The Random-Cluster Model
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
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the random-cluster model
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Geoffrey Grimmett
The Random-Cluster Model With 37 Figures
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Geoffrey R. Grimmett University of Cambridge Statistical Laboratory Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB United Kingdom E-mail: [email protected]
Library of Congress Control Number: 2006925087 Mathematics Subject Classification (2000): 60K35, 82B20, 82B43 ISSN 0072-7830 ISBN-10 3-540-32890-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-32890-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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FK Kees Fortuin (1971)
Piet Kasteleyn (1968)
Preface
The random-cluster model was invented by Cees [Kees] Fortuin and Piet Kasteleyn around 1969 as a unification of percolation, Ising, and Potts models, and as an extrapolation of electrical networks. Their original motivation was to harmonize the series and parallel laws satisfied 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 then. In 1980 was published the proof that pc = 12 for bond percolation on the square lattice, and this was followed soon by Harry Kesten’s monograph on twodimensional percolat
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