Scaling functions in the square Ising model

Abstract: We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $ F_{\pm}(r)= \lim_{scaling} {\cal M}{\pm}^{-2} < \sigma{0,0} \, \sigma_{M,N}> = \sum_{n} I_{n}(r)$. The integrals $ I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $ r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlev\'e VI equation in the scaling limit. Combinations of the (analytic at $ r= 0$) solutions of $ {\cal L}^{scal}_q$ sum to $ \exp(r^2/8)$. We show that the expression $ r^{1/4} \exp(r^2/8)$ is the scaling limit of the correlation function $ C(N, N)$ and $ C(N, N+1)$. The differential Galois groups of the factors occurring in the operators $ {\cal L}^{scal}_q$ are given.