Felix Flicker

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. We find 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. 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. We find various other phenomena in dimer models on other quasicrystals.

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