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The random-cluster model on the complete graph
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  • Published: September 1996

The random-cluster model on the complete graph

  • B. Bollobás1,
  • G. Grimmett2 &
  • S. Janson3 

Probability Theory and Related Fields volume 104, pages 283–317 (1996)Cite this article

  • 398 Accesses

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Summary

The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called ‘mean-field’. In this study of the mean-field random-cluster model with parametersp=λ/n andq, we show that its properties for any value ofq∈(0, ∞) may be derived from those of an Erdős-Rényi random graph. In this way we calculate the critical pointλ c (q) of the model, and show that the associated phase transition is continuous if and only ifq≦2. Exact formulae are given forλ C (q), the density of the largest component, the density of edges of the model, and the ‘free energy’. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq≧2. Equivalent results are obtained for a ‘fixed edge-number’ random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).

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Author information

Authors and Affiliations

  1. DPMMS, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, UK

    B. Bollobás

  2. Statistical Laboratory, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, UK

    G. Grimmett

  3. Department of Mathematics, Uppsala University, P.O. Box 480, S-751 06, Uppsala, Sweden

    S. Janson

Authors
  1. B. Bollobás
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  2. G. Grimmett
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  3. S. Janson
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Bollobás, B., Grimmett, G. & Janson, S. The random-cluster model on the complete graph. Probab. Th. Rel. Fields 104, 283–317 (1996). https://doi.org/10.1007/BF01213683

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  • Received: 07 December 1994

  • Revised: 08 September 1995

  • Issue Date: September 1996

  • DOI: https://doi.org/10.1007/BF01213683

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Mathematics Subject Classification (1991)

  • 05C80
  • 60K35
  • 82B20
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