Gradient fields on the lattice and their scaling limits

Random gradient fields are a class of statistical mechanics models that appear in a range of applications across mathematics and physics. In particular, they arise as models for random interfaces, in the study of elasticity and as a starting point for massless field theories. They are characterised by a Hamiltonian (energy function) that depends only on the discrete gradients of the field; the strength of the interaction is governed by a potential. Where the potential is a strictly convex function, many results have been obtained about these models over the last three decades. The field for non-convex potentials is less well surveyed. I will aim to give an accessible introduction to these models and their scaling limits, outline some of the main results for strictly convex potentials and then detail some recent advances for non-convex potentials using renormalisation group arguments. This is based on joint work with Stefan Adams.