Weakly non-planar dimers

TL;DR

This paper investigates a two-dimensional bipartite dimer model on a non-planar graph, showing that for small non-planar edge weights, the height function converges to a Gaussian Free Field, revealing new universality features.

Contribution

It introduces a non-planar dimer model where traditional methods fail and demonstrates convergence to a Gaussian Free Field using fermionic Renormalization Group analysis.

Findings

Height function scales to Gaussian Free Field for small non-planar edge weights

Model maps to interacting lattice fermions in Luttinger universality class

Non-planarity prevents the use of Kasteleyn's theory, requiring new analysis methods

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 $\mathbb Z^2$ via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight $\lambda$ of the non-planar edges is small enough, a suitably defined height function scales on large distances to the Gaussian Free Field with a $\lambda$-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, 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.