# 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