Classical dimers on Penrose tilings

We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be
identified as 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 − 50φ ≅ 0.098 in
the thermodynamic limit, with φ= ( 1 = √5)/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-neighbor 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 sublattices, leading to a minimum monomer density of ð7 − 4φÞ=5 ≈ 0.106 all of one charge.