Scaling limit of gradient models on $\mathbb{Z}^d$ with non-convex energy

Random fields of gradients are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. The models we consider are characterised by an imposed boundary tilt and the free energy (called surface tension in the context of random interface models) as a function of tilt. Of interest are, in particular, whether the surface tension is strictly convex and whether the large-scale behaviour of the model remains that of the massless free field (Gaussian universality class). Where the Hamiltonian (energy) of the system is determined by a strictly convex potential, good progress has been made on these questions over the last three decades. For models with non-convex energy fewer results are known. Open problems include the conjecture (verified recently in the strictly convex case) that, in any regime where the scaling limit is Gaussian, its covariance (diffusion) matrix should be given by the Hessian of surface tension as a function of tilt. I will survey some recent advances in this direction using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit on the torus and in infinite volume for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.