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

TL;DR

This paper investigates a continuum Potts model with particles in R^d, demonstrating uniqueness and exponential decay of correlations in restricted ensembles for large interaction ranges, paving the way for proving phase coexistence.

Contribution

It introduces a continuum Potts model with Kac interactions, establishing uniqueness and exponential decay results for restricted ensembles, which is novel in this context.

Findings

Uniqueness of Gibbs measures in restricted ensembles

Exponential decay of correlations for large interaction range

Existence of multiple coexisting phases in the model

Abstract

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.