Analyticity properties and a convergent expansion for the inverse correlation length of the low-temperatured-dimensional Ising model

Abstract

We show that the inverse correlation lengthm(z) of the truncated spin-spin correlation function of theZ ^d Ising model with + or — boundary conditions admits the representationm(z) = −(4d−4)ln z(1−δ_d1) + r(z) for smallz=e ^−β, i.e., large inverse temperatures$\beta > 0.r(z) = \Sigma _{n = 1}^\infty b_{n^{Z^n } }$ is ad-dependent analytic function atz = 0, already known in closed form ford = 1 and 2; ford = 3 b_n can be computed explicitly from a finite number of the Z^d limits of z = 0 Taylor series coefficients of the finite lattice correlation function at a finite number of points ofZ ^d.