Ising Model with Curie–Weiss Perturbation

Federico Camia, Jianping Jiang, Charles M. Newman

Research output: Contribution to journalArticlepeer-review


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 (US)
Article number5
JournalJournal of Statistical Physics
Issue number1
StatePublished - Jul 2022


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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