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 language | English |
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Article number | 5 |
Pages (from-to) | 1-23 |
Number of pages | 23 |
Journal | Journal of Statistical Physics |
Volume | 188 |
Issue number | 1 |
Early online date | 16 May 2022 |
DOIs | |
Publication status | Published - 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.
Funders | Funder number |
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National Science Foundation | DMS-1507019 |
National Science Foundation | |
National Natural Science Foundation of China | 11901394 |
National Natural Science Foundation of China |
Keywords
- Analytic density
- Curie–Weiss interaction
- High dimensions
- Ising model
- Periodic boundary conditions
- Pure imaginary zeros