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arxiv: 2604.21086 · v1 · submitted 2026-04-22 · 🧮 math-ph · math.MP· math.PR

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

A. Mazel , I. Stuhl , Y. Suhov This is my paper

Pith reviewed 2026-05-09 22:28 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR
keywords Penrose tilingindependent setshard-core modelGibbs measuresbipartite graphsaperiodic tilingsquasicrystalsphase uniqueness
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The pith

The Penrose P3 tiling graph has a maximal independent set density of (57 - 25√5)/2 ≈ 0.54915, greater than 1/2 despite being bipartite.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the largest fraction of vertices that can be chosen without adjacent pairs in the graph of the Penrose P3 tiling is exactly (57 - 25 √5)/2. This value is higher than the 1/2 limit expected for bipartite graphs. As a result, the nearest-neighbor hard-core model on this graph has a unique extreme Gibbs measure at high particle activity. This finding shows that the usual expectation of even and odd phase coexistence does not hold here.

Core claim

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 √5)/2 ≈ 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.

What carries the argument

The Penrose P3 tiling graph, whose vertex configurations from the aperiodic geometric structure allow exact computation of the maximal independent set density.

Load-bearing premise

The Penrose P3 tiling graph admits a maximal independent set with density exactly (57 - 25√5)/2 that exceeds 1/2 and controls the uniqueness of the Gibbs measure at high activity.

What would settle it

A demonstration that some independent set in the Penrose P3 graph has density strictly larger than (57 - 25√5)/2, or that the extreme Gibbs measure fails to be unique at large activity, would falsify the result.

Figures

Figures reproduced from arXiv: 2604.21086 by A. Mazel, I. Stuhl, Y. Suhov.

Figure 1
Figure 1. Figure 1: ). The corresponding graphs G are bipartite [4] and linearly repetitive [3] [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A fragment of the ground state for a P3 tiling [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The five basic patterns Our construction is based on the five basic collared patterns shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The five basic patterns with loops In [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The level-4 supertiles [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The level-4 equivalent supertiles The yellow edges form the boundaries of the patches. The supertiles in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The extended 0-atlas of a geometrical P3 tiling Applying the extended 0-atlas to a P3 supertiling constructed from supertiles in Fig￾ure 6, it is not hard to conclude that the supertile vertices are one-to-one mapped to the 1-corona and 2-corona in the original P3 tiling, which are shown in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The coronas in a P3 tiling Furthermore, the kite element from the supertiling extended 0-atlas is the only one whose center does not correspond to a patch center. The center of a deuce element from the extended 0-atlas is always the center of a “bat”. The center of a jack element from the extended 0-atlas is always the center of a “turtle”. The center of an ace element from the extended 0-atlas is always t… view at source ↗
Figure 9
Figure 9. Figure 9: Superimposed P3 and RKTT tilings The RKTT tiling obtained from a P3 tiling by adding all diagonals of the thin rhombi and joining them with the adjacent collinear unit edges into new, longer edges. Simul￾taneously, the thin rhombi edges which are incident to the corresponding joining vertex are not included into the RKTT and consequently are painted light gray. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: P3 and RKTT supertilings and patches with boundaries [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper proves that the maximum independent-set density in the infinite graph induced by the Penrose P3 tiling equals (57 - 25√5)/2 ≈ 0.54915, which exceeds 1/2 even though the graph is bipartite. It then concludes that the extreme Gibbs measure of the nearest-neighbor hard-core model is unique for sufficiently large activity, ruling out coexistence of even and odd phases.

Significance. If the derivation holds, the result supplies a concrete, exactly solvable example of an aperiodic bipartite graph whose substitution structure fixes an imbalance between the two color classes, yielding a unique high-activity Gibbs measure. This is of interest for statistical mechanics on quasicrystals and provides a falsifiable prediction for the density that can be checked against finite approximants.

major comments (1)
  1. §3, substitution-matrix analysis: the claim that the left Perron eigenvector directly supplies the exact class frequencies leading to (57 - 25√5)/2 must be shown to be independent of the choice of prototile origin; otherwise the density could depend on the ergodic measure selected.
minor comments (3)
  1. Abstract: the numerical approximation 0.54915 is given to five decimals; either replace it by the exact algebraic expression throughout or state the rounding explicitly.
  2. Notation: the definition of the Penrose P3 graph (vertex set, edge condition) should be stated once in §2 before any density calculations, to avoid implicit reliance on the tiling picture.
  3. Figure 2 (or equivalent): the finite patch used to illustrate the bipartition should be labeled with the two color classes and the computed local density to allow direct comparison with the global formula.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to confirm independence from prototile origin in the substitution analysis. We address the comment below and will revise the manuscript to make the argument fully explicit.

read point-by-point responses

  1. Referee: §3, substitution-matrix analysis: the claim that the left Perron eigenvector directly supplies the exact class frequencies leading to (57 - 25√5)/2 must be shown to be independent of the choice of prototile origin; otherwise the density could depend on the ergodic measure selected.

    Authors: We agree that explicit verification of independence is required for rigor. The P3 substitution rule is primitive. By the Perron-Frobenius theorem the positive left eigenvector is therefore unique up to scaling and supplies the unique tile frequencies with respect to the (unique) ergodic probability measure on the hull of Penrose tilings. Any choice of prototile origin corresponds to a point in the same measure class; the frequencies themselves are invariant under the substitution and hence independent of origin. In the revised manuscript we will add a short clarifying paragraph immediately after the eigenvector computation in §3, recalling primitivity and unique ergodicity of the Penrose tiling to conclude that the resulting independent-set density (57 - 25√5)/2 is well-defined for the infinite graph and does not depend on the selected ergodic measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation

full rationale

The paper computes the maximum independent-set density directly from the substitution frequencies and prototile measures of the Penrose P3 tiling, obtaining the explicit algebraic value (57 - 25√5)/2 > 1/2. Because the graph is bipartite, this density is simply the larger color-class proportion fixed by the unique invariant measure on the substitution system. The uniqueness of the high-activity hard-core Gibbs measure then follows from the general fact that any bipartite graph with maximum independent-set density strictly above 1/2 cannot support coexistence of even/odd phases. No step reduces to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the geometric calculation is independent of the statistical-mechanical conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the established geometric and combinatorial properties of the Penrose P3 tiling and standard definitions from graph theory and statistical mechanics. No free parameters, new entities, or ad-hoc assumptions beyond the tiling's structure are indicated.

axioms (1)
  • domain assumption The Penrose P3 tiling, viewed as a planar graph, is bipartite yet possesses a maximal independent set density of exactly (57 - 25√5)/2.
    Invoked directly in the abstract to contrast with the expected behavior of bipartite graphs.

pith-pipeline@v0.9.0 · 5379 in / 1288 out tokens · 44974 ms · 2026-05-09T22:28:54.285782+00:00 · methodology

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Reference graph

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