Massive Scaling Limit of the Ising Model: Subcritical Analysis and Isomonodromy

Abstract

We study the spin n-point functions of the planar Ising model on a simply connected domain \Omega discretised by the square lattice \delta\mathbb{Z}^{2} under near-critical scaling limit. While the scaling limit on the full-plane \mathbb{C} has been analysed in terms of a fermionic field theory, the limit in general \Omega has not been studied. We will show that, in a massive scaling limit wherein the inverse temperature is scaled \beta\sim\beta_{c}-m_{0}\delta for a constant m_{0}<0, the renormalised spin correlations converge to a continuous quantity determined by a boundary value problem set in \Omega. In the case of \Omega=\mathbb{C} and n=2, this result reproduces the celebrated formula of [WMTB76] involving the Painlev\'e III transcendent. To this end, we generalise the comprehensive discrete complex analytic framework used in the critical setting to the massive setting, which results in a perturbation of the usual notions of analyticity and harmonicity.