Abstract
We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice ℤd, d≥2. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abadi, M.: Exponential approximation for hitting times in mixing processes. Math. Phys. Electron. J. 7 (2001)
Abadi, M., Galves, A.: Inequalities for the occurrence of rare events in mixing processes. The state of the art. In: ‘Inhomogeneous random systems’ (Cergy-Pontoise, 2000), Markov Process. Related Fields 7(1), 97–112 (2001)
Asselah, A., Dai Pra, P.: Sharp estimates for the occurrence time of rare events for symmetric simple exclusion. Stochastic Process. Appl. 71(2), 259–273 (1997)
Asselah, A., Dai Pra, P.: Occurrence of rare events in ergodic interacting spin systems. Ann. Inst. H. Poincaré Probab. Statist. 33(6), 727–751 (1997)
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975
Bryc, W.: A remark on the connection between the large deviation principle and the central limit theorem. Statist. & Probab. Lett. 18, 253–256 (1993)
Chazottes, J.-R.: Dimensions and waiting time for Gibbs measures. J. Stat. Phys. 98(3/4), 305–320 (2000)
Chi, Z.: The first-order asymptotic of waiting times with distortion between stationary processes. IEEE Trans. Inform. Theory 47(1), 338–347 (2001)
Collet, P., Galves, A., Schmitt, B.: Fluctuations of repetition times for gibbsian sources. Nonlinearity 12, 1225–1237 (1999)
Comets, F.: Grandes déviations pour des champs de Gibbs sur ℤd. CRAS, t. 303(11), 511–513 (1986)
Dembo, A., Kontoyiannis, I.: Source coding, large deviations and approximate pattern matching. IEEE Trans. Inf. Theory 48(6), 1590–1615 (2002)
Dembo, A., Zeitouni, O.: Large Deviations Techniques & Applications. Applic. Math. 38, Berlin-Heidelberg-New York: Springer, 1998
Dobrushin, R.L., Shlosman, S.B.: Completely analytical interactions: constructive description. J. Stat. Phys. 46(5-6), 983–1014 (1987)
Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 271, New York: Springer- Verlag, 1985
Daniëls, H.A.M., van Enter, A.C.D.: Differentiability properties of the pressure in lattice systems. Commun. Math. Phys. 71(1), 65–76 (1980)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. Berlin: Walter de Gruyter & Co., 1988
Guyon, X.: Random Fields on a Network. Modeling, Statistics and Applications. New York-Berlin: Springer Verlag, 1995
Haydn, N., Luévano, J., Mantica, G., Vaienti, S.: Multifractal Properties of Return Time Statistics. Phys.Rev. Lett. 88(22), (2002)
Krengel, U.: Ergodic theorems. de Gruyter Studies in Mathematics 6, Berlin: Walter de Gruyter & Co., 1985
Föllmer, H.: On entropy and information gain in random fields. Z. Wahrsch. Theorie Verw. Gebiete 26, 207–217 (1973)
Martinelli, F.: An elementary approach to finite size conditions for the exponential decay of covariances in lattice spin models. In: On Dobrushin’s way. From probability theory to statistical physics, Am. Math. Soc. Transl. Ser. 2, 198, Providence, RI: Am. Math. Soc., 2000, pp. 169–181
Olla, S.: Large deviations for Gibbs random fields. Prob. Th. Rel. Fields 77, 343–357 (1988)
Ornstein, D., Weiss, B.: Entropy and recurrence rates for stationary random fields. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48(6), 1694–1697 (2002)
Plachky, D., Steinebach, J.A.: A theorem about probabilities of large deviations with an application to queuing theory. Periodica Math. Hungar. 6, 343–345 (1975)
Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. Encyclopdia of Mathematics and its Applications 5. Reading, Mass.: Addison-Wesley Publishing Co., 1978
Shields, P.C.: The ergodic theory of discrete sample paths. Providence, RI: AMS, 1996
Simon, B.: The statistical mechanics of lattice gases. Vol. I. Princeton Series in Physics. Princeton, NJ: Princeton University Press, 1993
Thouvenot, J.P.: Convergence en moyenne de l’information pour l’action de ℤ2. Z. Wahrsch. Theorie Verw. Gebiete 24, 135–137 (1972)
Wyner, A.J.: More on recurrence and waiting times. Ann. Appl. Probab. 9(3), 780–796 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
H. Spohn
Rights and permissions
About this article
Cite this article
Abadi, M., Chazottes, JR., Redig, F. et al. Exponential Distribution for the Occurrence of Rare Patterns in Gibbsian Random Fields. Commun. Math. Phys. 246, 269–294 (2004). https://doi.org/10.1007/s00220-004-1041-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1041-7