Ising Model with Curie–Weiss Perturbation

Consider the nearest-neighbor Ising model on Λ _n: = [- n, n] ^d∩ Z^d at inverse temperature β≥ 0 with free boundary conditions, and let Yn(σ):=∑u∈Λnσu be its total magnetization. Let X_n 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 X_n and Y_n respectively. We prove that for any d≥ 4 and β∈ [0 , β_c(d)] where β_c(d) is the critical inverse temperature, any subsequential limit (in distribution) of {Xn/E(Xn2):n∈N} has an analytic density (say, f_X) all of whose zeros are pure imaginary, and f_X has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of Y_n. 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 f_X and the high-dimensional critical Ising model with periodic boundary conditions are discussed.