Scaling functions in the square Ising model

Abstract

We show and give the linear differential operators ${\mathcal{L}}_q^{scal}$ 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 $F_{\pm }(r)={\lim }_{scaling}{\mathcal{M}}_{\pm }^{-2}<{\sigma }_{0,0}{\sigma }_{M,N}>={\sum }_n{I}_n(r)$. The integrals $I_n(r)$ are given in expansion around r= 0 in the basis of the formal solutions of ${\mathcal{L}}_q^{scal}$ with transcendental combination coefficients. We find that the expression $r^{1/4}\exp (r^2/8)$ is a solution of the Painlevé VI equation in the scaling limit. Combinations of the (analytic at $r=0$) solutions of ${\mathcal{L}}_q^{scal}$ 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 ${\mathcal{L}}_q^{scal}$ are given.

26 pages

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Cited by in corpus (2)

• Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics