On the two-point function of the Ising model with infinite range-interactions

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.