Electronic Communications in Probability

Fluctuations for block spin Ising models

Matthias Löwe and Kristina Schubert

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We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet et al. in [2]. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we prove a non-standard CLT for the magnetization.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 53, 12 pp.

Received: 22 June 2018
Accepted: 6 August 2018
First available in Project Euclid: 1 September 2018

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 82B05: Classical equilibrium statistical mechanics (general) 60G09: Exchangeability

Ising model Curie-Weiss model fluctuations Central Limit Theorem block model

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Löwe, Matthias; Schubert, Kristina. Fluctuations for block spin Ising models. Electron. Commun. Probab. 23 (2018), paper no. 53, 12 pp. doi:10.1214/18-ECP161. https://projecteuclid.org/euclid.ecp/1535767264

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  • [1] Arash A. Amini and Elizaveta Levina, On semidefinite relaxations for the block model, Ann. Statist. 46 (2018), no. 1, 149–179.
  • [2] Quentin Berthet, Philippe Rigollet, and Piyush Srivastavaz, Exact recovery in the ising blockmodel, Preprint, arXiv:1612.03880v1 (2016), 1–29.
  • [3] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.
  • [4] Guy Bresler, Efficiently learning Ising models on arbitrary graphs [extended abstract], STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing, ACM, New York, 2015, pp. 771–782.
  • [5] Guy Bresler, Elchanan Mossel, and Allan Sly, Reconstruction of Markov random fields from samples: some observations and algorithms, SIAM J. Comput. 42 (2013), no. 2, 563–578.
  • [6] Francesca Collet, Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach, J. Stat. Phys. 157 (2014), no. 6, 1301–1319.
  • [7] Pierluigi Contucci, Ignacio Gallo, and Giulia Menconi, Phase transitions in social sciences: Two-population mean field theory, International Journal of Modern Physics B 22 (2008), no. 14, 2199–2212.
  • [8] Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010, Corrected reprint of the second (1998) edition.
  • [9] Frank den Hollander, Large deviations, Fields Institute Monographs, vol. 14, American Mathematical Society, Providence, RI, 2000.
  • [10] Peter Eichelsbacher and Matthias Löwe, Stein’s method for dependent random variables occurring in statistical mechanics, Electron. J. Probab. 15 (2010), no. 30, 962–988.
  • [11] Richard S. Ellis, Entropy, large deviations, and statistical mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1985 original.
  • [12] Richard S. Ellis and Charles M. Newman, Limit theorems for sums of dependent random variables occurring in statistical mechanics, Z. Wahrsch. Verw. Gebiete 44 (1978), no. 2, 117–139.
  • [13] Richard S. Ellis and Charles M. Newman, The statistics of Curie-Weiss models, J. Statist. Phys. 19 (1978), no. 2, 149–161.
  • [14] Micaela Fedele and Francesco Unguendoli, Rigorous results on the bipartite mean-field model, J. Phys. A 45 (2012), no. 38, 385001, 18.
  • [15] Ignacio Gallo, Adriano Barra, and Pierluigi Contucci, Parameter evaluation of a simple mean-field model of social interaction, Math. Models Methods Appl. Sci. 19 (2009), no. suppl., 1427–1439.
  • [16] Ignacio Gallo and Pierluigi Contucci, Bipartite mean field spin systems. Existence and solution, Math. Phys. Electron. J. 14 (2008), Paper 1, 21.
  • [17] Chao Gao, Zongming Ma, Anderson Y. Zhang, and Harrison H. Zhou, Achieving optimal misclassification proportion in stochastic block models, J. Mach. Learn. Res. 18 (2017), Paper No. 60, 45.
  • [18] Barbara Gentz and Matthias Löwe, The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature, Probab. Theory Related Fields 115 (1999), no. 3, 357–381.
  • [19] Werner Kirsch and Gabor Toth, Two groups in a Curie-Weiss model, preprint, arXiv:1712.08477 (2017).
  • [20] Werner Kirsch and Gabor Toth, Two groups in a Curie-Weiss model with heterogeneous coupling, preprint (2018).
  • [21] Matthias Löwe, Kristina Schubert, and Franck Vermet, Block spin Ising models on random graphs, preprint, in preparation (2018).
  • [22] Elchanan Mossel, Joe Neeman, and Allan Sly, Belief propagation, robust reconstruction and optimal recovery of block models, Ann. Appl. Probab. 26 (2016), no. 4, 2211–2256.