Gibbs random fields with unbounded spins on unbounded degree graphs

Details Gibbs random fields with unbounded spins on unbounded degree graphs Gibbs random fields with unbounded spins on unbounded degree graphs

2009-04-21

arxiv.org/abs/0904.3207v1

Mathematics

Probability

Scientific paper

Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589].

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