Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees

  • Sander Dommers
  • Cristian Giardinà
  • Claudio Giberti
  • Remco van der Hofstad


We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs.


Random graphs Ising model Annealing Large deviations 



Sander Dommers has been supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity. We acknowledge financial support from the Italian Research Funding Agency (MIUR) through FIRB project “Stochastic processes in interacting particle systems: duality, metastability and their applications”, Grant No. RBFR10N90W. The work of Remco van der Hofstad is supported in part by the Netherlands Organisation for Scientific Research (NWO) through VICI Grant 639.033.806 and the Gravitation Networks Grant 024.002.003. C. Giberti and C. Giardinà acknowledge financial supports from “Fondo di Ateneo per la Ricerca 2015” and “Fondo di Ateneo per la Ricerca 2016”, Università di Modena e Reggio Emilia.


  1. 1.
    Bollobás, B.: Random Graphs, Volume 73 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2001)Google Scholar
  2. 2.
    Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Algorithms 31(1), 3–122 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124(6), 1377–1397 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Can, V.: Annealed limit theorems for the Ising model on random regular graphs. Available at arXiv:1701.08639 [math.PR], Preprint (2017)
  5. 5.
    Can, V.: Critical behavior of the annealed Ising model on random regular graphs. Available at arXiv:1701.08628 [math.PR], Preprint (2017)
  6. 6.
    Chung, F., Lu, L.: Connected components in random graphs with given expected degree sequences. Ann. Comb. 6(2), 125–145 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chung, F., Lu, L.: Complex Graphs and Nnetworks, Volume 107 of CBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC (2006)Google Scholar
  8. 8.
    Dembo, A., Montanari, A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20, 565–592 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dembo, A., Montanari, A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24(2), 137–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dembo, A., Zeitoni, O.: Large Deviations Techniques and Applications. Springer, New York (2009)Google Scholar
  11. 11.
    Dommers, S., Giardinà, C., van der Hofstad, R.: Ising models on power-law random graphs. J. Stat. Phys. 141(4), 638–660 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dommers, S., Giardinà, C., van der Hofstad, R.: Ising critical exponents on random trees and graphs. Commun. Math. Phys. 328(1), 355–395 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dommers, S., Giardinà, C., Giberti, C., Hofstad, R.D., Prioriello, M.: Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs. Commun. Math. Phys. 348(1), 221–263 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dommers, S., Külske, C., Schriever, P.: Continuous spin models on annealed generalized random graphs. Stoch. Process. Appl. 127(11), 3719–3753 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dorogovtsev, S., Goltsev, A., Mendes, J.: Critical phenomena in complex networks. Rev. Mod. Phys. 80(4), 1275–1335 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Ellis, R.S.: The theory of large deviations: from Boltzmann’s 1877 calculation to equilibrium macro states in 2D turbulence. Physica D 133, 106–136 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ellis, R.S.: Entropy. Large Deviations and Statistical Mechanics. Springer, New York (2006)zbMATHGoogle Scholar
  18. 18.
    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, 1623–1657 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.L.: Annealed central limit theorems for the Ising model on random graphs. ALEA Latin Am. J. Prob. Math. Stat. 13, 121–161 (2016)zbMATHGoogle Scholar
  20. 20.
    Janson, S.: Asymptotic equivalence and contiguity of some random graphs. Random Struct. Algorithms 36(1), 26–45 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Janson, S., Łuczak, T., Rucinski, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Leone, M., Vázquez, A., Vespignani, A., Zecchina, R.: Ferromagnetic ordering in graphs with arbitrary degree distribution. Eur. Phys. J. B Condens. Matter Complex Syst. 28(2), 191–197 (2002)Google Scholar
  23. 23.
    Norros, I., Reittu, H.: On a conditionally Poissonian graph process. Adv. Appl. Probab. 38(1), 59–75 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    van der Hofstad, R.: Random graphs and complex networks, vol. 1. Cambridge University Press, Cambridge, Cambridge Series in Statistical and Probabilistic Mathematics (2017)CrossRefzbMATHGoogle Scholar
  25. 25.
    van der Hofstad, R.: Stochastic processes on random graphs. In: Lecture Notes for Saint-Flour Summer School 2017 (2018)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Ruhr-University BochumBochumGermany
  2. 2.University of Modena and Reggio EmiliaModenaItaly
  3. 3.University of Modena and Reggio EmiliaReggio EmiliaItaly
  4. 4.Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations