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Annales scientifiques de l'ENS - Parutions - série 4, 42 (2009)

Parutions < série 4, 42

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 42, fascicule 5 (2009)

Julien Dubédat
Duality of Schramm-Loewner Evolutions
Annales scientifiques de l'ENS 42, fascicule 5 (2009), 697-724

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Résumé :
Dualité des évolutions de Schramm-Loewner
On démontre dans cette note une version de la dualité conjecturée pour les évolutions de Schramm-Loewner, en établissant des identités en distribution exactes entre certains arcs de SLE_ chordal, >4, et des versions appropriées de SLE_, =16/.

Mots-clefs : Évolution de Schramm-Loewner, dualité, fonction de partition

Abstract:
In this note, we prove a version of the conjectured duality for Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal SLE_, >4, and appropriate versions of SLE_, =16/.

Keywords: Schramm-Loewner Evolution, duality, partition function

Class. math. : 60G17; 60K35


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Zhan, Dapeng
Reversibility of chordal SLE
Ann. Probab. 36 (2008) 1472–1494
Math Reviews MR2435856
Zentralblatt 1157.60051
2
Dubédat, Julien
SLE and the free field: Partition functions and couplings
J. Amer. Math. Soc. 22 (2009) 995–1054
3
Pommerenke, Ch.
Boundary behaviour of conformal maps
Springer, 1992
Math Reviews MR1217706
4
Schramm, Oded and Wilson, David B.
SLE coordinate changes
New York J. Math. 11 (2005) 659–669
Math Reviews MR2188260
5
Zhan, Dapeng
Duality of chordal SLE
Invent. Math. 174 (2008) 309–353
Math Reviews MR2439609
Zentralblatt 1158.60047
6
Kozdron, Michael J. and Lawler, Gregory F.
The configurational measure on mutually avoiding SLE paths
in Universality and renormalization
Fields Inst. Commun. 50 (2007) 199–224
Math Reviews MR2310306
7
Kenyon, Richard
Dominos and the Gaussian free field
Ann. Probab. 29 (2001) 1128–1137
Math Reviews MR1872739
8
Kenyon, Richard W. and Propp, James G. and Wilson, David B.
Trees and matchings
Electron. J. Combin. 7 (2000) Research Paper 25
Math Reviews MR1756162
9
Mercat, Christian
Discrete Riemann surfaces and the Ising model
Comm. Math. Phys. 218 (2001) 177–216
Math Reviews MR1824204
10
Ray, D. B. and Singer, I. M.
Analytic torsion for complex manifolds
Ann. of Math. 98 (1973) 154–177
Math Reviews MR0383463
11
Bers, Lipman
Finite-dimensional Teichmüller spaces and generalizations
Bull. Amer. Math. Soc. (N.S.) 5 (1981) 131–172
Math Reviews MR621883
12
Forster, Otto
Lectures on Riemann surfaces
Springer, 1991
Math Reviews MR1185074
13
Belinson, A. A. and Schechtman, V. V.
Determinant bundles and Virasoro algebras
Comm. Math. Phys. 118 (1988) 651–701
Math Reviews MR962493
14
Kontsevich, M. L.
The Virasoro algebra and Teichmüller spaces
Functional Anal. Appl. 21 (1987) 156–157
Math Reviews MR902301
15
Jost, Jürgen
Bosonic strings: a mathematical treatment
Amer. Math. Soc., 2001
Math Reviews MR1999211
16
Kodaira, Kunihiko
Complex manifolds and deformation of complex structures
Springer, 2005
Math Reviews MR2109686
17
Gardiner, Frederick P.
Teichmüller theory and quadratic differentials
John Wiley Sons Inc., 1987 , A Wiley-Interscience Publication
Math Reviews MR903027
18
Fay, John D.
