The Annals of Applied Probability

Annealed limit theorems for the Ising model on random regular graphs

Van Hao Can

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In a recent paper, Giardinà et al. [ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121–161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of configuration model random graphs.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1398-1445.

Dates
Received: May 2017
Revised: September 2017
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566834

Digital Object Identifier
doi:10.1214/17-AAP1377

Mathematical Reviews number (MathSciNet)
MR3914548

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F5
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Ising model random graphs central limit theorems annealed measure

Citation

Can, Van Hao. Annealed limit theorems for the Ising model on random regular graphs. Ann. Appl. Probab. 29 (2019), no. 3, 1398--1445. doi:10.1214/17-AAP1377. https://projecteuclid.org/euclid.aoap/1550566834


Export citation

References

  • [1] Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47–97.
  • [2] Basak, A. and Dembo, A. (2017). Ferromagnetic Ising measures on large locally tree-like graphs. Ann. Probab. 45 780–823.
    Zentralblatt MATH: 1372.05032
    Digital Object Identifier: doi:10.1214/15-AOP1075
    Project Euclid: euclid.aop/1490947308
  • [3] Bianconi, G. (2012). Superconductor-insulator transition on annealed complex networks. Phys. Rev. E 85 061113.
  • [4] Camia, F., Garban, C. and Newman, C. M. (2015). Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43 528–571.
    Zentralblatt MATH: 1332.82012
    Digital Object Identifier: doi:10.1214/13-AOP881
    Project Euclid: euclid.aop/1422885569
  • [5] Can, V. H. (2017). Critical behavior of the annealed Ising model on random regular graphs. J. Stat. Phys. 169 480–503.
    Zentralblatt MATH: 1382.82007
    Digital Object Identifier: doi:10.1007/s10955-017-1879-7
  • [6] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
    Zentralblatt MATH: 1191.82025
    Digital Object Identifier: doi:10.1214/09-AAP627
    Project Euclid: euclid.aoap/1268143433
  • [7] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
    Zentralblatt MATH: 1205.05209
    Digital Object Identifier: doi:10.1214/09-BJPS027
    Project Euclid: euclid.bjps/1271770268
  • [8] Dembo, A., Montanari, A., Sly, A. and Sun, N. (2014). The replica symmetric solution for Potts models on $d$-regular graphs. Comm. Math. Phys. 327 551–575.
    Mathematical Reviews (MathSciNet): MR3183409
    Zentralblatt MATH: 1288.82009
    Digital Object Identifier: doi:10.1007/s00220-014-1956-6
  • [9] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
    Zentralblatt MATH: 1177.60035
  • [10] Dommers, S., den Hollander, F., Jovanovski, O. and Nardi, F. R. (2017). Metastability for Glauber dynamics on random graphs. Ann. Appl. Probab. 27 2130–2158.
    Zentralblatt MATH: 1377.60020
    Digital Object Identifier: doi:10.1214/16-AAP1251
    Project Euclid: euclid.aoap/1504080028
  • [11] Dommers, S., Giardinà, C., Giberti, C., van der Hofstad, R. and Prioriello, M. L. (2016). Ising critical behavior of inhomogeneous Curie–Weiss models and annealed random graphs. Comm. Math. Phys. 348 221–263.
    Zentralblatt MATH: 1359.82020
    Digital Object Identifier: doi:10.1007/s00220-016-2752-2
  • [12] Dommers, S., Giardinà, C. and van der Hofstad, R. (2014). Ising critical exponents on random trees and graphs. Comm. Math. Phys. 328 355–395.
    Zentralblatt MATH: 1292.82004
    Digital Object Identifier: doi:10.1007/s00220-014-1992-2
  • [13] Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F. (2002). Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E 66 016104.
  • [14] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York.
    Zentralblatt MATH: 0566.60097
  • [15] Ellis, R. S. and Newman, C. M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
    Zentralblatt MATH: 0364.60120
    Digital Object Identifier: doi:10.1007/BF00533049
  • [16] Giardinà, C., Giberti, C., van der Hofstad, R. and Prioriello, M. L. (2015). Quenched central limit theorems for the Ising model on random graphs. J. Stat. Phys. 160 1623–1657.
    Zentralblatt MATH: 1327.82013
    Digital Object Identifier: doi:10.1007/s10955-015-1302-1
  • [17] Giardinà, C., Giberti, C., van der Hofstad, R. and Prioriello, M. L. (2016). Annealed central limit theorems for the Ising model on random graphs. ALEA Lat. Am. J. Probab. Math. Stat. 13 121–161.
    Zentralblatt MATH: 1331.05193
  • [18] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [19] Ising, E. (1924). Beitrag zur Theorie des Ferro- und Paramagnetismus. Ph.D. thesis, Univ. Hamburg.
  • [20] Montanari, A., Mossel, E. and Sly, A. (2012). The weak limit of Ising models on locally tree-like graphs. Probab. Theory Related Fields 152 31–51.
    Zentralblatt MATH: 1242.82012
    Digital Object Identifier: doi:10.1007/s00440-010-0315-6
  • [21] Mossel, E. and Sly, A. (2013). Exact thresholds for Ising–Gibbs samplers on general graphs. Ann. Probab. 41 294–328.
    Zentralblatt MATH: 1270.60113
    Digital Object Identifier: doi:10.1214/11-AOP737
    Project Euclid: euclid.aop/1358951988
  • [22] Robbins, H. (1955). A remark on Stirling’s formula. Amer. Math. Monthly 62 26–29.
    Zentralblatt MATH: 0068.05404
  • [23] van der Hofstad, R. Random graphs and complex networks. Available at: http://www.win.tue.nl/~rhofstad/NotesRGCN.html.
    Zentralblatt MATH: 1361.05002

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more