Loss without recovery of Gibbsianness during diffusion of continuous spins

C. Külske, F.H.J. Redig

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    22 Citations (Scopus)

    Abstract

    We consider a specific continuous-spin Gibbs distribution µt=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under independent diffusions. For `high temperature' initial measures we prove that the time-evoved measure µt is Gibbsian for all t. For `low temperature' initial measures we prove that µt stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d=2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than their discrete-spin counterparts.
    Original languageEnglish
    Pages (from-to)428-456
    JournalProbability Theory and Related Fields
    Volume135
    Issue number3
    DOIs
    Publication statusPublished - 2006

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