Low-temperature dynamics of the Curie-Weiss Model: Periodic orbits, multiple histories, and loss of Gibbsianness

TL;DR

This paper analyzes the transition between Gibbsian and non-Gibbsian states in the Curie-Weiss model under Glauber dynamics, revealing how temperature changes induce symmetry-breaking and non-Gibbsian phenomena linked to periodic orbits.

Contribution

It provides a rigorous analysis of non-Gibbsian behavior during cooling in a mean-field model, highlighting the role of periodic orbits in the loss of Gibbsianness.

Findings

Gibbs measure persists at high initial temperatures.

Sharp transition to non-Gibbsian states during cooling.

Periodic orbits are linked to bad configurations causing non-Gibbsianness.

Abstract

We consider the Curie-Weiss model at a given initial temperature in vanishing external field evolving under a Glauber spin-flip dynamics corresponding to a possibly different temperature. 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 bigger than one 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, when the initial temperature is smaller than the temperature of the dynamics, and smaller than 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 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.