Ising model with Curie-Weiss perturbation

Consider the nearest-neighbor Ising model on $Λ_n:=[-n,n]^d\cap\mathbb{Z}^d$ at inverse temperature $β\geq 0$ with free boundary conditions, and let $Y_n(σ):=\sum_{u\inΛ_n}σ_u$ be its total magnetization. Let $X_n$ be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., \begin{equation*} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp[x^2/\left(2\langle Y_n^2 \rangle_{Λ_n,β}\right)]}{\left\langle\exp[Y_n^2/\left(2\langle Y_n^2\rangle_{Λ_n,β}\right)]\right\rangle_{Λ_n,β}}, \end{equation*} where $F_{X_n}$ and $F_{Y_n}$ are the distribution functions for $X_n$ and $Y_n$ respectively. We prove that for any $d\geq 4$ and $β\in[0,β_c(d)]$ where $β_c(d)$ is the critical inverse temperature, any subsequential limit (in distribution) of $\{X_n/\sqrt{\mathbb{E}\left(X_n^2\right)}:n\in\mathbb{N}\}$ 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 $d\geq 1$ and then for $β$ small, \begin{equation*} f_X(x)=K\exp(-C^4x^4), \end{equation*} where $C=\sqrt{Γ(3/4)/Γ(1/4)}$ and $K=\sqrt{Γ(3/4)}/(4Γ(5/4)^{3/2})$. Possible connections between $f_X$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

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