Advertisement

Mathematical Notes

, Volume 72, Issue 1–2, pp 83–89 | Cite as

Gibbs Measures and Markov Random Fields with Association \(I \)

  • A. M. Rakhmatullaev
  • U. A. Rozikov
Article
  • 40 Downloads

Abstract

We introduce the notions of a Gibbs measure with the corresponding potential with association \(I\) (where \(I\) is a subset of the set \(\{ 1,2,...,k\} \)) of a Markov random field with memory \(I\) and measure with association \(I\). It is proved that these three notions are equivalent.

Gibbs measure Markov random field finite graph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    C. J. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, 1974.Google Scholar
  2. 2.
    T. M. Liggett, Interacting Particle Systems, Springer-Verlag, New York-Berlin-Heidelberg, 1985.Google Scholar
  3. 3.
    V. A. Malyshev and R. A. Minlos, Gibbs Random Fields [in Russian], Nauka, Moscow, 1985.Google Scholar
  4. 4.
    Ya. G. Sinai, The Theory of Phase Transitions [in Russian], Nauka, Moscow, 1980.Google Scholar
  5. 5.
    W. Gibbs, Elementary Principles of Statistical Mechanics, Yale University, 1902.Google Scholar
  6. 6.
    R. L. Dobrushin, “The description of a random field by means of conditional probabilities and conditions for its regularity,” Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.] (1968), no. 2, 201–229.Google Scholar
  7. 7.
    F. Spitzer, “Markov random fields and Gibbs ensembles,” Ann. Math. Monthly, 78 (1971), 142–154.Google Scholar
  8. 8.
    R. L. Dobrushin, “The study of Gibbs states for three-dimensional lattice systems,” Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.], XVIII (1973), no. 2, 261–279.Google Scholar
  9. 9.
    N. N. Ganikhodzhaev and U. A. Rozikov, “The description of periodic extreme Gibbs measures of certain models on the Cayley tree,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 111 (1997), no. 1, 109–117.Google Scholar
  10. 10.
    U. A. Rozikov, “The structure of partitions of a group representation for the Cayley tree and their applications for describing periodic Gibbs distributions,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 112 (1997), no. 1, 170–176.Google Scholar
  11. 11.
    U. A. Rozikov, “The description of the limit Gibbs measures of certain models on the Bethe lattice,” Sibirsk. Mat. Zh. [Siberian Math. J.], 39 (1998), no. 2, 427–435.Google Scholar
  12. 12.
    U. A. Rozikov, “The construction of the uncountable number of measures of the inhomogeneous Izing model,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 118 (1999), no. 1, 95–104.Google Scholar
  13. 13.
    N. N. Ganikhodzhaev and U. A. Rozikov, “On unordered phases of certain models on the Cayley tree,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 190 (1999), no. 2, 31–42.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • A. M. Rakhmatullaev
    • 1
  • U. A. Rozikov
    • 1
  1. 1.Mathematics InstituteAcademy of Sciences of UzbekistanTashkent

Personalised recommendations