Theta functions on Riemann surfaces
Springer, 1973 Lecture Notes in Mathematics, Vol. 352
Math Reviews MR0335789
19
Mumford, David
Tata lectures on theta. I
Birkhäuser, 1983
Math Reviews MR688651
20
Griffiths, Phillip and Harris, Joseph
Principles of algebraic geometry
John Wiley Sons Inc., 1994 Reprint of the 1978 original
Math Reviews MR1288523
21
Mumford, David
Tata lectures on theta. II
Birkhäuser, 1984
Math Reviews MR742776
22
Schramm, Oded and Sheffield, Scott
Contour lines of the two-dimensional discrete Gaussian free field
Acta Math. 202 (2009) 21–137
Math Reviews MR2486487
23
Sonoda, Hidenori
Functional determinants on punctured Riemann surfaces and their application to string theory
Nuclear Phys. B 294 (1987) 157–192
Math Reviews MR909430
24
Burghelea, D. and Friedlander, L. and Kappeler, T.
Meyer-Vietoris type formula for determinants of elliptic differential operators
J. Funct. Anal. 107 (1992) 34–65
Math Reviews MR1165865
25
Forman, Robin
Determinants of Laplacians on graphs
Topology 32 (1993) 35–46
Math Reviews MR1204404
26
Forman, Robin
Functional determinants and geometry
Invent. Math. 88 (1987) 447–493
Math Reviews MR884797
27
Rosenberg, Steven
The Laplacian on a Riemannian manifold
Cambridge Univ. Press, 1997 An introduction to analysis on manifolds
Math Reviews MR1462892
28
Le Jan, Yves
Markov loops, determinants and Gaussian fields
arXiv:math.PR/0612112
29
Handbook of mathematical functions with formulas, graphs, and mathematical tables
Dover Publications Inc., 1992 Reprint of the 1972 edition
Math Reviews MR94b:00012
30
Ahlfors, Lars V.
Complex analysis
McGraw-Hill Book Co., 1978 An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics
Math Reviews MR80c:30001
31
Ahlfors, Lars V.
Conformal invariants: topics in geometric function theory
McGraw-Hill Book Co., 1973 McGraw-Hill Series in Higher Mathematics
Math Reviews MR50 \#10211
32
Alvarez, Orlando
Theory of strings with boundaries: fluctuations, topology and quantum geometry
Nuclear Phys. B 216 (1983) 125–184
Math Reviews MR701643
33
Alvarez-Gaumé, Luis and Bost, Jean-Benoît and Moore, Gregory and Nelson, Philip and Vafa, Cumrun
Bosonization on higher genus Riemann surfaces
Comm. Math. Phys. 112 (1987) 503–552
Math Reviews MR908551
34
Appell, Paul and Kampé de Fériet, Joseph
Fonctions hypergéométriques et hypersphériques. Polynômes d'Hermite
Gauthier-Villars, 1926
35
Erdélyi, Arthur and Magnus, Wilhelm and Oberhettinger, Fritz and Tricomi, Francesco G.
Higher transcendental functions. Vol. I
Robert E. Krieger Publishing Co. Inc., 1981 Based on notes left by Harry Bateman
Math Reviews MR84h:33001a
36
Bauer, Michel and Bernard, Denis and Kytölä, Kalle
Multiple Schramm-Loewner evolutions and statistical mechanics martingales
J. Stat. Phys. 120 (2005) 1125–1163
Math Reviews MR2187598
37
Beffara, Vincent
The dimension of SLE curves
(2002) arXiv:math.PR/0211322
38
Beffara, Vincent
Hausdorff dimensions for SLE6
Ann. Probab. 32 (2004) 2606–2629
Math Reviews MR2078552
39
Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.
Infinite conformal symmetry in two-dimensional quantum field theory
Nuclear Phys. B 241 (1984) 333–380
Math Reviews MR86m:81097
40
Beliaev, Dmitri
Harmonic measure on random fractals
PhD thesis, KTH (2005)
41
Bertoin, Jean
Excursions of a BES0(d) and its drift term (0<d<1)
Probab. Theory Related Fields 84 (1990) 231–250
Math Reviews MR91e:60229
42
Bertoin, Jean
Complements on the Hilbert transform and the fractional derivative of Brownian local times
J. Math. Kyoto Univ. 30 (1990) 651–670
Math Reviews MR92f:60139
43
Biane, Philippe
Quelques propriétés du mouvement brownien dans un cône
Stochastic Process. Appl. 53 (1994) 233–240
Math Reviews MR95j:60129
44
Biane, Philippe
Intertwining of Markov semi-groups, some examples
in Séminaire de Probabilités, XXIX
Lecture Notes in Math. 1613 (1995) 30–36
Math Reviews MR98k:46117
45
Camia, Federico and Newman, Charles M.
