# Classical Dimers on Penrose Tilings

## TL;DR

This paper investigates the classical dimer model on Penrose tilings, revealing the absence of perfect matchings, a characteristic monomer density, and the fractal structure of maximum matchings, with implications for charge distribution and tiling properties.

## Contribution

It introduces the first analysis of dimer coverings on Penrose tilings, including algorithms for maximum matchings and insights into their topological and charge properties.

## Key findings

- Penrose tilings lack perfect matchings.
- Maximum matchings have a monomer density of approximately 0.098.
- Maximum matchings form a connected manifold under local rearrangements.

## Abstract

We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of $81-50\varphi\approx 0.098$ in the thermodynamic limit, with $\varphi=\left(1+\sqrt{5}\right)/2$ the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings, and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements. We show that dart-kite Penrose tilings feature an imbalance of charge between bipartite sub-lattices, leading to a minimum monomer density of $\left(7-4\varphi\right)/5\approx 0.106$ all of one charge.