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.
Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics
Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates
S.R.S. Varadhan
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Geoffrey Grimmett
The Random-Cluster Model With 37 Figures
ABC
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, broadcasti