Chains with Complete Connections and One-Dimensional Gibbs Measures
Gregory Maillard (Laboratoire de mathematiques Raphael Salem, Universite de Rouen)
Abstract
We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 145--176
Publication Date: February 25, 2004
DOI: 10.1214/EJP.v9-149
References
- Berbee, H. (1987), Chains with infinte connections: Uniqueness and Markov representation, Probab. Theory Related Fields 76, 243-253. MR 89c:60052
- Bressaud, X. FernÃ¡ndez, R. and Galves, A. (1999), Decay of correlations for non HËlderian dynamics. A coupling approach, Electron. J. Probab. 4. MR 2000j:60049
- Bowen, R. (1975), Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York. MR 56 #1364
- Coelho, Z. and Quas, A. N. (1998), Criteria for $overline d$-continuity, Trans. Amer. Math. Soc. 350, 3257-3268. MR 99d:28028
- Dachian, S. and Nahapetian, B.S. (2001), Description of random fields by means of one-point conditional distributions and some applications, Markov Process. Related Fields 7, 193-214. MR 2002f:60102
- Dobrushin, R.L. (1968), Description of a random field by means of conditional probabilities and conditions of its regularity, Theory of probability and its applications 13, 197-224. MR 37 #6989
- FernÃ¡ndez, R. and Maillard, G. (2003), Chains with complete connections. General theory, uniqueness, loss of memory and decay of correlations, (math.PR/0305026)
- FernÃ¡ndez, R. and Pfister, C.-E. (1997), Global specifications and nonquasilocality of projections of Gibbs measures, Ann. Probab. 25, 1284-1315. MR 98h:60066
- Geogii, H.-O. (1974), Stochastische Felder und ihre Anwendung auf Interaktionssysteme, Lecture Notes, Institut fÂ¸r Angewandte Mathematik, Universitâ°t Heidelberg.
- Georgii, H.-O. (1988), Gibbs Measures and Phase Transitions, Walter de Gruyter & Co., Berlin, Vol. 9, Berlin-New York. MR 89k:82010
- Goldstein, S. Kuik, R. Lebovitz, J. L. and Maes, C. (1989), >From PCA's to equilibrium systems and back, Comm. Math. Phys. 125, 71-79. MR 91b:82031
- Harris, T. E. (1955), On chains of infinite order, Pacific J. Math. 5, 707-724. MR 17,755b
- Johansson, A. and Ã·berg, A. (2002), Square summability of variations of $g$-functions and uniqueness of $g$-measures, Preprint.
- Kalikow, S. (1990), Random Markov processes and uniform martingales, Israel J. Math. 71, 33-54. MR 92a:60152
- Keane, M. (1972), Strongly mixing $g$-measures, Invent. Math. 16, 309-324. MR 46 #9295
- Keller, G. (1998), Equilibrium states in ergodic theory, London Mathematical Society Student Texts, Vol. 42, Cambridge University Press, Cambridge. MR 99e:28022
- Lalley, S. P. (1986), Regenerative representation for one-dimensional Gibbs states, Ann. Prob. 14, 1262-1271. MR 88c:60076
- Lanford, O. E. (1973), Entropy and equilibrium states in classical statistical mechanics, Lenard, A. (ed.), Statiscal mechanics and mathematical problems, Battelle Seattle Rencontres 1971, {LPN}h 20, pp. 1-113.
- Ledrappier, F. (1974), Principe variationnel et systÃ¨mes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30, 185-202. MR 53 #8384
- Onicescu, O. and Mihoc, G. (1935), Sur les chaÃ®nes statistiques, C. R. Acad. Sci. Paris 200, 511-512.
- Ruelle, D. (1978), Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co. MR 80g:82017
- Stenflo, Ã·. (2003), Uniqueness in $g$-measures, Nonlinearity 16, 403-410. MR 2004a:28027
- Walters, P. (1975), Ruelle's operator theorem and $g$-measures, Trans. Amer. Math. Soc. 214, 375-387. MR 54 #515
This work is licensed under a Creative Commons Attribution 3.0 License.