Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles

Abstract

In this paper we study a continuum version of the Potts model, where particles are points in R d , 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 γ −1 , γ > 0. In mean field, for any inverse temperature β there is a value of the chemical potential λ β 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, [9], we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper, [8], we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S + 1 extremal DLR measures.

Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

“…We start with the basic definitions. They are quite standard and consistent with those of the companion paper [7].…”

“…In the following sections we will see that the proof of Theorem 1.1 follows from Theorem 3.2 and Theorem 3.1 of [7] using the same general arguments as in [16] for the analogous proof in the LMP model. In Part II we will prove Theorem 3.2.…”

“…Proof of Theorem 3.8. By Theorem 3.2 the contour weights satisfy the assumptions in Theorem 3.1 of [7] which can then be applied. It then follows that there is a coupling dP(q , Γ , q , Γ ) of dp…”

“…Despite the simplicity of the model its thermodynamics, which is defined by minimizing F mf β,λ (ρ) over ρ ∈ R S + , has a rather interesting structure. In [10] and [7] it is proved that the resulting phase diagram is characterized by a critical curve λ = λ β , β > 0, as in Figure 1. so that the total density of the state ρ (S+1) is smaller than the total density in any of the ordered critical points ρ (k) , k ≤ S, which is in fact equal to b * .…”

“…When λ > λ β , only the ordered states survive and there are S minimizers, when λ < λ β , only the disordered state survives and there is a unique minimizer. Therefore when crossing vertically the critical curve the total density jumps, a phenomenon which can be related to magnetostriction as argued in [7]. where q = (..., r i , s i , ...), i = 1, .., n, r i ∈ R d , s i ∈ {1, .., S}, is a finite configuration of particles with spin; λ the chemical potential and V γ (r i , r j ) = γ d V (γr i , γr j ), V (r, r ) a symmetric probability kernel, say with range 1.…”