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The Random-Cluster Model

Geoffrey R. Grimmett (Autor)

Buch | Hardcover

XIII, 378 Seiten

2006
Springer Berlin (Verlag)
978-3-540-32890-2 (ISBN)

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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.

PhD (Oxford 1974) under supervision of John Hammersley and Dominic Welsh. Member of the Mathematics Department of Bristol University (1976-1992), and subsequently appointed to the Professorship of Mathematics Statistics at Cambridge University. Author of around 100 articles and five books in probability and related fields, including Percolation (Springer 1999), Probability and Random Processes (with David Stirzaker, Oxford University Press 2001). Managing Editor of "Probability Theory and Related Fields", 2001-2005.

Random-Cluster Measures.- Monotonic Measures.- Fundamental Properties.- Infinite-Volume Measures.- Phase Transition.- In Two Dimensions.- Duality in Higher Dimensions.- Dynamics of Random-Cluster Models.- Flows in Poisson Graphs.- On Other Graphs.- Graphical Methods for Spin Systems.

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 Häggström, International Statistical Review, Vol. 75 (2), 2007)

Erscheint lt. Verlag	13.6.2006
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Zusatzinfo	XIII, 378 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	710 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Themenwelt	Naturwissenschaften ► Physik / Astronomie
Schlagworte	ferromagnetism • Geometry • Ising Model • percolation • Potts model • Random-cluster model • Zufall / Random (Statistik) • Zufall (Statistik)
ISBN-10	3-540-32890-4 / 3540328904
ISBN-13	978-3-540-32890-2 / 9783540328902
Zustand	Neuware