Advertisement

Random-Cluster Correlation Inequalities for Gibbs Fields

  • Alberto Gandolfi
Article
  • 9 Downloads

Abstract

In this note we prove a correlation inequality for local variables of a Gibbs field based on the connectivity in a random cluster representation of the non overlap configuration distribution of two independent copies of the field. As a consequence, we show that absence of a particular type of percolation (of Machta–Newman–Stein blue bonds) implies uniqueness of Gibbs distribution in EA Spin Glasses. In dimension two this could constitute a step towards a proof that the critical temperature is zero.

Keywords

Correlation inequalities Gibbs fields Random cluster representation Disagreement percolation Spin glasses 

Mathematics Subject Classification

60G60 60J99 82B44 82D30 

References

  1. 1.
    Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Newman, C.: Disordered Ising systems and random cluster representations. In: Grimmett, G. (ed.) Probability and Phase Transition, pp. 247–260. Kluwer, Dordrecht (1994)CrossRefGoogle Scholar
  3. 3.
    van den Berg, J., Gandolfi, A.: BK-type inequalities and generalized random-cluster representations. Probab. Theory Relat. Fields 157(1–2), 157–181 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Machta, J., Newman, C.M., Stein, D.L.: The percolation signature of the spin glass transition. J Stat Phys 130, 113–128 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lebowitz, J.L.: GHS and other inequalities. Commun. Math. Phys. 35(2), 87–92 (1974)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Reimer, D.: Proof of the Van den Berg-Kesten conjecture. Comb. Probab. Comput. 9, 27–32 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gandolfi, A.: FKG (and other inequalities) via (generalized) FK representation (and iterated folding). Preprint (2018)Google Scholar
  8. 8.
    Stein, D.L., Newman, C.M.: Spin Glasses and Complexity. Princeton University Press, Princeton (2013)CrossRefMATHGoogle Scholar
  9. 9.
    Tanaka, S., Tamura, R., Chakrabarti, B.K.: Quantum Spin Glasses, Annealing and Computation. Cambridge University Press, Cambridge, UK (2017)MATHGoogle Scholar
  10. 10.
    Gandolfi, A., Lenarda, P.: A note on Gibbs and Markov random fields with constraints and their moments. Math. Mech. Complex Syst. 4(3–4), 407–422 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ruelle, D.: Thermodynamic Formalism. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  12. 12.
    van den Berg, J.: A uniqueness condition for Gibbs measures, with application to the 2-dimensional ising antiferromagnet. Commun. Math. Phys. 152, 161–166 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dobrushin, R.L.: The problem of uniqueness of a Gibbs random field and the problem of phase transition. Funct. Anal. Appl. 2, 302–312 (1968)CrossRefMATHGoogle Scholar
  14. 14.
    Simon, B.: A remark on Dobrushin’s uniqueness theorem. Commun. Math. Phys. 68(2), 183–185 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of a Gibbs field. In: Fritz, J., Jaffe, A., Szasz, D. (eds.) Statistical Mechanics and Dynamical Systems, pp. 371–403. Birkhauser, Boston (1985)CrossRefGoogle Scholar
  16. 16.
    Van Den Berg, J., Maes, C.: Disagreement percolation in the study of Markov fields. Ann. Probab. 22(2), 749–763 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Berg, J., van den Steif, J.E.: Percolation and the hard-core lattice gas model. Stoch. Process. Appl. 49(2), 179–197 (1994)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Georgii, H.O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988)CrossRefMATHGoogle Scholar
  19. 19.
    van den Berg, J.: A constructive mixing condition for \(2\)-D Gibbs measures with random interactions. Ann. Probab. 25(3), 1316–1333 (1994)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Science divisionNYU Abu DhabiAbu DhabiUnited Arab Emirates
  2. 2.Dipartimento di Matematica e Informatica U. DiniUniversità di FirenzeFirenzeItaly

Personalised recommendations