Elsevier

Physica

Volume 57, Issue 4, 15 February 1972, Pages 536-564
Physica

On the random-cluster model: I. Introduction and relation to other models

https://doi.org/10.1016/0031-8914(72)90045-6Get rights and content

Abstract

The random-cluster model is defined as a model for phase transitions and other phenomena in lattice systems, or more generally in systems with a graph structure. The model is characterized by a (probability) measure on a graph and a real parameter κ. By specifying the value of κ to 1, 2, 3, 4, … is shown that the model covers the percolation model, the Ising model, the Ashkin-Teller-Potts model with 3, 4, … states per atom, respectively, and thereby, contains information on graph-colouring problems; in the limit κ ↓ 0 it describes linear resistance networks. It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.

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