We consider interacting dimers on the two-dimensional square lattice.
The non-interacting model is ``integrable'' and solvable via
Kasteleyn theory, but interactions destroy integrability and have
prevented up to now a rigorous understanding.
By using Constructive Renormalization Group
we prove that all the moments of the height difference converge to those
of the Gaussian Free Field. Remarkably, dimer-dimer correlation
functions are instead not universal and
decay with a critical exponent that depends on the interaction
strength. The height difference takes the form of a non-local
fermionic operator, consisting of a sum of fermionic monomials along
an arbitrary path; this path-independence plays a crucial role
eliminating spurious ultraviolet divergences.
Work in collaboration with F. Toninelli and A. Giuliani (arXiv:1406.7710)