The Random-Cluster Model

  • Geoffrey R. Grimmett
Book

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

Table of contents

  1. Front Matter
    Pages i-xiii
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  2. Random-Cluster Measures
    Pages 1-18
  3. Monotonic Measures
    Pages 19-36
  4. Fundamental Properties
    Pages 37-66
  5. Infinite-Volume Measures
    Pages 67-97
  6. Phase Transition
    Pages 98-132
  7. In Two Dimensions
    Pages 133-166
  8. Duality in Higher Dimensions
    Pages 167-221
  9. Dynamics of Random-Cluster Models
    Pages 222-256
  10. Flows in Poisson Graphs
    Pages 257-275
  11. On Other Graphs
    Pages 276-319
  12. Graphical Methods for Spin Systems
    Pages 320-340
  13. Back Matter
    Pages 341-379
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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

  • Geoffrey R. Grimmett
    • 1
  1. 1.Statistical Laboratory, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-32891-9
  • Copyright Information Springer Berlin Heidelberg 2006
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-32890-2
  • Online ISBN 978-3-540-32891-9
  • Series Print ISSN 0072-7830
  • About this book