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1 results

  • Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs

  • Part of
  • Yuri Kondratiev, Yuri Kozitsky, Tanja Pasurek
  • Journal:
    Journal of Applied Probability / Volume 47 / Issue 3 / September 2010
    Published online by Cambridge University Press:
    14 July 2016, pp. 856-875
    Print publication:
    September 2010
    • Article
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  • Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).