Hostname: page-component-84b7d79bbc-l82ql Total loading time: 0 Render date: 2024-07-31T06:05:04.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs

Part of: Graph theory Equilibrium statistical mechanics Special 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 theory 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 05C07: Vertex degrees
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 856 - 875
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Albeverio, S. and Zegarliński, B. (1990). Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds. Commun. Math. Phys. 132, 39–71.Google Scholar
[2] Albeverio, S., Kondratiev, Y., Kozitsky, Y. and Röckner, M. (2009). The Statistical Mechanics of Quantum Lattice Systems (EMS Tracts Math. 8). European Mathematical Society, Zürich.CrossRefGoogle Scholar
[3] Bassalygo, L. A. and Dobrushin, R. L. (1986). Uniqueness of a Gibbs field with a random potential—an elementary approach. Teor. Veroyat. Primen. 31, 651–670 (in Russian). English translation: Theory Prob. Appl. 31, 572-589.Google Scholar
[4] Burioni, R. and Cassi, D. (2005). Random walks on graphs: ideas, techniques and results. J. Phys. A 38, R45–R78.CrossRefGoogle Scholar
[5] Burioni, R., Cassi, D. and Vezzani, A. (2004). Random walks and physical models on infinite graphs: an introduction. In Random Walks and Geometry, ed. Kaimanovich, V. A., Walter de Gruyter, Berlin.Google Scholar
[6] Cassandro, M., Olivieri, E., Pellegrinotti, A. and Presutti, E. (1977/78). Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrscheinlichkeitsth. 41, 313–334.Google Scholar
[7] Clark, L. and Gutman, I. (2008). The exponent in the general Randić index. J. Math. Chem. 43, 32–44.Google Scholar
[8] De Angelis, G. F., de Falco, D. and Di Genova, G. (1986). Random fields on Riemannian manifolds: a constructive approach. Commun. Math. Phys. 103, 297–303.Google Scholar
[9] Fujiwara, K. (1995). Eigenvalues of Laplacians on a closed Riemannian manifold and its nets. Proc. Amer. Math. Soc. 123, 2585–2594.Google Scholar
[10] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin.Google Scholar
[11] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18, eds. Domb, C. and Lebowitz, J. L., Academic Press, San Diego, CA, pp. 1–142,Google Scholar
[12] Häggström, O. (2000). Markov random fields and percolation on general graphs. Adv. Appl. Prob. 32, 39–66.Google Scholar
[13] Häggström, O., Schonmann, R. H. and Steif, J. E. (2000). The Ising model on diluted graphs and strong amenability. Ann. Prob. 28, 1111–1137.Google Scholar
[14] Hattori, K., Hattori, T. and Watanabe, H. (1987). Gaussian field theories on general networks and the spectral dimensions. Progr. Theoret. Phys. Suppl. 92, 108–143.CrossRefGoogle Scholar
[15] Kozitsky, Y. and Pasurek, T. (2007). Euclidean Gibbs measures of interacting quantum anharmonic oscillators. J. Statist. Phys. 127, 985–1047.Google Scholar
[16] Lebowitz, J. L. and Presutti, E. (1976). Statistical mechanics of systems of unbounded spins. Commun. Math. Phys. 50, 195–218.Google Scholar
[17] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys. 125, 337–353.Google Scholar
[18] Lyons, R. (2000). Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, 1099–1126.Google Scholar
[19] Pasurek, T. (2007). Theory of Gibbs measures with unbounded spins: probabilistic and analytic aspects. , Universität Bielefeld.Google Scholar
[20] Ruelle, D. (1976). Probability estimates for continuous spin systems. Commun. Math. Phys. 50, 189–194.Google Scholar
[21] Simon, B. (1974). The P(ɸ)2 Euclidean (Quantum) Field Theory.} Princeton University Press.Google Scholar
[22] Wald, R. M. (1994). Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press.Google Scholar