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 \(J_{x}=\psi (x)\textsf{e}^{-\rho (x)}\) 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 \(\beta =\beta _\textrm{sat}(s)\) (where \(\beta _\textrm{sat}(s)\) 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 \([0,\beta _\textrm{sat}(s)))\). Moreover, we prove that the two-point function satisfies up-to-constants Ornstein-Zernike asymptotics for \(\beta =\beta _\textrm{sat}(s)\) on \(\mathbb {Z}\). As far as we know, this constitutes the first result on the behaviour of the two-point function at \(\beta _\textrm{sat}(s)\). 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 and their proofs are built on different results and ideas developed in Duminil-Copin and Tassion (Commun Math Phys 359(2):821–822, 2018) and Aoun et al. in (Commun Math Phys 386:433–467, 2021).