Geoffrey R. Grimmett

The Random-Cluster Model (eBook, PDF)

• Format: PDF

Produktbeschreibung

The random-cluster model has emerged in recent years as a key tool in the mathematical study of ferromagnetism. It may be viewed as an extension of percolation to include Ising and Potts models, and its analysis is a mix of arguments from probability and geometry. This systematic study includes accounts of the subcritical and supercritical phases, together with clear statements of important open problems. There is an extensive treatment of the first-order (discontinuous) phase transition, as well as a chapter devoted to applications of the random-cluster method to other models of statistical physics.

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 378
• Erscheinungstermin: 13. Dezember 2006
• Englisch
• ISBN-13: 9783540328919
• Artikelnr.: 37369369

Autorenporträt

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.

Inhaltsangabe

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.

Rezensionen

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)