Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures
Download PDF
Download PDF
  • Published: September 1989

Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures

  • F. Comets1 nAff2 

Probability Theory and Related Fields volume 80, pages 407–432 (1989)Cite this article

  • 411 Accesses

  • Metrics details

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. 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)

    Google Scholar 

  2. 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)

    Google Scholar 

  3. Chayes, J.T., Chayes, L., Fröhlich J.: the low temperature behaviour of disordered magnets. Comm. Math. Phys.100, 399–436 (1985)

    Google Scholar 

  4. Comets, F.: Grandes déviations pour des champs de Gibbs sur ℤd. C.R. Acad. Sci., Paris, Série 1,303, 511–514 (1986)

    Google Scholar 

  5. Comets, F., Eisele, T., Schatzman, M.: On secondary bifurcation for some nonlinear convolution equation. Trans. Am. Math. Soc.296, 661–702 (1986)

    Google Scholar 

  6. Csiszár, I.:I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3, 146–158 (1975)

    Google Scholar 

  7. 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)

  8. 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)

    Google Scholar 

  9. Eisele, T., Ellis, R.S.: Symmetry breaking and random waves for magnetic systems on a circle. Z. Wahrscheinlichkeitstheor. verw. Geb.63, 297–348 (1983)

    Google Scholar 

  10. Ekeland, I., Temam, R.: Convex analysis and variational problems. Amsterdam: North Holland 1976

    Google Scholar 

  11. Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  12. 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)

    Google Scholar 

  13. Enter, A.C.D. van, Griffiths, R.: The order parameter in a spin glass. Comm. Math. Phys.90, 319–327 (1983)

    Google Scholar 

  14. Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969

    Google Scholar 

  15. Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab.16, 961–977 (1988)

    Google Scholar 

  16. Fröhlich, J., Imbrie, J.Z.: Improved perturbation expansion for disordered systems: beating Griffiths singularities. Comm. Math. Phys.96, 145–180 (1984)

    Google Scholar 

  17. Hemmen, J.L. van, Enter, A.C.D. van, Canisius, J.: On a classical spin glass model. Z. Phys. B50, 311–336 (1983)

    Google Scholar 

  18. Khanin, K.M., Sinaï, Ya.G.: Existence of free energy for models with long-range random Hamiltonians. J. Stat. Phys.20, 573–584 (1979)

    Google Scholar 

  19. 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

    Google Scholar 

  20. Ledrappier, F.: Pressure and variational principle for random Ising model. Comm. Math. Phys.56, 297–302 (1977)

    Google Scholar 

  21. Mc Coy, B.M., Wu, T.T.: The two dimensional Ising model. Cambridge: Harvard University Press 1973

    Google Scholar 

  22. Olla, S.: Large deviation for Gibbs random fields. Probab. Th. Rel. Fields77, 343–357 (1988)

    Google Scholar 

  23. Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967

    Google Scholar 

  24. Peretto, P.: Collective properties of neural networks: a statistical physics approach. Biol. Cybern.50, 51–62 (1984)

    Google Scholar 

  25. Prum, B.: Processus sur un réseau et mesures de Gibbs; applications. Paris: Messor 1986

    Google Scholar 

  26. Sherrington, D., Kirckpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett.35, 1792–1796 (1975)

    Google Scholar 

  27. Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Comm. Pure Appl. Math.19, 261–286 (1966)

    Google Scholar 

  28. Varadhan, S.R.S.: Large deviations and applications Philadelphia: SIAM 1984

    Google Scholar 

  29. 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)

    Google Scholar 

  30. Zabell, S.L.: Rates of convergence for conditional expectations. Ann. Probab.8, 928–941 (1980)

    Google Scholar 

Download references

Author information

Author notes
  1. F. Comets

    Present address: Laboratoire de Statistiques Appliquée, UA CNRS 743, Mathématique, Université Paris-Sud, Bâtiment 425, F-91405, Orsay Cedex, France

Authors and Affiliations

  1. Université Paris X-Nanterre, 200 Av. Republique, F-92000, Nanterre, France

    F. Comets

Authors
  1. F. Comets
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints 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

Download citation

  • Received: 21 May 1987

  • Revised: 30 May 1988

  • Issue Date: September 1989

  • DOI: https://doi.org/10.1007/BF01794432

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Conditional Probability
  • Mathematical Biology
  • Short Range
  • Deviation Rate
  • Deviation Estimate
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature