Summary
Let (X i ,Y i ) ∈ℤd, be independent identically distributed random variables with arbitrary distribution. We show that, for almost every(Y i ) i , the conditional law of the empirical field given(Y i ) i satisfies to large deviation inequalities. This applies to the study of Gibbs measures with random interaction, in the case of some mean-field models as well as of short range summable interaction. We show that the pressure is nonrandom, and is given by a variational formula. These random Gibbs measures have the same large deviation rate, which does not depend on the particular realization of the interaction; their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula.
References
Campanino, M., Olivieri, E., van Enter, A.C.D.: One dimensional spin glasses with potential decay 1/r 1+ɛ. Absence of phase transition and cluster properties. Comm. Math. Phys.108, 241–255 (1987)
Cassandro, M., Olivieri, E., Tirozzi, B: Infinite differentiability for the one-dimensional spin system with long range interaction. Comm. Math. Phys.87, 229–252 (1982)
Chayes, J.T., Chayes, L., Fröhlich J.: the low temperature behaviour of disordered magnets. Comm. Math. Phys.100, 399–436 (1985)
Comets, F.: Grandes déviations pour des champs de Gibbs sur ℤd. C.R. Acad. Sci., Paris, Série 1,303, 511–514 (1986)
Comets, F., Eisele, T., Schatzman, M.: On secondary bifurcation for some nonlinear convolution equation. Trans. Am. Math. Soc.296, 661–702 (1986)
Csiszár, I.:I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3, 146–158 (1975)
Dacunha-Castelle, D.: Formule de Chernoff pour une suite de variables réelles. In: Séminaire d'Orsay «grandes déviations et applications statistiques’. Astérisque68, 19–24 (1978)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov expectations for large time. IV. Comm. Pure Appl. Math.36, 183–212 (1983)
Eisele, T., Ellis, R.S.: Symmetry breaking and random waves for magnetic systems on a circle. Z. Wahrscheinlichkeitstheor. verw. Geb.63, 297–348 (1983)
Ekeland, I., Temam, R.: Convex analysis and variational problems. Amsterdam: North Holland 1976
Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Berlin Heidelberg New York: Springer 1985
Enter, A.C.D. van, Fröhlich, J.: Absence of symmetry breaking forn-vector spin glass models in two dimensions. Comm Math. Phys.98, 425–432 (1985)
Enter, A.C.D. van, Griffiths, R.: The order parameter in a spin glass. Comm. Math. Phys.90, 319–327 (1983)
Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969
Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab.16, 961–977 (1988)
Fröhlich, J., Imbrie, J.Z.: Improved perturbation expansion for disordered systems: beating Griffiths singularities. Comm. Math. Phys.96, 145–180 (1984)
Hemmen, J.L. van, Enter, A.C.D. van, Canisius, J.: On a classical spin glass model. Z. Phys. B50, 311–336 (1983)
Khanin, K.M., Sinaï, Ya.G.: Existence of free energy for models with long-range random Hamiltonians. J. Stat. Phys.20, 573–584 (1979)
Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. In: Statistical mechanics, and mathematical problems. (Lect. Notes Phys. vol 20, pp. 1–113) Berlin Heidelberg New York: Springer 1973
Ledrappier, F.: Pressure and variational principle for random Ising model. Comm. Math. Phys.56, 297–302 (1977)
Mc Coy, B.M., Wu, T.T.: The two dimensional Ising model. Cambridge: Harvard University Press 1973
Olla, S.: Large deviation for Gibbs random fields. Probab. Th. Rel. Fields77, 343–357 (1988)
Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967
Peretto, P.: Collective properties of neural networks: a statistical physics approach. Biol. Cybern.50, 51–62 (1984)
Prum, B.: Processus sur un réseau et mesures de Gibbs; applications. Paris: Messor 1986
Sherrington, D., Kirckpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett.35, 1792–1796 (1975)
Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Comm. Pure Appl. Math.19, 261–286 (1966)
Varadhan, S.R.S.: Large deviations and applications Philadelphia: SIAM 1984
Vuillermot, P.A.: Thermodynamics of quenched random spin systems and applications to the problem of phase transition in magnetic (spin) glasses. J. Phys. A. Math. Gen.10, 1319–1333 (1977)
Zabell, S.L.: Rates of convergence for conditional expectations. Ann. Probab.8, 928–941 (1980)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Comets, F. Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Th. Rel. Fields 80, 407–432 (1989). https://doi.org/10.1007/BF01794432
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01794432
Keywords
- Conditional Probability
- Mathematical Biology
- Short Range
- Deviation Rate
- Deviation Estimate