The Edwards-Sokal coupling is a well known method that links the uniqueness
of Gibbs measure in the Potts model with the lack of percolation under the
associated random cluster measure. Comparing stochastically this random
cluster measure with the Bernoulli bond percolation measure, it is possible
to estimate the critical temperature of the Potts model. This coupling has
also been used in other models such as the Ashkin-Teller model, the
disordered ferromagnetic Potts model, the Edwards-Anderson spin-glass model
and the Widom-Rowlinson model. These issues are developed in [1] and [2].
In this talk we show how to use this method to prove the positive
correlation property for the q-state clock model and to find a temperature
regime where there is lack of uniqueness of Gibbs measure.
This is a joint work with P. Ferrari and I. Armendáriz.
[1] Geoffrey Grimmett; The random-cluster model, volumen 333 de Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], Springer-Verlag, Berlin, 2006
[2] Georgii, Hans-Otto and Häggström, Olle and Maes, Christian;The random
geometry of equilibrium phases, volumen 18 de Phase transitions and critical
phenomena, Academic Press, San Diego, 2001.