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

Loop statistics in the toroidal honeycomb dimer model