Recoverable systems and the maximal hard-core model on the triangular lattice

Abstract

In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity.

Figures (8)

• Figure 1: Maximal independent sets in ${\mathbb Z}^2$ and ${\mathbb A}$.
• Figure 2: Connectivity and contours
• Figure 3: A coloring of the vertices of ${\mathbb A}$
• Figure 4: Illustrating the contour erasing procedure.
• Figure 5: Two possible cases with an empty triangle (filled in black), five of whose neighbor triangles (filled in gray) are also empty. Each of these arrangements forces an empty hexagon, so they cannot arise in a MIS. This shows that the degree of the graph associated with a contour is at most 4.

Theorems & Definitions (19)

• Definition 1.1: The maximal hard-core model
• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4