Description of random fields by means of one-point finite-conditional distributions

  • A. S. DalalyanEmail author
  • B. S. Nahapetian
Statistical Physics


The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ℤ ν and the conditional probability measures at one point given the values on a finite subset of the lattice ℤ ν . We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.


Random field one-point conditional distribution mixing properties 

MSC2010 numbers

82C70 60G60 


  1. 1.
    S. Yu. Dashyan, B. S. Nahapetian, “Description of random fields by means of one-point conditional distributions and some applications”, Markov Process. Related Fields, 7(2), 193–214 (2001).MathSciNetGoogle Scholar
  2. 2.
    S. Yu. Dashyan, B. S. Nahapetian, “Description of specifications by means of probability distributions in small volumes under condition of very weak positivity”, J. Statist. Phys., 117(1–2), 281–300 (2004).MathSciNetGoogle Scholar
  3. 3.
    S. Yu. Dashyan, B. S. Nahapetian, “On Gibbsianness of random fields. Markov Process. Related Fields, 15(1), 81–104 (2009).MathSciNetGoogle Scholar
  4. 4.
    S. Yu. Dashyan, B. S. Nahapetian, “Inclusion-exclusion description of random fields”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 30(6), 50–61 (1995).Google Scholar
  5. 5.
    R. L. Dobrushin, “The problem of uniquenessof a Gibbsian random field and the problem of phase transitions”, Funktsional. Anal. i Prilojen., 2(4), 44–57 (1968).Google Scholar
  6. 6.
    R. L. Dobrushin, “Description of a random field by means of conditional probabilities and conditions for its regularity”, Teor. Verojatnost. i Primenen., 13, 201–229 (1968).MathSciNetGoogle Scholar
  7. 7.
    P. Doukhan, Mixing. Properties and examples, Lecture Notes in Statistics, 85 (Springer-Verlag, New York, 1994).zbMATHGoogle Scholar
  8. 8.
    R. Fernández, R., G. Maillard, “Construction of a specification from its singleton part”, ALEA Lat. Am. J. Probab. Math. Stat., 2, 297–315 (2006).MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images”, IEEE Transl. Pattern Anal.Mach. Intell, 6, 721–741 (1984).zbMATHCrossRefGoogle Scholar
  10. 10.
    B. Nahapetian, Limit theorems and some applications in statistical physics, Teubner-Texte zur Mathematik 123 (B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991).zbMATHGoogle Scholar
  11. 11.
    N. Shental, A. Zomet, T. Hertz, Y. Weiss, “Learning and inferring image segmentations using the GBP typical cut algorithm”, in 2 (Proceedings of ICCV, October 13–16, 2003).Google Scholar

Copyright information

© Allerton Press, Inc. 2011

Authors and Affiliations

  1. 1.Ecole des Ponts ParisTechUniversité Paris-EstParisFrance
  2. 2.Institute of MathematicsNational Academy of Sciences of ArmeniaYerevanArmenia

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