TY - JOUR

T1 - Ising Model with Curie–Weiss Perturbation

AU - Camia, Federico

AU - Jiang, Jianping

AU - Newman, Charles M.

N1 - 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.

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - Analytic density

KW - Curie–Weiss interaction

KW - High dimensions

KW - Ising model

KW - Periodic boundary conditions

KW - Pure imaginary zeros

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U2 - 10.1007/s10955-022-02935-1

DO - 10.1007/s10955-022-02935-1

M3 - Article

AN - SCOPUS:85130150759

VL - 188

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

M1 - 5

ER -