Rigorous analysis of singularities and absence of analytic continuation at first-order phase-transition points in lattice-spin models.

We report about two new rigorous results on the nonanalytic properties of thermodynamic potentials at first-order phase transition. For lattice models (d>or=2) with arbitrary finite state space, finite-range interactions which have two ground states, we prove that the pressure has no analytic continuation at the first-order phase-transition point, under the only further assumptions that the Peierls condition is satisfied for the ground states and that the temperature is sufficiently low. For Ising models with Kac potentials J(gamma)(x)=gamma(d)phi(gammax), where 00) and analyticity in the mean field limit (gamma SE pointing arrow 0).