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The Wulff Crystal in Ising and Percolation Models : Ecole d'Etï½ de Probabilitï½s de Saint-Flour XXXIV - 2004 - RaphaÃ«l Cerf

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The Wulff Crystal in Ising and Percolation Models

Ecole d'Etï½ de Probabilitï½s de Saint-Flour XXXIV - 2004

By: RaphaÃ«l Cerf, Jean Picard (Editor)

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Paperback | 12 May 2006

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Three series of lectures were given at the 34th Probability Summer School in Saint-Flour (July 6-24, 2004), by the Professors Cerf, Lyons and Slade. We have decided to publish these courses separately. This volume contains the course of Professor Cerf. We cordially thank the author for his performance at the summer school, and for the redaction of these notes. 69 participants have attended this school. 35 of them have given a short lecture. The lists of participants and of short lectures are enclosed at the end of the volume. The Saint-Flour Probability Summer School was founded in 1971. Here are the references of Springer volumes which have been published prior to this one. All numbers refer to theLecture Notes in Mathematics series, except S-50 which refers to volume 50 of the Lecture Notes in Statistics series. 1971: vol 307 1980: vol 929 1990: vol 1527 1997: vol 1717 1973: vol 390 1981: vol 976 1991: vol 1541 1998: vol 1738 1974: vol 480 1982: vol 1097 1992: vol 1581 1999: vol 1781 1975: vol 539 1983: vol 1117 1993: vol 1608 2000: vol 1816 1976: vol 598 1984: vol 1180 1994: vol 1648 2001: vol 1837 & 1851 1977: vol 678 1985/86/87: vol 1362 & S-50 2002: vol 1840 1978: vol 774 1988: vol 1427 1995: vol 1690 2003: vol 1869 1979: vol 876 1989: vol 1464 1996: vol 1665 2004: vol 1878 & 1879 Further details can be found on the summer school web site

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Non-Fiction Mathematics Probability & Statistics Optimisation Linear Programming Science Physics Calculus & Mathematical Analysis

Introduction
Phase coexistence and subadditivity	p. 3
Water and oil	p. 3
Subadditivity	p. 5
Cramér's theorem	p. 6
Presentation of the models
Ising model	p. 15
Construction of the model	p. 15
First asymptotics	p. 16
Phase transition	p. 18
Proofs of the heuristics	p. 18
Bernoulli percolation	p. 25
The probability space	p. 25
Order on ¿	p. 26
Phase transition	p. 26
FK or random cluster model	p. 31
Finite volume FK measures	p. 31
Phase transition	p. 32
FK Ising coupling	p. 33
Boundary conditions	p. 41
Main results
The Wulff crystal	p. 45
Ising model	p. 45
Bernoulli percolation	p. 55
FK percolation	p. 58
What do we know about the Wulff crystal?	p. 60
Bibliographical comments	p. 62
Large deviation principles
Large deviation theory	p. 67
Main definitions	p. 67
I-tightness	p. 68
Contraction principle	p. 72
Varadhan's lemma	p. 73
Surface large deviation principles	p. 75
Surface energy	p. 75
The empirical magnetization	p. 78
Minimal surfaces	p. 79
The cluster shapes	p. 82
FK percolation	p. 83
Volume large deviations	p. 85
Bernoulli percolation	p. 85
FK percolation	p. 92
Ising model	p. 96
Fundamental probabilistic estimates
Coarse graining	p. 105
The good blocks	p. 105
Extension to FK measures	p. 110
The rescaled lattice	p. 112
Two rough estimates	p. 114
Decoupling	p. 117
Half-space clusters	p. 117
Decoupling lemma	p. 121
Surface tension	p. 129
Existence	p. 129
Finite volume definition	p. 133
Basic properties	p. 136
Separating sets	p. 141
What do we know about the surface tension?	p. 145
Interface estimate	p. 147
Interface lemma	p. 147
Near the boundary	p. 152
Percolation setting	p. 153
Lower bound	p. 155
Basic geometric tools
Sets of finite perimeter	p. 159
Basic definitions	p. 159
Covering and differentiating	p. 160
Caccioppoli sets	p. 164
Two technical results	p. 167
Surface energy	p. 173
Definition	p. 175
Lowersemicontinuity and compactness	p. 178
Covering	p. 178
Polyhedral approximation	p. 181
The Wulff theorem	p. 189
Statement of the theorem	p. 189
The anisotropic isoperimetric inequality	p. 190
The proof of Brothers and Morgan	p. 192
Stability of the Wulff crystal	p. 197
Final steps of the proofs
LDP for the cluster shapes	p. 203
Coarse grained image	p. 204
Exponential contiguity	p. 206
Local upper bound	p. 209
Lower bound	p. 211
Enhanced upper bound	p. 215
A lemma from discrete geometry	p. 215
Uniform large deviation upper bounds	p. 216
Conclusion of the proof	p. 221
Extension to FK percolation	p. 227
LDP for FK percolation	p. 229
Coarse grained image	p. 229
Exponential contiguity	p. 233
Local upper bound	p. 235
Lower bound	p. 236
LDP for Ising	p. 241
Coarse grained image	p. 241
Exponential contiguity	p. 243
Local upper bound	p. 246
Lower bound	p. 249
References	p. 253
Index	p. 259
List of participants	p. 261
List of short lectures	p. 263
Table of Contents provided by Publisher. All Rights Reserved.

The Wulff Crystal in Ising and Percolation Models

Ecole d'Etï½ de Probabilitï½s de Saint-Flour XXXIV - 2004

At a Glance

Paperback

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Defective items

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