Weakly nonplanar dimers

Abstract

We study a model of fully packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from Z² via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight λ of the nonplanar edges is small enough, a suitably defined height function scales on large distances to the Gaussian free field with a λ-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of nonplanarity, Kasteleyn’s theory does not apply: the model is not integrable. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, which we then analyze via fermionic renormalization group methods.