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© 2006

The Random-Cluster Model

  • Sequel to G. Grimmett's famous and influential book on Percolation (Grundlehren der mathematischen Wissenschaften 321)

  • Author is leader in the field, and known as masterly expositor

  • Includes some history of the subject from both mathematics and physics angles

Book

Part of the Grundlehren der mathematischen Wissensch book series (GL, volume 333)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Pages 19-36
  3. Pages 98-132
  4. Pages 133-166
  5. Pages 257-275
  6. Pages 276-319
  7. Back Matter
    Pages 341-379

About this book

Introduction

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.

Keywords

Ising model Potts model Random-cluster model ferromagnetism geometry percolation

Authors and affiliations

  1. 1.Statistical Laboratory, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

About the authors

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.

Bibliographic information

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