Exponential Decay of Correlations in the 2D Random Field Ising Model

  • Michael AizenmanEmail author
  • Matan Harel
  • Ron Peled


An extension of the Ising spin configurations to continuous functions is used for an exact representation of the random field Ising model’s order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.


Random field Ising model Quenched disorder 2D Disagreement percolation Exponential decay Anti-concentration bounds 



We gratefully acknowledge the following support. The work of MA was supported in parts by the NSF Grant DMS-1613296, and the Weston Visiting Professorship at the Weizmann Institute of Science. The work of MH and RP was supported in part by Israel Science Foundation Grant 861/15 and the European Research Council starting Grant 678520 (LocalOrder). MH was supported by the Zuckerman Postdoctoral Fellowship. We thank the Faculty of Mathematics and Computer Science and the Faculty of Physics at WIS for the hospitality enjoyed there during work on this project.


  1. 1.
    Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds for random curves. Duke Math. J. 99, 419–453 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aizenman, M., Greenblatt, R.L., Lebowitz, J.L.: Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems. J. Math. Phys. 53, 023301 (2012)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aizenman, M., Peled, R.: A power-law upper bound on the correlations in the \(2D\) random field Ising model. Preprint arXiv:1808.08351 (2018)
  4. 4.
    Aizenman, M., Wehr, J.: Rounding of first-order phase transitions in systems with quenched disorder. Phys. Rev. Lett. 62, 2503 (1989)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Berretti, A.: Some properties of random Ising models. J. Stat. Phys. 38, 483–496 (1985)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bricmont, J., Kupiainen, A.: Lower critical dimension for the random-field Ising model. Phys. Rev. Lett. 59(16), 1829–1832 (1987)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bricmont, J., Kupiainen, A.: The hierarchical random field Ising model. J. Stat. Phys. 51, 1021–1032 (1988)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Commun. Math. Phys. 116(4), 539–572 (1988)ADSCrossRefGoogle Scholar
  10. 10.
    Camia, F., Jiang, J., Newman, C.M.: A note on exponential decay in the random field Ising model. J. Stat. Phys. 173, 268–284 (2018)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chatterjee, S.: On the decay of correlations in the random field Ising model. Commun. Math. Phys. 362, 253–267 (2018)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cohen-Alloro, O., Peled, R.: Rarity of extremal edges in random surfaces and other theoretical applications of cluster algorithms. Preprint arXiv:1711.00259 (2017)
  13. 13.
    Derrida, B., Shnidman, Y.: Possible line of critical points for a random field Ising model in dimension 2. J. Phys. Lett. 45, 577–581 (1984)CrossRefGoogle Scholar
  14. 14.
    Ding, J., Xia, J.: Exponential decay of correlations in the two-dimensional random field Ising model at zero temperature. Preprint arXiv:1902.03302 (2019)
  15. 15.
    Grinstein, G., Ma, S.-K.: Roughening and lower critical dimension in the random-field Ising model. Phys. Rev. Lett. 49, 685 (1982)ADSCrossRefGoogle Scholar
  16. 16.
    Imbrie, J.: The ground state of the three-dimensional random-field Ising model. Commun. Math. Phys. 98, 145–176 (1985)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Imbrie, J., Fröhlich, J.: Improved perturbation expansion for disordered systems: beating Griffiths singularities. Commun. Math. Phys. 96, 145–180 (1984)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Imry, Y., Ma, S.-K.: Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399 (1975)ADSCrossRefGoogle Scholar
  19. 19.
    Sheffield, S.: Random surfaces. Astérisque (2005)Google Scholar
  20. 20.
    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)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s)  2019

Authors and Affiliations

  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Weizmann Institute of ScienceRehovotIsrael
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations