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, where particles are points in R-d supercript stop, d >= 2, with a spin which may take S >= 3 possible values. Particles with different spins repel each other via a Kac pair potential of range gamma(-1), gamma > 0. In mean field, for any inverse temperature beta there is a value of the chemical potential lambda (beta) at which S+1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase 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 (Dobrushin and Shlosman in J. Stat. Phys. 46(5-6):983-1014, 1987), we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for gamma small enough which implies uniqueness and exponential decay of correlations. In a second paper (De Masi et al. in Coexistence of ordered and disordered phases in Potts models in the continuum, 2008), we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.