Ising Model with Curie–Weiss Perturbation

Consider the nearest-neighbor Ising model on $$\Lambda _n:=[-n,n]^d\cap {\mathbb {Z}}^d$$ at inverse temperature $$\beta \ge 0$$ with free boundary conditions, and let $$Y_n(\sigma ):=\sum _{u\in \Lambda _n}\sigma _u$$ be its total magnetization. Let $$X_n$$ magnetization perturbed by a critical Curie–Weiss interaction, i.e., $$\begin{aligned} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp [x^2/\left( 2\langle Y_n^2 \rangle _{\Lambda _n,\beta }\right) ]}{\left\langle \exp [Y_n^2/\left( Y_n^2\rangle ]\right\rangle }}, \end{aligned}$$ where $$F_{X_n}$$ $$F_{Y_n}$$ are distribution functions for $$Y_n$$ respectively. We prove that any $$d\ge 4$$ \in [0,\beta _c(d)]$$ _c(d)$$ is temperature, subsequential limit (in distribution) of $$\{X_n/\sqrt{{\mathbb {E}}\left( X_n^2\right) }:n\in {N}}\}$$ has an analytic density (say, $$f_X$$ ) all whose zeros pure imaginary, explicit expression in terms asymptotic behavior moment generating function . also 1$$ then $$ small, f_X(x)=K\exp (-C^4x^4), $$C=\sqrt{\Gamma (3/4)/\Gamma (1/4)}$$ $$K=\sqrt{\Gamma (3/4)}/(4\Gamma (5/4)^{3/2})$$ Possible connections between high-dimensional periodic conditions discussed.