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The asymmetric random cluster model and comparison of Ising and Potts models
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  • Published: July 2001

The asymmetric random cluster model and comparison of Ising and Potts models

  • Kenneth S. Alexander1 

Probability Theory and Related Fields volume 120, pages 395–444 (2001)Cite this article

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Abstract.

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model.

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Authors and Affiliations

  1. Department of Mathematics, DRB 155, University of Southern California, Los Angeles, CA 90089-1113, USA. e-mail: alexandr@math.usc.edu, , , , , , US

    Kenneth S. Alexander

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  1. Kenneth S. Alexander
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Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001

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Alexander, K. The asymmetric random cluster model and comparison of Ising and Potts models. Probab Theory Relat Fields 120, 395–444 (2001). https://doi.org/10.1007/PL00008788

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  • Issue Date: July 2001

  • DOI: https://doi.org/10.1007/PL00008788

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  • Mathematics Subject Classification (2000): Primary: 60K35; Secondary: 82B20, 82B43
  • Key words or phrases: Random cluster model – Potts model – Ising model – Potts lattice gas – FKG property – Dilution – Critical point
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