Loss without recovery of Gibbsianness during diffusion of continuous spins

C. Külske, F.H.J. Redig

    Research output: Contribution to journal › Article › Academic › peer-review

    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

    Fingerprint

    Recovery
    Gibbs Distribution
    Double-well Potential
    Continuous-time Model
    Gibbs Measure
    Spin Models
    External Field
    Magnetic Field
    Tend
    Temperature
    Magnetic field
    Continuous time

    Cite this

    Külske, C. ; Redig, F.H.J. / Loss without recovery of Gibbsianness during diffusion of continuous spins. In: Probability Theory and Related Fields. 2006 ; Vol. 135, No. 3. pp. 428-456.
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    Külske, C & Redig, FHJ 2006, 'Loss without recovery of Gibbsianness during diffusion of continuous spins', Probability Theory and Related Fields, vol. 135, no. 3, pp. 428-456. https://doi.org/10.1007/s00440-005-0469-9

    Loss without recovery of Gibbsianness during diffusion of continuous spins. / Külske, C.; Redig, F.H.J.

    In: Probability Theory and Related Fields, Vol. 135, No. 3, 2006, p. 428-456.

    Research output: Contribution to journal › Article › Academic › peer-review

    TY - JOUR

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    AU - Redig, F.H.J.

    PY - 2006

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    N2 - 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.

    AB - 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.

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    DO - 10.1007/s00440-005-0469-9

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    Külske C, Redig FHJ. Loss without recovery of Gibbsianness during diffusion of continuous spins. Probability Theory and Related Fields. 2006;135(3):428-456. https://doi.org/10.1007/s00440-005-0469-9

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