Scaling of Density Fluctuations and Hyperuniformity in One-dimensional Substitution Tilings

Substitution tilings include periodic, quasiperiodic, limit periodic, and other self-similar structures generated by iterated subdivision and rescaling of a finite set of tiles. We study the scaling of density fluctuations associated with a broad class of substitution rules in one dimension. We show that a simple, heuristic argument for the rate of decay of the integrated Fourier intensity Z(k) at small values of the wavenumber k correctly predicts the scaling of the variance σ²(R) in the number of points contained in intervals of length 2R. The exponent α, defined by Z~k^α+1, is determined by the ratio of the second largest and largest eigenvalues of the substitution matrix and can vary between -1 and 3, where α>0 implies a hyperuniform distribution of tile vertices. The hyperuniform class includes tilings that are periodic, quasiperiodic, or limit periodic, including a new class of limit-periodic tilings for which Z approaches zero faster than any power law. Tilings with continuous diffraction spectra may be hyperuniform or may even exhibit stronger fluctuations than a Poisson system.