The Full Scaling Limit of Two-Dimensional Critical Percolation
(2005) arXiv:math.PR/0504036
46
Carmona, Philippe and Petit, Frédérique and Yor, Marc
Beta-gamma random variables and intertwining relations between certain Markov processes
Rev. Mat. Iberoamericana 14 (1998) 311–367
Math Reviews MR2000b:60173
47
Cardy, John L.
Critical percolation in finite geometries
J. Phys. A 25 (1992) L201–L206
Math Reviews MR92m:82048
48
Cardy, John L.
Conformal invariance and percolation
(2001) arXiv:math-ph/0103018
49
Cardy, John L.
Crossing formulae for critical percolation in an annulus
J. Phys. A 35 (2002) L565–L572
Math Reviews MR2003i:82042
50
Cardy, John
Stochastic Loewner evolution and Dyson's circular ensembles
J. Phys. A 36 (2003) L379–L386
Math Reviews MR2004294
51
Cardy, John L.
Corrigendum: ``Stochastic Loewner evolution and Dyson's circular ensembles'' [J. Phys. A 36 (2003), no. 24, L379–L386 ]
J. Phys. A 36 (2003)
Math Reviews MR2025835
52
Cardy, John L.
ADE and SLE
(2006) arXiv:math-ph/0610030
53
Camia, Federico and Newman, Charles M.
The Full Scaling Limit of Two-Dimensional Critical Percolation
arXiv:math.PR/0504036
54
Chandrasekharan, K.
Elliptic functions
Springer, 1985
Math Reviews MR87e:11058
55
Charlaix, E. and Guyon, E. and Roux, S.
Permeability of a random array of fractures of widely varying apertures
Trans. Porous Media 2 (1987) 31–43
56
Branges, Louis (de)
A proof of the Bieberbach conjecture
Acta Math. 154 (1985) 137–152
Math Reviews MR772434
57
Di Francesco, Philippe and Mathieu, Pierre and Sénéchal, David
Conformal field theory
Springer, 1997
Math Reviews MR1424041
58
Dotsenko, Vl. S. and Fateev, V. A.
Conformal algebra and multipoint correlation functions in 2D statistical models
Nuclear Phys. B 240 (1984) 312–348
Math Reviews MR762194
59
Dubédat, Julien
SLE and triangles
Electron. Comm. Probab. 8 (2003) 28–42
Math Reviews MR2004c:60266
60
Dubédat, Julien
Reflected planar Brownian motions, intertwining relations and crossing probabilities
Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 539–552
Math Reviews MR2086013
61
Dubédat, Julien
Critical percolation in annuli and SLE6
Comm. Math. Phys. 245 (2004) 627–637
Math Reviews MR2 045 686
62
Dubédat, Julien
SLE(,) martingales and duality
Ann. Probab. 33 (2005) 223–243
Math Reviews MR2118865
63
Dubédat, Julien
Excursion decompositions for SLE and Watts' crossing formula
Probab. Theory Related Fields 134 (2006) 453–488
Math Reviews MR2226888
64
Dubédat, Julien
Commutation relations for SLE
Comm. Pure Applied Math. 60 (2007) 1792–1847
65
Dubédat, Julien
Euler integrals for commuting SLEs
Journal Statist. Phys. 123 (2006) 1183–1218
66
Dubédat, Julien
SLE partition functions, -regularization and Virasoro representations
in preparation (2006)
67
Dubédat, Julien
Construction of the c=-2 Malliavin-Kontsevich-Suhov measure
in preparation (2006)
68
Dubédat, Julien
Analytic torsion and dimers on surfaces
in preparation (2006)
69
Zhan, Dapeng
Stochastic Loewner Evolution in doubly connected domains
Probab. Theory Related Fields (2003)
70
Zhan, Dapeng
Random Loewner chains in Riemann surfaces
PhD thesis (2004)
71
Friedrich, Roland and Kalkkinen, J.
