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Journal of Statistical Physics

, Volume 122, Issue 4, pp 647–670 | Cite as

Random Cluster Models on the Triangular Lattice

  • L. Chayes
  • H. K. Lei
Article

Abstract

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the q > 1 random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.

Key Words

percolation random cluster models Potts models star–triangle relations FKG inequalities 

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Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • L. Chayes
    • 1
  • H. K. Lei
    • 1
  1. 1.Department of MathematicsUCLALos AngelesUSA

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