Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

We compute exactly the spin-spin correlation functions 〈σ0,0σM,N〉 for the two-dimensional Ising model on a square lattice in zero magnetic field for T>Tc and T<Tc. We then analyze the correlation functions in the scaling limit T→Tc,M²+N²→∞ such that (T-Tc) is fixed. In this scaling limit $〈{{σ}}0,0{{σ}}M,N〉={R}^{{-}1/4}{F}_{±{}}(t)+{R}^{{-}5/4}{F}_{1±{}}(t)+o({R}^{{-}5/4})$, where $t$ is the scaling variable ${R}{{ξ}}$ and ${F}_{±{}}(t)$ and ${F}_{1±{}}(t)$ are the scaling functions (${ξ}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for ${F}_{±{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0±{}}$ and ${C}_{1±{}}$ in the expansion ${k}BT{χ}(T)={C}_{0±{}}{|1{-}{{T}c}{T}|}^{{-}7/4}+{C}_{1±{}}{|1{-}{{T}c}{T}|}^{{-}3/4}+O(1)$ of the zero-field susceptibility ${χ}(T)$ as $T{→}{T}c^{±{}}$.