Skip to main content
SpringerLink
Log in

Fluctuations of the Magnetization for Ising Models on Dense Erdős–Rényi Random Graphs

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

Abstract

We analyze Ising/Curie–Weiss models on the (directed) Erdős–Rényi random graph on N vertices in which every edge is present with probability p. These models were introduced by Bovier and Gayrard (J Stat Phys 72(3–4):643–664, 1993). We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime \(\beta <1\) when \(p=p(N)\) satisfies \(p^3N^2\rightarrow +\,\infty \).

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. Fröhlich, J.: Mathematical aspects of the physics of disordered systems. In: Phénomènes critiques, systèmes aléatoires, théories de jauge, Part I, II (Les Houches, 1984), pp. 725–893. North-Holland, Amsterdam. With the collaboration of A. Bovier and U. Glaus (1986)

  2. Georgii, H.-O.: Spontaneous magnetization of randomly dilute ferromagnets. J. Stat. Phys. 25(3), 369–396 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bovier, A., Gayrard, V.: The thermodynamics of the Curie–Weiss model with random couplings. J. Stat. Phys. 72(3–4), 643–664 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Ellis, R.S., Newman, C.M.: The statistics of Curie–Weiss models. J. Stat. Phys. 19(2), 149–161 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44(2), 117–139 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eisele, T., Ellis, R.S.: Multiple phase transitions in the generalized Curie–Weiss model. J. Stat. Phys. 52(1–2), 161–202 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Classics in Mathematics. Springer-Verlag, Berlin (2006). (Reprint of the 1985 original)

    Book  MATH  Google Scholar 

  8. Eichelsbacher, P., Löwe, M.: Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15(30), 962–988 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatterjee, S., Shao, Q.-M.: Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21(2), 464–483 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dembo, A., Montanari, A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dembo, A., Montanari, A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24(2), 137–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dommers, S., Giardinà, C., van der Hofstad, R.: Ising models on power-law random graphs. J. Stat. Phys. 141(4), 638–660 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Dommers, S., Giardinà, C., van der Hofstad, R.: Ising critical exponents on random trees and graphs. Commun. Math. Phys. 328(1), 355–395 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.L.: Quenched central limit theorems for the Ising model on random graphs. J. Stat. Phys. 160(6), 1623–1657 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Dommers, S., Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.L.: Ising critical behavior of inhomogeneous Curie–Weiss models and annealed random graphs. Commun. Math. Phys. 348(1), 221–263 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.: Annealed central limit theorems for the Ising model on random graphs. Alea Latin Am. J. Probab. Math. Stat. 13(1), 121–161 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Löwe, M., Schubert, K., Vermet, F.: Multi-group binary choice with social interaction and a random communication structure: a random graph approach (2019). arXiv:1904.11890

  18. Löwe, M., Schubert, K.: Fluctuations for block spin Ising models. Electron. Commun. Probab. 23, 12 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Davis, B., McDonald, D.: An elementary proof of the local central limit theorem. J. Theor. Probab. 8(3), 693–701 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Zakhar Kabluchko & Matthias Löwe

  2. Fakultät für Mathematik, TU Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Kristina Schubert

Authors
  1. Zakhar Kabluchko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias Löwe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kristina Schubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Löwe.

Additional information

Communicated by Hal Tasaki.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of the first two authors was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure. The third author has been supported by the German Research Foundation (DFG) via Research Training Group RTG 2131 High dimensional phenomena in probability—fluctuations and discontinuity.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kabluchko, Z., Löwe, M. & Schubert, K. Fluctuations of the Magnetization for Ising Models on Dense Erdős–Rényi Random Graphs. J Stat Phys 177, 78–94 (2019). https://doi.org/10.1007/s10955-019-02358-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-019-02358-5

Keywords

Mathematics Subject Classification

Search

Navigation