Ising Model with Curie–Weiss Perturbation

Federico Camia, Jianping Jiang*, Charles M. Newman

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Consider the nearest-neighbor Ising model on Λ n: = [- n, n] d∩ Zd at inverse temperature β≥ 0 with free boundary conditions, and let Yn(σ):=∑u∈Λnσu be its total magnetization. Let Xn be the total magnetization perturbed by a critical Curie–Weiss interaction, i.e., dFXndFYn(x):=exp[x2/(2⟨Yn2⟩Λn,β)]〈exp[Yn2/(2⟨Yn2⟩Λn,β)]〉Λn,β,where FXn and FYn are the distribution functions for Xn and Yn respectively. We prove that for any d≥ 4 and β∈ [0 , βc(d)] where βc(d) is the critical inverse temperature, any subsequential limit (in distribution) of {Xn/E(Xn2):n∈N} has an analytic density (say, fX) all of whose zeros are pure imaginary, and fX has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of Yn. We also prove that for any d≥ 1 and then for β small, fX(x)=Kexp(-C4x4),where C=Γ(3/4)/Γ(1/4) and K=Γ(3/4)/(4Γ(5/4)3/2). Possible connections between fX and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

Original languageEnglish
Article number5
Pages (from-to)1-23
Number of pages23
JournalJournal of Statistical Physics
Volume188
Issue number1
Early online date16 May 2022
DOIs
Publication statusPublished - Jul 2022

Bibliographical note

Funding Information:
The research of the second author was partially supported by NSFC grant 11901394 and that of the third author by US-NSF grant DMS-1507019. The authors thank two anonymous reviewers for useful comments and suggestions.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Funding

The research of the second author was partially supported by NSFC grant 11901394 and that of the third author by US-NSF grant DMS-1507019. The authors thank two anonymous reviewers for useful comments and suggestions.

FundersFunder number
National Science FoundationDMS-1507019
National Science Foundation
National Natural Science Foundation of China11901394
National Natural Science Foundation of China

    Keywords

    • Analytic density
    • Curie–Weiss interaction
    • High dimensions
    • Ising model
    • Periodic boundary conditions
    • Pure imaginary zeros

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