Random-Cluster Representations in the Study of Phase Transitions

O. Haggstrom

1998, v.4, №3, 275-321


The Fortuin - Kasteleyn random-cluster model has, during the last ten years, proved to be a highly useful probabilistic device for studying the phase transition behaviour of Ising and Potts models. In this survey paper, a detailed description is given of how this is accomplished. It is then shown how the same set of ideas can be used to study phase transitions in various other systems: the Ising model with random interactions, the Ashkin - Teller model, subshifts of finite type, and certain point processes. The tools used include coupling, stochastic domination, and percolation.

Keywords: random-cluster model,Ising model,Potts model,percolation,coupling,stochastic domination,Ashkin\tire Teller model,subshift of finite type,Widom - Rowlinson model


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