Ising Model Scaling Functions at Short Distance

I will sketch a proof that the short distance behavior of the even scaling functions for the Ising model that arise by taking the scaling limit of the Ising correlation functions from below the critical temperature is given by the Luther–Peschel formula [6] (see below). The fact that the Luther–Peschel formula is consistent with conformal field theory insights into the correlations for the large scale limit of the critical Ising model has lead to the conviction in the Physics community that this formula is right [4]. The critical scaling limit for the Ising model is obtained by sitting right at T c (the critical temperature) and extracting the large scale limit of the correlations at this fixed temperature. However, as far as I am aware there is no mathematical proof that the large scale limit of the critical Ising model even exists (with the exception of the asymptotics of the diagonal two point correlation at the critical point [15]). In this note we will not address the question of the critical (massless) scaling limit for the Ising model but instead we will consider the massive scaling limit obtained by looking at the lattice correlations at the scale of the correlation length as the temperature approaches the critical temperature. This scaling limit was first considered in [14] and [11] and a mathematical proof of its existence was established in [7]. What we will sketch a proof of here is that the short distance asymptotics of this scaling limit for the even scaling function from below T c is given by the Luther–Peschel formula. There is a conjecture in the Physics literature called the scaling hypothesis which suggests that the large scale asymptotics at the critical temperature should agree with the short distance behaviour of the massive scaling theory. A rather detailed version of this scaling hypothesis has been proved for the two point function [12],[13]. Modulo the fact that this scaling hypothesis can only be literally correct for the scaling limit from above T c (since the critical Ising correlations vanish for an odd number of spins on the lattice and the odd scaled correlations from below T c are non-zero) this hypothesis does suggest a connection between the result we consider here and the conjectured result for the critical scaling limit. Incidentally, the technique to be explained below should extend to an analysis of the short distance asymptotics of the …