Abstract
Can one detect a phase transition from a single, large sample of a Gibbs measure? What information does one get on the other Gibbs distributions with the same potential? We approach these questions via Erdős-Rényi laws. In particular we prove almost-sure limit theorems for sets of empirical distributions of sub-samples of the given one: for suitable sub-samples size this set converges to the set of stationary Gibbs measures. Moreover we formulate Erdős-Rényi laws for general families of random variables with suitable large deviation principles.
Similar content being viewed by others
References
Arratia, R., Waterman, M.S.: The Erdős-Rényi strong law for pattern matching with a given proportion of mismatches. Ann. Probab.17, 1152–1169 (1989)
Book, S.A.: An extension of the Erdős-Rényi law of large numbers. Proc. Am. Math. Soc.48, 438–446 (1975)
Comets, F.: Grandes déviations pour des champs de Gibbs surZ d. C.R. Acad. Sc. Paris, Série I303, 511–514 (1986)
Csörgő, S.: Erdős-Rényi laws. Ann. Statist.7, 772–787 (1979a)
Csörgő, S.: Badahur efficiency and Erdős-Rényi maxima. Sankhya Ser. A41, 141–144 (1979b)
Deheuvels, P., Devroye, L., Lynch, J.: Exact convergence rate in the limit theorems of Erdős-Rényi an Shepp. Ann. Probab.14, 209–233 (1986)
Dembo, A., Karlin, S.: Strong limit theorems of empirical functionals for large exceedance of partial sums of i.i.d. variables. Ann. Probab.19, 1737–1755 (1991b)
Dembo, A., Karlin, S.: Strong limit theorems of empirical distributions for large segmental exceedances of partial sums of Markov variables. Ann. Probab.19, 1756–1767 (1991b)
Deuschel, J.D., Stroock, D.W.: Large deviations. Boston: Academic Press (1989)
Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction: A global shape from local interaction. Transl. of Math. Monographs104. Providence: Am. Math. Soc. (1992)
Erdős, P., Rényi, A.: On a new law of large numbers. J. Analyse Math.23, 103–111 (1970)
Erdős, P., Révész, P.: On the length of the longest head-run. Topics in Information theory Keszthely (Hyngary) Coll. Math. Soc. János Bolyai16, 219–228 (1975)
Follmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab.16, 961–977 (1988)
Freidlin, M.I., Wentzell, A.D.: Random perturbation of dynamical systems. Berlin, Heidelberg, New York: Springer (1984)
Georgii, H.O.: Large deviations and maximum entropy principle for interacting random fields onZ d. Preprint Univers. Munich (1992)
Georgii, H.O.: Gibbs measures and phase transitions. Berlin: de Gruyter (1988)
Olla, S.: Large deviations for Gibbs random fields. Probab. Th. Rel. Fields77, 343–357 (1988)
Samarova, S.S.: On the length of the longest head-run for a Markov chain with two states. Th. Prob. Appl.26, 498–509 (1981)
Author information
Authors and Affiliations
Additional information
Communicated by M. Aizenman
URA CNRS 756 (Centre de Mathématiques Appliquées, Ecole Polytechnique) et 1321 (Statistique et Modèles Aléatoires, Université Paris 7)
Rights and permissions
About this article
Cite this article
Comets, F.M. Erdős-Rényi laws for Gibbs measures. Commun.Math. Phys. 162, 353–369 (1994). https://doi.org/10.1007/BF02102022
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102022