  In this article, we prove some results concerning the truncated two-point
function of the infinite-range Ising model above and below the critical
temperature. More precisely, if the coupling constants are of the form $J_{x}=
\psi(x)e^{ -\rho(x)}$ with $\rho$ some norm and $\psi$ an subexponential
correction, we show under appropriate assumptions that given
$s\in\mathbb{S}^{d-1}$, the Laplace transform of the two-point function in the
direction $s$ is infinite for $\beta=\beta_{\text{sat}}(s)$ (where
$\beta_{\text{sat}}(s)$ is a the biggest value such that the inverse
correlation length $\nu_{\beta}(s)$ associated to the truncated two-point
function is equal to $\rho(s)$ on $[0,\beta_{\text{sat}}(s)))$. Moreover, we
prove that the two-point function satisfies Ornstein-Zernike asymptotics for
$\beta=\beta_{\text{sat}}(s)$ on $\mathbb{Z}$. As far as we know, this
constitutes the first result on the behaviour of the two-point function at
$\beta_{\text{sat}}(s)$. Finally, we show that there exists $\beta_{0}$ such
that for every $\beta>\beta_{0}$, $\nu_{\beta}(s)=\rho(s)$. All the results are
new.
