Skip to main content
SpringerLink
Log in

A Note on Exponential Decay in the Random Field Ising Model

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  2. Alexander, K.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110, 441–471 (1998)

    Article  MathSciNet  Google Scholar 

  3. Berretti, A.: Some properties of random Ising models. J. Stat. Phys. 38, 483–496 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bezuidenhout, C.E., Grimmett, G.R., Kesten, H.: Strict inequality for critical values of Potts models and random-cluster processes. Commun. Math. Phys. 158, 1–16 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  5. Blanchard, Ph, Gandolfo, D., Laanait, L., Satz, H.: On the Kertész line: thermodynamic versus geometric criticality. J. Phys. A 41, 1–9 (2008)

    Article  Google Scholar 

  6. Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Commun. Math. Phys. 116, 539–572 (1988)

    Article  ADS  Google Scholar 

  7. Chatterjee, S.: On the decay of correlations in the random field Ising model. Commun. Math. Phys. 362, 253–267 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  8. Chayes, L., Schonmann, R.H.: Mixed percolation as a bridge between site and bond percolation. Ann. Probab. 10, 1182–1196 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Duminil-Copin, H., Raoufi, A., Tassion, V.: Sharp phase transition for the random-cluster and Potts model via decision trees. arXiv:1705.03104v1 (2017)

  10. Edwards, R.G., Sokal, A.S.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  12. Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys. 8, 484–489 (1967)

    Article  ADS  Google Scholar 

  13. Grimmett, G.: Comparison and disjoint-occurrence inequalities for random-cluster models. J. Stat. Phys. 78, 1311–1324 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  14. Grimmett, G.: The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften, vol. 333. Springer, Berlin (2006)

    Google Scholar 

  15. Higuichi, Y.: Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions. Probab. Theory Relat. Fields 97, 1–33 (1993)

    Article  MathSciNet  Google Scholar 

  16. Imry, Y., Ma, S.: Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399–1401 (1975)

    Article  ADS  Google Scholar 

  17. Kertész, J.: Existence of weak singularities when going around the liquid-gas critical point. Physica A 161, 58–62 (1989)

    Article  ADS  Google Scholar 

  18. Lubetzky, E., Sly, A.: Critical Ising on the square lattice mixes in polynomial time. Commun. Math. Phys. 313, 815–836 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  19. Newman, C.M.: Topics in Disordered Systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1997)

    Google Scholar 

  20. Ruiz, J., Wouts, M.: On the Kertész line: some rigorous bounds. J. Math. Phys. 49, 053303 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  21. von Dreifus, H., Klein, A., Perez, J.F.: Taming Griffiths’ singularities: infinite differentiability of quenched correlation functions. Commun. Math. Phys. 170, 21–39 (1995)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of JJ was partially supported by STCSM Grant 17YF1413300 and that of CMN by US-NSF Grant DMS-1507019. The authors thank Dan Stein and Janek Wehr for useful discussions. They also thank the Institute of Applied Mathematics of the Chinese Academy of Sciences, where some of the work reported here was done.

Author information

Authors and Affiliations

  1. Division of Science, NYU Abu Dhabi, Saadiyat Campus, Abu Dhabi, United Arab Emirates

    Federico Camia

  2. Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    Federico Camia

  3. NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

    Jianping Jiang & Charles M. Newman

  4. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA

    Charles M. Newman

Authors
  1. Federico Camia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianping Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Charles M. Newman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianping Jiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s10955-018-2140-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-2140-8

Keywords

Search

Navigation