On conformal field theory and stochastic Loewner evolution
Nuclear Phys. B 687 (2004) 279–302
Math Reviews MR2059141
72
Friedrich, Roland and Werner, Wendelin
Conformal restriction, highest-weight representations and SLE
Comm. Math. Phys. 243 (2003) 105–122
Math Reviews MR2 020 222
73
Farkas, H. M. and Kra, I.
Riemann surfaces
Springer, 1992
Math Reviews MR1139765
74
Fomin, Sergey
Loop-erased walks and total positivity
Trans. Amer. Math. Soc. 353 (2001) 3563–3583
Math Reviews MR1837248
75
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 MR99c:60216
76
Grimmett, Geoffrey
Percolation
Springer, 1999
Math Reviews MR2001a:60114
77
Itô, Kiyosi and McKean, Henry P. Jr.
Diffusion processes and their sample paths
Springer, 1974 Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125
Math Reviews MR49 \#9963
78
Kac, Mark
Can one hear the shape of a drum?
Amer. Math. Monthly 73 (1966) 1–23
Math Reviews MR0201237
79
Kenyon, Richard
The asymptotic determinant of the discrete Laplacian
Acta Math. 185 (2000) 239–286
Math Reviews MR1819995
80
Kenyon, Richard
Local statistics of lattice dimers
Ann. Inst. H. Poincaré Probab. Statist. 33 (1997) 591–618
Math Reviews MR99b:82039
81
Kenyon, Richard
Conformal invariance of domino tiling
Ann. Probab. 28 (2000) 759–795
Math Reviews MR2002e:52022
82
Kenyon, Richard
An introduction to the dimer model
in School and Conference on Probability Theory
ICTP Lect. Notes, XVII (2004) 267–304
Math Reviews MR2198850
83
Kenyon, Richard and Wilson, David
Boundary Partitions in Trees and Dimers
(2006) arXiv:math.PR/0608422
84
Kesten, Harry
Percolation theory for mathematicians
Birkhäuser, 1982
Math Reviews MR84i:60145
85
Kesten, Harry
Scaling relations for 2D-percolation
Comm. Math. Phys. 109 (1987) 109–156
Math Reviews MR879034
86
Kleban, Peter and Zagier, Don
Crossing probabilities and modular forms
J. Statist. Phys. 113 (2003) 431–454
Math Reviews MR2 013 692
87
Kontsevich, Maxim and Suhov, Yuri
On Malliavin measures, SLE and CFT
(2006) arXiv:math-ph/0609056
88
Kozdron, Michael J. and Lawler, Gregory F.
Estimates of random walk exit probabilities and application to loop-erased random walk
Electron. J. Probab. 10 (2005) 1442–1467
Math Reviews MR2191635
89
Lawler, Gregory F.
An introduction to the stochastic Loewner evolution
in Random walks and geometry
(2004) 261–293
Math Reviews MR2087784
90
Lawler, Gregory F.
Loop-erased random walk
in Perplexing problems in probability
Progr. Probab. 44 (1999) 197–217
Math Reviews MR2000k:60092
91
Lawler, Gregory F.
