Height fluctuations and universality relations in interacting dimer models

Two-dimensional dimer models are popular models, which are used to describe either the liquid phase of dense anisotropic molecules or, thanks to a well-known mapping between dimer configurations and discrete height functions, the rough phase of fluctuating random surfaces. The last few years witnessed important progresses in the understanding of the critical phase of dimer systems, including the proof of existence and conformal covariance of the scaling limit, and the proof of convergence of the discrete height field to the massless Gaussian Free Field (GFF), due to Kenyon, Okounkov and Sheffield. In this talk I will review some aspects of the theory of critical dimer models, which is based, in large part, on the celebrated Kasteleyn solution for `non-interacting' dimers, combined with discrete holomorphicity methods. I will also discuss a novel approach to *interacting* dimer models, based on constructive Renormalization Group techniques, which recently allowed us to prove the convergence of the discrete height function to the GFF, in the presence of non-integrable perturbations of the dimers' Gibbs measure, as well as the validity of a universality relation between the renormalized variance of the GFF and the critical exponent of the dimer-dimer correlations (in collaboration with V. Mastropietro and F. Toninelli)