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Spectral structure and quantitative size for Penrose tiling Laplacians

Establish that the spectrum of the graph Laplacian for the three Penrose tiling models T1, T2, and T3 has infinitely many connected components and zero Lebesgue measure, and determine the quantitative values of the box-counting dimensions (approximately 0.95, 0.86, and 0.92 for T1, T2, T3) and capacities (approximately 1.50, 3.08, and 2.27 for T1, T2, T3).

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Background

Using SCI-based algorithms under assumption (S2), the authors compute convergent spectral covers for Penrose tiling Laplacians and present evidence for intricate spectral features, including many gaps and fractal characteristics.

They formulate conjectures asserting zero measure and infinite connectivity, together with specific approximate values for box-counting dimensions and logarithmic capacities, which, if proven, would significantly advance the quantitative spectral theory of two-dimensional quasicrystals.

References

Penrose tiles: We conjecture that for the spectrum of the graph Laplacian of the three tilings, $T_1$, $T_2$, and $T_3$, the number of connected components is infinite and the Lebesgue measure is zero. We also conjecture that the box-counting dimensions are approximately $0.95$, $0.86$, and $0.92$, whilst the capacities are approximately $1.50$, $3.08$, $2.27$, for $T_1$, $T_2$, and $T_3$, respectively.

— Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals (2407.20353 - Colbrook et al., 29 Jul 2024) in Section 7: Final Remarks (Conjectures)