Abstract.
We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure μ on the space X of finite-alphabet configurations on ℤd, d≥1, can be weakly approximated by Markov measures μ n with supp(μ n )=X and with the entropies h(μ n )→h(μ). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.
References
Cajar, H.: Billingsley Dimension in Probability Spaces. Lect. Notes in Math. 892, Springer-Verlag, Berlin (1981)
Dobrushin, R.L.: Shannon’s theorems for channels with synchronization errors (Russian). Problemy Peredachi Informacii. 3, 18–36; English translation in Problems of Information Transmission. 3, 11–26 (1967)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter (1988)
Gray, R.M., Ornstein, D.S., Dobrushin, R.L.: Block synchronization, sliding-block coding, invulnerable sources and zero errors codes for discrete noisy channels. Ann. Probab. 8, 639–674 (1980)
Gurevich, B.M., Tempelman, A.A.: Hausdorff dimension of the set of generic points for Gibbs measures (Russian). Funktsional. Anal. i Prilozhen. 36, 68–71; English translation in Functional Analysis and its Applications. 36, 225–227 (2002)
Gurevich, B.M., Tempelman, A.A.: Hausdorff dimension of sets of generic points for Gibbs measures. J. Statist. Phys. 108, 1281–1301 (2002)
Gurevich, B.M., Tempelman, A.A.: On sets of time and space means for continuous functions on a configuration space (Russian). Uspekhi Mat. Nauk. 58, 161–162 (2003); English translation in Russian Math. Surveys. 58, 370–371 (2003)
Israel, R.B.: Convexity in the Theory of Lattice Gases. Princeton Univ. Press, Princeton, NJ (1979)
Israel, R.B.: Generic triviality of phase diagrams in spaces of long-range interactions. Commun. Math. Phys. 106, 459–466 (1986)
Olivier, E.: Dimension de Billingsley d’ensembles satur’es. C.R. Acad. Sci. Paris. 328, Série I, 13–16 (1999)
Ruelle, D.: Thermodynamic Formalism. Addison-Wesley, Reading, MA (1978)
Sokal, A.D.: More surprises in the general theory of lattice systems. Commun. Math. Phys. 86, 327–336 (1982)
Acknowledgments.
(1). The authors are deeply indebted to the remarkable doctors, Prof. A. I. Vorobyov and E. A. Gilyazitdinova. Without their skill and their concern for the first author this paper would have not been written. (2). We are grateful to B. Weiss for calling our attention to the ‘‘synchronizing prefix’’ method used in Section 3. (3). The remarks by the referees resulted in essential improvement of the text of the paper. (4). The first author is supported by the RFBR grant number 02-01-00444.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classifications (2000): Primary 28D20, 37C85, 60G60; secondary 82B20
Dedicated to Professor A. I. Vorobyov, member of the Russian Academy of Sciences and Director of the Hematology Research Center of the Russian Academy of Medical Sciences, on the occasion of his 75th birthday
Rights and permissions
About this article
Cite this article
Gurevich, B., Tempelman, A. Markov approximation of homogeneous lattice random fields. Probab. Theory Relat. Fields 131, 519–527 (2005). https://doi.org/10.1007/s00440-004-0383-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0383-6
Keywords
- Homogeneous random fields
- Markov fields
- Equilibrium measures