Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles

In this paper we study a continuum version of the Potts model. Particles are points in R^d, with a spin which may take S possible values, S being at least 3. Particles with different spins repel each other via a Kac pair potential. In mean field, for any inverse temperature there is a value of the chemical potential at which S+1 distinct phases coexist. For each mean field pure phase, we introduce a restricted ensemble which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, we get uniqueness and exponential decay of correlations when the range of the interaction is large enough. In a second paper, we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.Comment: 72 pages, 1 figur

Coexistence of ordered and disordered phases in Potts models in the continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may take $S\ge 3$ possible values. Particles with different spins repel each other via a Kac pair potential of range \ga^{-1}, \ga>0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist $S+1$ mutually distinct DLR measures.Comment: 57 pages, 1 figur

First-Order Phase Transition in Potts Models with finite-range interactions

We consider the $Q$-state Potts model on $\mathbb Z^d$, $Q\ge 3$, $d\ge 2$, with Kac ferromagnetic interactions and scaling parameter \ga. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for \ga small enough there is a value of the temperature at which coexist $Q+1$ Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for $d=2$, Q=3, in contrast with the case of nearest-neighbor interactions for which available results indicate a second order phase transition. Putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction.Comment: Soumis pour publication a Journal of statistical physics - version r\'{e}vis\'{e}