  Consider the nearest-neighbor Ising model on
$\Lambda_n:=[-n,n]^d\cap\mathbb{Z}^d$ at inverse temperature $\beta\geq 0$ with
free boundary conditions, and let $Y_n(\sigma):=\sum_{u\in\Lambda_n}\sigma_u$
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_{\Lambda_n,\beta}\right)]}{\left\langle\exp[Y_n^2/\left(2\langle
Y_n^2\rangle_{\Lambda_n,\beta}\right)]\right\rangle_{\Lambda_n,\beta}},
\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
$\beta\in[0,\beta_c(d)]$ where $\beta_c(d)$ 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
$\beta$ small, \begin{equation*}
  f_X(x)=K\exp(-C^4x^4), \end{equation*} where
$C=\sqrt{\Gamma(3/4)/\Gamma(1/4)}$ and
$K=\sqrt{\Gamma(3/4)}/(4\Gamma(5/4)^{3/2})$. Possible connections between $f_X$
and the high-dimensional critical Ising model with periodic boundary conditions
are discussed.
