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

TL;DR

This paper determines the maximum independent set density in a Penrose tiling graph and shows the uniqueness of the Gibbs measure for a related particle model at high activity levels.

Contribution

It establishes the exact maximal independent set density in a Penrose P3 tiling and proves the uniqueness of the Gibbs measure for the nearest-neighbor hard-core model at high activity.

Findings

Maximal independent set density is approximately 0.54915.

The Gibbs measure is unique for large particle activity.

Even and odd phase coexistence does not occur in this model.

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.