On the Two-Point Function of the Ising Model with Infinite-Range Interactions

Abstract

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 $$J_{x}=\psi (x)\textsf{e}^{-\rho (x)}$$ J x = ψ ( x ) e - ρ ( x ) with $$\rho $$ ρ some norm and $$\psi $$ ψ an subexponential correction, we show under appropriate assumptions that given $$s\in \mathbb {S}^{d-1}$$ s ∈ S d - 1 , the Laplace transform of the two-point function in the direction s is infinite for $$\beta =\beta _\textrm{sat}(s)$$ β = β sat ( s ) (where $$\beta _\textrm{sat}(s)$$ β sat ( s ) is a the biggest value such that the inverse correlation length $$\nu _{\beta }(s)$$ ν β ( s ) associated to the truncated two-point function is equal to $$\rho (s)$$ ρ ( s ) on $$[0,\beta _\textrm{sat}(s)))$$ [ 0 , β sat ( s ) ) ) . Moreover, we prove that the two-point function satisfies up-to-constants Ornstein-Zernike asymptotics for $$\beta =\beta _\textrm{sat}(s)$$ β = β sat ( s ) on $$\mathbb {Z}$$ Z . As far as we know, this constitutes the first result on the behaviour of the two-point function at $$\beta _\textrm{sat}(s)$$ β sat ( s ) . Finally, we show that there exists $$\beta _{0}$$ β 0 such that for every $$\beta >\beta _{0}$$ β > β 0 , $$\nu _{\beta }(s)=\rho (s)$$ ν β ( s ) = ρ ( 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).