WIAS Preprint No. 2546, (2018)

Extremal decomposition for random Gibbs measures


  • Cotar, Codina
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Külske, Christof

2010 Mathematics Subject Classification

  • 82B44 60K35


  • Gibbs measures, disordered systems, extremal decomposition, metastates




The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.

Appeared in

  • Electron. Comm. Probab., 23 (2018), pp. 1--12, DOI https://doi.org/10.1214/18-ECP200 with the new title ``Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures''

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