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

We compute exactly the spin-spin correlation functions $〈{\sigma }_{0,0}{\sigma }_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>\mathrm{}{T}_c$ and $T<\mathrm{}{T}_c$. We then analyze the correlation functions in the scaling limit $T\to T_c,M^2+N^2\to \infty$ such that $(T-T_c)$ is fixed. In this scaling limit $〈{\sigma }_{0,0}{\sigma }_{M,N}〉=R^{-\frac{1}{4}}{F}_{\pm }\mathrm{}\left(t\right)+R^{-\frac{5}{4}}{F}_{1\pm }\mathrm{}\left(t\right)+o\left(R^{-\frac{5}{4}}\right)$, where $t$ is the scaling variable $\frac{R}{\xi }$ and $F_{\pm }\mathrm{}\left(t\right)\mathrm{}$ and $F_{1\pm }\mathrm{}\left(t\right)\mathrm{}$ are the scaling functions ($\xi$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlevé function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for $F_{\pm }\mathrm{}\left(t\right)\mathrm{}$ (good to ten significant digits) is also given. As an application we computer the coefficients $C_{0\pm }$ and $C_{1\pm }$ in the expansion $k_{B}T\chi \mathrm{}\left(T\right)=C_{0\pm }{|1-\frac{T_c}{T}|}^{-\frac{7}{4}}+C_{1\pm }{|1-\frac{T_c}{T}|}^{-\frac{3}{4}}+O\left(1\right)\mathrm{}$ of the zero-field susceptibility $\chi \mathrm{}\left(T\right)\mathrm{}$ as $T\to T_c^{\pm }$.