The scaling limit of the 2D discrete Gaussian model at high temperature
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.