  The dimer model on a graph embedded in the torus can be interpreted as a
collection of random self-avoiding loops. In this paper, we consider the
uniform toroidal honeycomb dimer model. We prove that when the mesh of the
graph tends to zero and the aspect of the torus is fixed, the winding number of
the collection of loops converges in law to a two-dimensional discrete Gaussian
distribution. This is known to physicists in more generality from their
analysis of toroidal two-dimensional critical loop models and their mapping to
the massless free field on the torus. This paper contains the first
mathematical proof of this more general physics result in the specific case of
the loop model induced by a toroidal dimer model.
