Abstract
In this report we describe the class of interactions A for which the corresponding Gibbs field is unique and possess every possible virtue one can imagine. We mean by this the well-known regularity properties of Gibbs fields in the high-temperature region. This class of interactions is defined axiomatically by ten (!) very natural properties. But the reader should not be confused by their great amount because all of them turn out to be equivalent! This fact alone shows that the class considered is natural. We call the potentials of the class completely analitical. The boundary of A corresponds to phase transition surface. Furthermore, the region A can be defined constructively by a certain algorithm. Here we use the word “constructive” in the same sense, as we used it in report [1], which is conceptually close to this one. For the sake of simplicity, we restrict ourselves to random fields on Z V, with finite single-spin space S and finite-range, translation-invariant interactions with finite values. The ideas of the present report and those of [1] can be applied in more complex cases; in particular, we study perturbations of Gaussian fields in [2]. Throughout this report we shall use the notations of [1], §1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Dobrushin R. L., Shlosman S. B. Constructive criterion for the uniqueness of Gibbs field, this volume.
Dobrushin R. L. A new method of the study of the Gibbs perturbations of Gaussian fields (in preparation).
Shlosman S. B. Mediation on Czech models: uniqueness, half-space non-uniqueness and analyticity properties (in preparation).
Lebowitz J. L., Penrose O. Analytic and Clustering properties of thermondinamic functions and distribu-tion functions for classical lattice and continuum sys tems. Comm. Math. Phys. 11, 99–124 (1968).
Gallavotti G., Miracle-Sole S. On the Cluster property above the critical temperature in lattice gase. Comm. Math. Phys. 12, 269–274 (1969).
Duneau M., Jagolnitzer D., Souillard B. Decrease properties of truncated correlation functions and analyticity properties for classical lattice and continuous systems. Comm. Math. Phys. 31, 191–208 (1973).
Duneau M., Souillard B., Jagolnitzer D. Analyticity and strong Cluster properties for classical gases with finite ränge interactions. Comm. Math. Phys. 35, 307–320 (1974).
Duneau M., Souillard B. Cluster properties of lattice and continuum systems. Comm. Math. Phys. 47, 155–166 (1976).
Dobrushin R. L. Analyticity of correlation functions in one-dimensional classical systems with slowly de-creasing potentials. Comm. Math. Phys. 23, 269–289 (1973).
Dobrushin R. L. Analyticity of correlation functions in one-dimensional classical systems with power decay of the potential. Matern. Sbornik 94, No. 1, 16–48 (1974).
Malyshev V. A. Cluster expansions in lattice models of Statistical physics and quantum field theory. Uspehi Mat. Nauk 35, No. 2, 3–53 (1980).
Malyshev V. A., Minlos K. A. Gibbsian random fields (the method of Cluster expansion). Moscow, 1985 (in print).
Dobrushin K. L., Pechersky E. A. Uniqueness conditions for finitely dependent random fields. In “Random Fields Vol. 1, North-Holland, Amsterdam-Oxford-N-Y, 232–262 (1981).
Simon B. Correlation inequalities and the decay of cor relation in ferromagnets. Comm. Math. Phys. 77, 111–126 (1980).
Lieb E. M. A refinement of Simon’s correlation inequal ity. Comm. Math. Phys. 77, 127–136 (1980).
Statuleviöus V. A. On large deviations. Z. Wahrsch. verw. Geb. 6, 123–144 (1966).
Dobrushin R. L., Nahapetian B. S. Strong convexity of the pressure for lattice systems of classical Statistical physics. Teor. Mat. Fiz. 20, No. 2, 223–234 (1974).
Dobrushin R. L., Tirozzi B. The central limit theorem and the problem of equivalence of ensembles. Comm. Math. Phys. 54, 173–192 (1977).
Gruber S., Hintermann A., Merlini D. Analyticity and uniqueness of the invariant equilibrium states for general spin classical systems. Comm. Math. Phys. 40, 83–99 (1975).
Van Hove L., Sur 1’integrale de configuration pour les systems de perticules ä une dimension. Physica 16, 137–143 (1950).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dobrushin, R.L., Shlosman, S.B. (1985). Completely Analytical Gibbs Fields. In: Fritz, J., Jaffe, A., Szász, D. (eds) Statistical Physics and Dynamical Systems. Progress in Physics, vol 10. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6653-7_21
Download citation
DOI: https://doi.org/10.1007/978-1-4899-6653-7_21
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-6655-1
Online ISBN: 978-1-4899-6653-7
eBook Packages: Springer Book Archive