The Discrete Gaussian model, I. Renormalisation group flow at high temperature

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature.