The discrete Gaussian model is a random lattice field model imitating the Gaussian free field but restricted to taking integer values. Given a lattice, assigning an integer value to each lattice site would give a configuration, and the probability of exhibiting a certain configuration is weighted by measuring the total amount of gradients of the configuration. Because of its relation with some fundamental problems in physics, such as U(1) gauge field theory and the Kosterlitz-Thouless phase transition in XY model, this model had drawn attention from a number of mathematical physicists.
Despite the growing understanding of this topic recently, studying the exact limiting behaviour of related models often turn out to be challenging. In this talk, I will describe why the scaling limit of the 2D discrete Gaussian model at high temperature is lucky enough for its scaling limit can be described with great precision. The method consists firstly of `smoothing’ out the model and secondly of running a renormalisation group argument.