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We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure µ = ν. Both ν and µ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:(1) For all ν and µ, νS(t) is Gibbs for small t.(2) If both ν and µ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.(3) If ν has a low non-zero temperature and a zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.(4) If ν has a low non-zero temperature and a non-zero magnetic field and µ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where µ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

We prove that various SO(n)-invariant n-vector models with interactions which have a deep and narrow enough minimum have a first-order transition in the temperature. The result holds in dimension two or more, and is independent on the nature of the low-temperature phase.Recently Blöte, Guo and Hilhorst [2], extending earlier work by Domany, Schick and Swendsen [4] on 2-dimensional classical XY-models, performed a numerical study of 2-dimensional n-vector models with non-linear interactions. For sufficiently strong values of the non-linearity, they found the presence of a first-order transition in temperature. In [4] a heuristic explanation of this first-order behavior, based on a similarity with the high-q Potts model, was suggested, explaining the numerical results. A further confirmation of this transition was found by Caracciolo and Pellisetto [1], who 1

We consider various sufficiently nonlinear sigma models for nematic liquid crystal ordering of RP N −1 type and of lattice gauge type with continuous symmetries. We rigorously show that they exhibit a first-order transition in the temperature. The result holds in dimension 2 or more for the RP N −1 models and in dimension 3 or more for the lattice gauge models. In the twodimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.

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