Conformally invariant processes in the plane
Amer. Math. Soc., 2005
Math Reviews MR2129588
92
Le Gall, Jean-François
Mouvement brownien, cônes et processus stables
Probab. Theory Related Fields 76 (1987) 587–627
Math Reviews MR89b:60193
93
Le Gall, Jean-François
Une approche élémentaire des théorèmes de décomposition de Williams
in Séminaire de Probabilités, XX, 1984/85
Lecture Notes in Math. 1204 (1986) 447–464
Math Reviews MR89g:60219
94
Langlands, Robert and Pouliot, Philippe and Saint-Aubin, Yvan
Conformal invariance in two-dimensional percolation
Bull. Amer. Math. Soc. (N.S.) 30 (1994) 1–61
Math Reviews MR94e:82056
95
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
Values of Brownian intersection exponents. I. Half-plane exponents
Acta Math. 187 (2001) 237–273
Math Reviews MR2002m:60159a
96
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
Conformal invariance of planar loop-erased random walks and uniform spanning trees
Ann. Probab. 32 (2004) 939–995
Math Reviews MR2044671
97
Lawler, Gregory and Schramm, Oded and Werner, Wendelin
Conformal restriction: the chordal case
J. Amer. Math. Soc. 16 (2003) 917–955
Math Reviews MR1 992 830
98
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
On the scaling limit of planar self-avoiding walk
in Fractal geometry and application. A jubilee of Benoit Mandelbrot
AMS Proc. Symp. Pure Math. (2002)
99
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
Values of Brownian intersection exponents. III. Two-sided exponents
Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 109–123
Math Reviews MR2003d:60163
100
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
One-arm exponent for critical 2D percolation
Electron. J. Probab. 7 (2002)
Math Reviews MR2002k:60204
101
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
Values of Brownian intersection exponents. II. Plane exponents
Acta Math. 187 (2001) 275–308
Math Reviews MR2002m:60159b
102
Lawler, Gregory F. and Schramm, Oded and Werner, Wendelin
Analyticity of intersection exponents for planar Brownian motion
Acta Math. 189 (2002) 179–201
Math Reviews MR1961197
103
Lawler, Gregory F. and Werner, Wendelin
The Brownian loop soup
Probab. Theory Related Fields 128 (2004) 565–588
Math Reviews MR2045953
104
Lawler, Gregory F.
Conformally invariant processes in the plane
Amer. Math. Soc., 2005
Math Reviews MR2129588
105
Lawler, Gregory F.
The Laplacian-b random walk and the Schramm-Loewner evolution
to appear in Illinois J. Math.
106
Maier, Robert S.
On crossing event formulas in critical two-dimensional percolation
J. Statist. Phys. 111 (2003) 1027–1048
Math Reviews MR1 975 920
107
Mandelbrot, Benoit B.
The fractal geometry of nature
W. H. Freeman and Co., 1982
Math Reviews MR665254
108
McKean, Henry P. Jr. and Singer, I. M.
Curvature and the eigenvalues of the Laplacian
J. Differential Geometry 1 (1967) 43–69
Math Reviews MR0217739
109
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
110
Pitman, J. W.
One-dimensional Brownian motion and the three-dimensional Bessel process
Advances in Appl. Probability 7 (1975) 511–526
Math Reviews MR51 \#11677
111
Pitman, Jim and Yor, Marc
Bessel processes and infinitely divisible laws
in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980)
Lecture Notes in Math. 851 (1981) 285–370
Math Reviews MR82j:60149
112
Pitman, Jim and Yor, Marc
A decomposition of Bessel bridges
Z. Wahrsch. Verw. Gebiete 59 (1982) 425–457
Math Reviews MR84a:60091
113
Pemantle, Robin
Choosing a spanning tree for the integer lattice uniformly
Ann. Probab. 19 (1991) 1559–1574
Math Reviews MR92g:60014
114
Pinson, Haru T.
Critical percolation on the torus
J. Statist. Phys. 75 (1994) 1167–1177
Math Reviews MR95g:82046
115
Polyakov, A. M.
Quantum geometry of fermionic strings
Phys. Lett. B 103 (1981) 211–213
Math Reviews MR623210
116
Polyakov, A. M.
Quantum geometry of bosonic strings
Phys. Lett. B 103 (1981) 207–210
Math Reviews MR623209
117
Quillen, D.
Determinants of Cauchy-Riemann operators on Riemann surfaces
Functional Anal. Appl. 19 (1985) 31–34
118
Raghavan, R. and Henley, Christopher L. and Arouh, Scott L.
New two-color dimer models with critical ground states
J. Statist. Phys. 86 (1997) 517–550
Math Reviews MR98e:82018
119
Rogers, L. C. G. and Pitman, J. W.
