On the Gibbs Phase Rule in the Pirogov–Sinai Regime

Abstract

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.