# Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry

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Communications in Mathematical Physics

, Volume 271, Issue 2, pp 431–454

First Online: 08 February 2007Received: 14 October 2005Accepted: 14 September 2006

Abstract

We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity bad points. We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature \\beta^{-1}\,{\geq}\,1\, we prove that the time-evolved measure is always Gibbsian. For \\frac{2}{3}\,{\leq}\,\beta^{-1}{ < }1\, the time-evolved measure loses its Gibbsian character at a sharp transition time. For \\beta^{-1}\,{ < }\,\frac{2}{3}\, we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.

Communicated by J.L. Lebowitz

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Author: **Christof Külske - Arnaud Le Ny**

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