Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits, Multiple Histories, and Loss of Gibbsianness

Abstract

We consider the Curie-Weiss model at initial temperature 0<β −1≤∞ in vanishing external field evolving under a Glauber spin-flip dynamics with temperature 0<β′−1≤∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity (bad points). We provide a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending earlier work for the case of independent spin-flip dynamics.For initial temperature β −1>1 we prove that the time-evolved measure stays Gibbs forever, for any (possibly low) temperature of the dynamics.In the regime of heating to low-temperatures from even lower temperatures, 0<β −1<min {β′−1,1} we prove that the time-evolved measure is Gibbs initially and becomes non-Gibbs after a sharp transition time. We find this regime is further divided into a region where only symmetric bad configurations exist, and a region where this symmetry is broken.In the regime of further cooling from low-temperatures, β′−1<β −1<1 there is always symmetry-breaking in the set of bad configurations. These bad configurations are created by a new mechanism which is related to the occurrence of periodic orbits for the vector field which describes the dynamics of Euler-Lagrange equations for the path large deviation functional for the order parameter.To our knowledge this is the first example of the rigorous study of non-Gibbsian phenomena related to cooling, albeit in a mean-field setup.