The discrete Gaussian model, II. Infinite-volume scaling limit at high temperature

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integervalued.In two dimensions, at sufficiently high temperature, we show that the scaling limit of the infinite-volume gradient Gibbs state with zero mean is a multiple of the Gaussian free field.This article is the second in a series on the Discrete Gaussian model, extending the methods of the first paper by the analysis of general external fields (rather than macroscopic test functions on the torus).As a byproduct, we also obtain a scaling limit for mesoscopic test functions on the torus.