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Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs

Part of:Equilibrium statistical mechanicsGraph theorySpecial processes

Published online by Cambridge University Press:  14 July 2016

Yuri Kondratiev,
Yuri Kozitsky and
Tanja Pasurek
Yuri Kondratiev*
Affiliation:
Universität Bielefeld
Yuri Kozitsky*
Affiliation:
Uniwersytet Marii Curie-Skłodowskiej
Tanja Pasurek*
Affiliation:
Universität Bielefeld
*
∗ Postal address: Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany.
∗∗ Postal address: Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland. Email address: jkozi@hektor.umcs.lublin.pl
∗ Postal address: Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany.
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Abstract

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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).

Keywords

Generalized Randić indexGibbs measureGibbs specificationunbounded spinunbounded degree graphDLR equation

MSC classification

Secondary:60K35: Interacting random processes; statistical mechanics type models; percolation theory82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs05C07: Vertex degrees

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010, pp. 856 - 875
Copyright
Copyright © Applied Probability Trust 2010 

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