The Random-Cluster Model

By: Geoffrey R. Grimmett

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Hardcover | 1 June 2006

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396 Pages

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Therandom-clustermodelwasinventedbyCees[Kees]FortuinandPietKasteleyn around 1969 as a uni?cation of percolation, Ising, and Potts models, and as an extrapolation of electrical networks. Their original motivation was to harmonize the series and parallel laws satis?ed by such systems. In so doing, they initiated a study in stochastic geometry which has exhibited beautiful structure in its own right, and which has become a central tool in the pursuit of one of the oldest challenges of classical statistical mechanics, namely to model and analyse the ferromagnet and especially its phase transition. The importance of the model for probability and statistical mechanics was not fully recognized until the late 1980s. There are two reasons for this period of dormancy. Although the early publications of 1969-1972 contained many of the basic properties of the model, the emphasis placed there upon combinatorial aspects may have obscured its potential for applications. In addition, many of the geometrical arguments necessary for studying the model were not known prior to 1980, but were developed during the 'decade of percolation' that began 1 then. In 1980 was published the proof that p = for bond percolation on the c 2 square lattice, and this was followed soon by Harry Kesten's monograph on t- dimensionalpercolation. Percolationmovedintohigherdimensionsaround1986, and many of the mathematical issues of the day were resolved by 1989. Interest in the random-cluster model as a tool for studying the Ising/Potts models was rekindled around 1987.

Industry Reviews

From the reviews:

"This book constitutes an authoritative treatise of the random cluster model and its applications. It will be indispensable for anyone working on this and related subjects and can provide the basis for a graduate course or seminar." (Anton Bovier, Zentralblatt MATH, Vol. 1122 (24), 2007)

"This text is the first reference book devoted to the random-cluster model. The author knows the matter very well, since he is one of the first mathematicians who worked on this model. ... This is a very useful book, because it is very well written and collects results that are currently disseminated in the literature." (Olivier Garet, Mathematical Reviews, Issue 2007 m)

"The random-cluster model was discovered in the late 1960s by P. W. Kasteleyn and his doctoral student C. M. Fortuin ... . can be used in two ways: as an easily accessible introduction to the field for the newcomer, and as a valuable reference work of the present state of the art for the expert. The book is a must for anyone interested in percolation and mathematical Statistical Mechanics, and will certainly continue the success ... ." (Hans-Otto Georgii, Combinatorics, Probability and Computing, Vol. 17, 2008)

"Readership: Probabilists, ranging from graduate students in general to experts on the random-cluster and related models. ... The random-cluster model is a generalization of standard bond percolation. ... In the random-cluster model, the probability distribution is perturbed by a factor q raised to the number of connected components. ... The clarity of the exposition makes the book ideal for graduate students and other novices to the model." (Olle Haeggstroem, International Statistical Review, Vol. 75 (2), 2007)

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Published: 17th November 2010

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Random-Cluster Measures	p. 1
Introduction	p. 1
Random-cluster model	p. 4
Ising and Potts models	p. 6
Random-cluster and Ising/Potts models coupled	p. 8
The limit as q [down arrow] 0	p. 13
Basic notation	p. 15
Monotonic Measures	p. 19
Stochastic ordering of measures	p. 19
Positive association	p. 25
Influence for monotonic measures	p. 30
Sharp thresholds for increasing events	p. 33
Exponential steepness	p. 35
Fundamental Properties	p. 37
Conditional probabilities	p. 37
Positive association	p. 39
Differential formulae and sharp thresholds	p. 40
Comparison inequalities	p. 43
Exponential steepness	p. 49
Partition functions	p. 53
Domination by the Ising model	p. 57
Series and parallel laws	p. 61
Negative association	p. 63
Infinite-Volume Measures	p. 67
Infinite graphs	p. 67
Boundary conditions	p. 70
Infinite-volume weak limits	p. 72
Infinite-volume random-cluster measures	p. 78
Uniqueness via convexity of pressure	p. 85
Potts and random-cluster models on infinite graphs	p. 95
Phase Transition	p. 98
The critical point	p. 98
Percolation probabilities	p. 102
Uniqueness of random-cluster measures	p. 107
The subcritical phase	p. 110
Exponential decay of radius	p. 113
Exponential decay of volume	p. 119
The supercritical phase and the Wulff crystal	p. 122
Uniqueness when q < 1	p. 131
In Two Dimensions	p. 133
Planar duality	p. 133
The value of the critical point	p. 138
Exponential decay of radius	p. 143
First-order phase transition	p. 144
General lattices in two dimensions	p. 152
Square, triangular, and hexagonal lattices	p. 154
Stochastic Lowner evolutions	p. 164
Duality in Higher Dimensions	p. 167
Surfaces and plaquettes	p. 167
Basic properties of surfaces	p. 169
A contour representation	p. 173
Polymer models	p. 179
Discontinuous phase transition for large q	p. 182
Dobrushin interfaces	p. 195
Probabilistic and geometric preliminaries	p. 199
The law of the interface	p. 202
Geometry of interfaces	p. 208
Exponential bounds for group probabilities	p. 215
Localization of interface	p. 218
Dynamics of Random-Cluster Models	p. 222
Time-evolution of the random-cluster model	p. 222
Glauber dynamics	p. 224
Gibbs sampler	p. 225
Coupling from the past	p. 227
Swendsen-Wang dynamics	p. 230
Coupled dynamics on a finite graph	p. 232
Box dynamics with boundary conditions	p. 237
Coupled dynamics on the infinite lattice	p. 240
Simultaneous uniqueness	p. 255
Flows in Poisson Graphs	p. 257
Potts models and flows	p. 257
Flows in the Ising model	p. 262
Exponential decay for the Ising model	p. 273
The Ising model in four and more dimensions	p. 274
On Other Graphs	p. 276
Mean-field theory	p. 276
On complete graphs	p. 277
Main results for the complete graph	p. 281
The fundamental proposition	p. 284
The size of the largest component	p. 286
Proofs of main results for complete graphs	p. 289
The nature of the singularity	p. 295
Large deviations	p. 296
On a tree	p. 299
The critical point for a tree	p. 305
(Non-)uniqueness of measures on trees	p. 313
On non-amenable graphs	p. 315
Graphical Methods for Spin Systems	p. 320
Random-cluster representations	p. 320
The Potts model	p. 321
The Ashkin-Teller model	p. 326
The disordered Potts ferromagnet	p. 330
The Edwards-Anderson spin-glass model	p. 333
The Widom-Rowlinson lattice gas	p. 337
The Origins of FK(G)	p. 341
List of Notation	p. 350
References	p. 353
Index	p. 373
Table of Contents provided by Ingram. All Rights Reserved.

The Random-Cluster Model

At a Glance

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Industry Reviews

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You Can Find This Book In

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