Extremal decomposition for random Gibbs measures
- Cotar, Codina
- Jahnel, Benedikt
- 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.
- 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''