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 Y_n(σ ) :=∑u_∈Λnσ_u be its total magnetization. Let X_n be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., <disp-formula id="Equ156"> d/F_Xnd F_{Y n} (x ) :=exp[x/²/(2 ⟨Y_n²⟩_Λn,β)] «exp[Y_n²/close=")" open="(">2 ⟨Y_n²⟩_Λn,β»]Λ_n,β, </disp-formula>where F_Xn and F_{Y n} 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 {X_n/√{E (X_n²) }: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, <disp-formula id="Equ157"> f_X(x ) =K exp(-C⁴x⁴) , </disp-formula>where C =√{Γ (3 /4 )/Γ (1 /4 ) } and K =√{Γ (3 /4 ) }/(4 Γ (5^{/4 ) 3 /2}) . Possible connections between f_X and the high-dimensional critical Ising model with periodic boundary conditions are discussed.