Two-point-function dichotomy below the critical temperature

Let $s\in\mathbb{S}^{d-1}$, let $n$ tend to infinity, let $\beta_{\rm sat}^{*}(s)$ be the saturation threshold for the truncated two-point function, let $\beta_{\rm c}$ be the critical inverse temperature, let $J_{ns}$ be the coupling at displacement $ns$, and let $\nu_{\beta}(x)$ denote the inverse correlation length. Two-point-function dichotomy conjecture. The following asymptotics hold:

\begin{enumerate} \item \text{For }\beta>\beta_{\rm sat}^{*}(s),\text{ there exists }C>0\text{ such that }\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}=CJ_{ns}(1+o_n(1)). \item \text{For }\beta\in(\beta_{\rm c},\beta_{\rm sat}^{*}(s)),\text{ there exists }C>0\text{ such that }\langle\sigma_{0};\sigma_{ns}\rangle_{\beta}=Cn^{-\frac{d-1}{2}}e^{-\nu_{\beta}(x)}(1+o_n(1)). \end{enumerate}

This extends the expected distinction between a one-jump regime and an Ornstein–Zernike regime below the critical temperature. The source presents it as an expectation, and no resolution is supplied.