A Nearest-Neighbor Hard-Core Model on a Penrose Graph

Abstract

We prove that the maximal graph-density of an independent set in a Penrose P3 tiling considered as a planar non-directed graph is equal to $(57-25\sqrt{5})/2\approx 0.54915$ despite the fact that the graph is bipartite. Accordingly, the extreme Gibbs measure of the nearest-neighbor hard core particle model on this graph is unique for sufficiently large values of the particle activity. This invalidates a natural expectation to observe the coexistence of even and odd phases.