Markov functions
Ann. Probab. 9 (1981) 573–582
Math Reviews MR82j:60133
120
Rohde, Steffen and Schramm, Oded
Basic properties of SLE
Ann. of Math. 161 (2005) 883–924
Math Reviews MR2153402
121
Rogers, L. C. G. and Williams, David
Diffusions, Markov processes, and martingales. Vol. 1
John Wiley Sons Ltd., 1994 Foundations
Math Reviews MR96h:60116
122
Revuz, Daniel and Yor, Marc
Continuous martingales and Brownian motion
Springer, 1999
Math Reviews MR2000h:60050
123
Schramm, Oded
Scaling limits of loop-erased random walks and uniform spanning trees
Israel J. Math. 118 (2000) 221–288
Math Reviews MR2001m:60227
124
Schramm, Oded
A percolation formula
Electron. Comm. Probab. 6 (2001) 115–120
Math Reviews MR2002h:60227
125
Schramm, Oded
Conformally invariant scaling limits (an overview and a collection of problems)
To appear in the ICM 2006 Madrid Proceedings (2006)
126
Segal, Graeme
The definition of conformal field theory
in Topology, geometry and quantum field theory
London Math. Soc. Lecture Note Ser. 308 (2004) 421–577
Math Reviews MR2079383
127
Saito, Mutsumi and Sturmfels, Bernd and Takayama, Nobuki
Gröbner deformations of hypergeometric differential equations
Springer, 2000
Math Reviews MR1734566
128
Smirnov, Stanislav and Werner, Wendelin
Critical exponents for two-dimensional percolation
Math. Res. Lett. 8 (2001) 729–744
Math Reviews MR2003i:60173
129
Sansone, Giovanni and Gerretsen, Johan
Lectures on the theory of functions of a complex variable. II: Geometric theory
Wolters-Noordhoff Publishing, Groningen, 1969
Math Reviews MR41 \#3714
130
Schramm, Oded and Sheffield, Scott
The harmonic explorer and its convergence to SLE(4).
(2003) arXiv:math.PR/0310210
131
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
132
Smirnov, Stanislav
Towards conformal invariance of 2d lattice models
To appear in the ICM 2006 Madrid Proceedings (2006)
133
Taylor, S. J. and Wendel, J. G.
The exact Hausdorff measure of the zero set of a stable process
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966) 170–180
Math Reviews MR35 \#1090
134
Varadhan, S. R. S. and Williams, R. J.
Brownian motion in a wedge with oblique reflection
Comm. Pure Appl. Math. 38 (1985) 405–443
Math Reviews MR87c:60066
135
Markov, Y. and Tarasov, V. and Varchenko, A.
The determinant of a hypergeometric period matrix
Houston J. Math. 24 (1998) 197–220
Math Reviews MR1690421
136
Villat, Henri
Le problème de Dirichlet dans une aire annulaire
Rend. circ. mat. Palermo (1912) 134–175
137
Virág, Bálint
Brownian beads
Probab. Theory Related Fields 127 (2003) 367–387
Math Reviews MR2 018 921
138
Werner, Wendelin
Critical exponents, conformal invariance and planar Brownian motion
in European Congress of Mathematics, Vol. II (Barcelona, 2000)
Progr. Math. 202 (2001) 87–103
Math Reviews MR2003f:60181
139
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
140
Werner, Wendelin
Girsanov's transformation for SLE(,) processes, intersection exponents and hiding exponents
Ann. Fac. Sci. Toulouse Math. 13 (2004) 121–147
Math Reviews MR2060031
141
Werner, Wendelin
Conformal restriction and related questions
Probab. Surv. 2 (2005) 145–190
Math Reviews MR2178043
142
Werner, Wendelin
SLEs as boundaries of clusters of Brownian loops
C. R. Math. Acad. Sci. Paris 337 (2003) 481–486
Math Reviews MR2023758
143
Werner, Wendelin
The conformally invariant measure on self-avoiding loops
(2005) arXiv:math.PR/0511605
144
Werner, Wendelin
SLE, conformal restriction, loops
in European Congress of Mathematics
(2005) 515–528
Math Reviews MR2185763
145
Williams, David
Path decomposition and continuity of local time for one-dimensional diffusions. I
Proc. London Math. Soc. 28 (1974) 738–768
Math Reviews MR50 \#3373
146
Watts, G. M. T.
A crossing probability for critical percolation in two dimensions
J. Phys. A 29 (1996) L363–L368
Math Reviews MR98a:82059
147
Wilson, David Bruce
Generating random spanning trees more quickly than the cover time
in Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996)
(1996) 296–303
Math Reviews MR1 427 525
148
Yoshida, Masaaki
Fuchsian differential equations
Friedr. Vieweg Sohn, 1987
Math Reviews MR90f:32025