June 2023 Fluctuations in mean-field Ising models
Nabarun Deb, Sumit Mukherjee
Author Affiliations +
Ann. Appl. Probab. 33(3): 1961-2003 (June 2023). DOI: 10.1214/22-AAP1857


In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately dN regular graph GN on N vertices. In particular, if GN satisfies a “spectral gap” condition, we show that whenever dNN, the fluctuations are universal and the same as that of the Curie–Weiss model in the entire ferromagnetic parameter regime. We give a counterexample to demonstrate that the condition dNN is tight, in the sense that the limiting distribution changes if dNN except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to dNN1/3. Our results include universal fluctuations of the average magnetization in Ising models on regular graphs, Erdős–Rényi graphs (directed and undirected), stochastic block models, and sparse regular graphons. In fact, our results apply to general matrices with nonnegative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we obtain Berry–Esseen bounds for these fluctuations, exponential concentration for the average of spins, tight error bounds for the mean-field approximation of the partition function, and tail bounds for various statistics of interest.

Funding Statement

SM gratefully acknowledges the partial support of NSF (DMS-1712037) during this research.


The authors would like to thank the Editor, the Associate Editor, and the two anonymous reviewers for their constructive suggestions that helped improve the presentation of this paper.


Download Citation

Nabarun Deb. Sumit Mukherjee. "Fluctuations in mean-field Ising models." Ann. Appl. Probab. 33 (3) 1961 - 2003, June 2023. https://doi.org/10.1214/22-AAP1857


Received: 1 January 2021; Revised: 1 July 2021; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583662
zbMATH: 1517.82013
Digital Object Identifier: 10.1214/22-AAP1857

Primary: 82B20
Secondary: 82B26

Keywords: Berry–Esseen bound , Ising model , mean-field , Partition function , regular graphs

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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