and statistics (Saint-Flour, 1996)
Lecture Notes in Math. 1665 (1997) 153–300
Math Reviews MR1490045
Zentralblatt 884.60089
18
Grimmett, Geoffrey
Percolation
Springer, 1999
Math Reviews MR1707339
Zentralblatt 948.60092
19
Grimmett, Geoffrey
The random-cluster model
Springer, 2006
Math Reviews MR2243761
Zentralblatt 1122.60087
20
Harris, T. E.
A lower bound for the critical probability in a certain percolation process
Proc. Cambridge Philos. Soc. 56 (1960) 13–20
Math Reviews MR0115221
Zentralblatt 122.36403
21
Karatzas, Ioannis and Shreve, Steven E.
Brownian motion and stochastic calculus
Springer, 1991
Math Reviews MR1121940
Zentralblatt 734.60060
22
Kesten, Harry
The critical probability of bond percolation on the square lattice equals 12
Comm. Math. Phys. 74 (1980) 41–59
Math Reviews MR575895
23
Kesten, Harry
Analyticity properties and power law estimates of functions in percolation theory
J. Statist. Phys. 25 (1981) 717–756
Math Reviews MR633715
Zentralblatt 512.60095
24
Kesten, Harry
Percolation theory for mathematicians
Birkhäuser, 1982
Math Reviews MR692943
Zentralblatt 522.60097
25
Lawler, Gregory F.
Conformally invariant processes in the plane
Amer. Math. Soc., 2005
Math Reviews MR2129588
Zentralblatt 1074.60002
26
Lawler, Gregory F.
Schramm-Loewner evolutions (SLE)
IAS/Park City Math. Ser. 16 (2009) 231–295
27
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
One-arm exponent for critical 2D percolation
Electron. J. Probab. 7m, no 2 (2002)
Math Reviews MR1887622
Zentralblatt 1015.60091
28
Menshikov, M. V.
Coincidence of critical points in percolation problems
Dokl. Akad. Nauk SSSR 288 (1986) 1308–1311
Math Reviews MR852458
29
Nienhuis, Bernard
Exact critical point and critical exponents of O(n) models in two dimensions
Phys. Rev. Lett. 49 (1982) 1062–1065
Math Reviews MR675241
30
Nienhuis, Bernard
Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas
J. Statist. Phys. 34 (1984) 731–761
Math Reviews MR751711
Zentralblatt 595.76071
31
Russo, Lucio
A note on percolation
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978) 39–48
Math Reviews MR0488383
Zentralblatt 363.60120
32
Russo, Lucio
On the critical percolation probabilities
Z. Wahrsch. Verw. Gebiete 56 (1981) 229–237
Math Reviews MR618273
Zentralblatt 457.60084
33
Schramm, Oded
Scaling limits of loop-erased random walks and uniform spanning trees
Israel J. Math. 118 (2000) 221–288
Math Reviews MR1776084
Zentralblatt 968.60093
34
Seymour, P. D. and Welsh, D. J. A.
Percolation probabilities on the square lattice
Ann. Discrete Math. 3 (1978) 227–245 Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977)
Math Reviews MR0494572
Zentralblatt 405.60015
35
Smirnov, Stanislav
Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 239–244
Math Reviews MR1851632
Zentralblatt 985.60090
36
Smirnov, Stanislav
Towards conformal invariance of 2D lattice models
in International Congress of Mathematicians. Vol. II
(2006) 1421–1451
Math Reviews MR2275653
Zentralblatt 1112.82014
37
Smirnov, Stanislav
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
à paraître dans Ann. Math
38
Smirnov, Stanislav and Werner, Wendelin
Critical exponents for two-dimensional percolation
Math. Res. Lett. 8 (2001) 729–744
Math Reviews MR1879816
Zentralblatt 1009.60087
39
Werner, Wendelin
Random planar curves and Schramm-Loewner evolutions
in Lectures on probability theory and statistics
Lecture Notes in Math. 1840 (2004) 107–195
Math Reviews MR2079672
Zentralblatt 1057.60078
40
Werner, Wendelin
Lectures on two-dimensional critical percolation
IAS/Park City Math. Ser. 16 (2009) 297–360