4.1 Article

Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

Journal

Publisher

INT UNION CRYSTALLOGRAPHY
DOI: 10.1107/S2053273318015528

Keywords

substitution tiling; hyperuniformity; diffraction; limit-periodic tilings; non-Pisot tilings; quasiperiodic tilings

Funding

  1. National Science Foundation [DMR-1714722]

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This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent alpha governing the scaling of Fourier intensities at small wavenumbers, tilings with alpha > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with alpha arbitrarily close to any given value between -1 and 3. Limit-periodic tilings can be constructed with alpha between -1 and 1 or with Fourier intensities that approach zero faster than any power law.

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