Classical Dimers on Penrose Tilings

We study the classical dimer model on rhombic Penrose tilings, whose
edges and vertices may be identified with those of a bipartite graph.
Wefind that Penrose tilings do not admit perfect matchings (dimer
coverings free of unmatched vertices, or 'monomer' defects).
Instead,their maximum matchings (with the fewest monomers) have a monomer
density of 81-50*(the golden ratio), approximately 0.098, in the
thermodynamic limit.
Such matchings divide the tiling into a fractal of
nested closed regions.
We devise a simple algorithm for generating
maximum matchings, and demonstrate that maximum matchings form a
connected manifold under local monomer-dimer rearrangements.
Turning to related structures, we show that dart-kite Penrose tilings instead
feature an imbalance of charge between bipartite sub-lattices, leading
to a minimum monomer density of (7-4*(the golden ratio))/5, around
0.106, all of one charge.
With summer student Jerome Lloyd we show that
the Ammann-Beenker tiling admits perfect matchings. Various other
phenomena are shown by dimer models on other quasicrystals.

F. Flicker, S. H. Simon, and S. A. Parameswaran, arXiv 1902.02799