On the Gibbs Phase Rule in the Pirogov–Sinai Regime

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We consider extended Pirogov–Sinai models including lattice and continuum particle systems with Kac potentials. Call Λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term −λα. We suppose that a Pirogov–Sinai phase transition with order parameter α occurs at Λ = 0, and that there are two distinct classes of DLR measures, the plus and the minus DLR measures, with the expectation of α respectively positive and negative in the two classes. We then prove that Λ = 0 is the only point in an interval I of values of Λ centered at 0 where this occurs, namely the expected value of α is positive, respectively negative, in all translational invariant DLR measures at {Λ > 0}⊓I and {Λ<0}⊓I.

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