Cerf / Paris-Sud / Orsay The Wulff Crystal in Ising and Percolation Models

1. Auflage 2006
ISBN: 978-3-540-34806-1
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 267 Seiten

Probability Theory and Stochastic Processes - Short description currently not available.

Autoren/Hrsg.

Cerf, Raphaël

Paris-Sud, Université

Orsay,

France,

Inhaltsverzeichnis

1;Foreword;6
2;Preface;7
3;Contents;9
4;Part I Introduction;13
4.1;1 Phase coexistence and subadditivity;14
4.1.1;1.1 Water and oil;14
4.1.2;1.2 Subadditivity;16
4.1.3;1.3 Cram ´ er’s theorem;17
5;Part II Presentation of the models;24
5.1;2 Ising model;25
5.1.1;2.1 Construction of the model;25
5.1.2;2.2 First asymptotics;26
5.1.3;2.3 Phase transition;28
5.1.4;2.4 Proofs of the heuristics;28
5.2;3 Bernoulli percolation;35
5.2.1;3.1 The probability space;35
5.2.2;3.2 Order on;36
5.2.3;3.3 Phase transition;36
5.3;4 FK or random cluster model;40
5.3.1;4.1 Finite volume FK measures;40
5.3.2;4.2 Phase transition;41
5.3.3;4.3 FK Ising coupling;42
5.3.4;4.4 Boundary conditions;50
6;Part III Main results;52
6.1;The Wulff crystal;53
6.1.1;5.1 Ising model;53
6.1.2;5.2 Bernoulli percolation;63
6.1.3;5.3 FK percolation;66
6.1.4;5.4 What do we know about the Wul. crystal?;68
6.1.5;5.5 Bibliographical comments;70
7;Part IV Large deviation principles;73
7.1;6 Large deviation theory;74
7.1.1;6.1 Main de.nitions;74
7.1.2;6.2;75
7.1.3;–tightness;75
7.1.4;6.3 Contraction principle;79
7.1.5;6.4 Varadhan’s lemma;80
7.2;7 Surface large deviation principles;82
7.2.1;7.1 Surface energy;82
7.2.2;7.2 The empirical magnetization;85
7.2.3;7.3 Minimal surfaces;86
7.2.4;7.4 The cluster shapes;89
7.2.5;7.5 FK percolation;90
7.3;8 Volume large deviations;92
7.3.1;8.1 Bernoulli percolation;92
7.3.2;8.2 FK percolation;99
7.3.3;8.3 Ising model;103
8;Part V Fundamental probabilistic estimates;110
8.1;9 Coarse graining;111
8.1.1;9.1 The good blocks;111
8.1.2;9.2 Extension to FK measures;116
8.1.3;9.3 The rescaled lattice;118
8.1.4;9.4 Two rough estimates;120
8.2;10 Decoupling;123
8.2.1;10.1 Half–space clusters;123
8.2.2;10.2 Decoupling lemma;127
8.3;11 Surface tension;134
8.3.1;11.1 Existence;134
8.3.2;11.2 Finite volume de.nition;138
8.3.3;11.3 Basic properties;141
8.3.4;11.4 Separating sets;146
8.3.5;11.5 What do we know about the surface tension?;150
8.4;12 Interface estimate;151
8.4.1;12.1 Interface lemma;151
8.4.2;12.2 Near the boundary;156
8.4.3;12.3 Percolation setting;157
8.4.4;12.4 Lower bound;159
9;Part VI Basic geometric tools;160
9.1;13 Sets of finite perimeter;161
9.1.1;13.1 Basic de.nitions;161
9.1.2;13.2 Covering and di.erentiating;162
9.1.3;13.3 Caccioppoli sets;166
9.1.4;13.4 Two technical results;169
9.2;14 Surface energy;175
9.2.1;14.1 De.nition;177
9.2.2;14.2 Lowersemicontinuity and compactness;180
9.2.3;14.3 Covering;180
9.2.4;14.4 Polyhedral approximation;183
9.3;15 The Wuff theorem;191
9.3.1;15.1 Statement of the theorem;191
9.3.2;15.2 The anisotropic isoperimetric inequality;192
9.3.3;15.3 The proof of Brothers and Morgan;194
9.3.4;15.4 Stability of the Wul. crystal;199
10;Part VII Final steps of the proofs;202
10.1;16 LDP for the cluster shapes;203
10.1.1;16.1 Coarse grained image;204
10.1.2;16.2 Exponential contiguity;206
10.1.3;16.3 Local upper bound;209
10.1.4;16.4 Lower bound;211
10.2;17 Enhanced upper bound;215
10.2.1;17.1 A lemma from discrete geometry;215
10.2.2;17.2 Uniform large deviation upper bounds;216
10.2.3;17.3 Conclusion of the proof;221
10.2.4;17.4 Extension to FK percolation;227
10.3;18 LDP for FK percolation;229
10.3.1;18.1 Coarse grained image;229
10.3.2;18.2 Exponential contiguity;233
10.3.3;18.3 Local upper bound;235
10.3.4;18.4 Lower bound;236
10.4;19 LDP for Ising;240
10.4.1;19.1 Coarse grained image;240
10.4.2;19.2 Exponential contiguity;242
10.4.3;19.3 Local upper bound;245
10.4.4;19.4 Lower bound;248
11;References;252
12;Index;258
13;List of participants;260
14;List of short lectures;262