Scaling of diffraction intensities near the origin: some rigorous results

Abstract

The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation behaviour known under the term hyperuniformity. Here, we consider one-dimensional systems with pure point, singular continuous and absolutely continuous diffraction spectra, which include perfectly ordered cut and project and inflation point sets as well as systems with stochastic disorder.

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

“…In particular, these matrices satisfy B ð1Þ ¼ B and B ðnÞ ð0Þ ¼ M n for all n 2 N, where M is the substitution matrix from equation 1, as well as the relations B ðnþmÞ ðyÞ ¼ B ðnÞ ðyÞ B ðmÞ ð n yÞ ð 19Þ for any m; n 2 N. Note that B ðnÞ ðyÞ defines a matrix cocycle, called the internal cocycle, which is related to the usual inflation cocycle (in physical space) by an application of the ?-map to the displacement matrices of the powers of the inflation rule; compare Baake, Gä hler & Mañ ibo (2019), Baake & Grimm (2019b), and see Bufetov & Solomyak (2018, 2020 for a similar approach. Note also that jj < 1, which means that n approaches 0 exponentially fast as n !…”

“…The matrix B is obtained by first taking the ?-map of the setvalued displacement matrix T of equation 4and then its inverse Fourier transform. For this reason, B is called the internal Fourier matrix (Baake & Grimm, 2019b), to distinguish it from the Fourier matrix of the renormalization approach in physical space (Baake & Gä hler, 2016;Baake, Frank et al, 2019); see Bufetov & Solomyak (2018, 2020 for various extensions with more flexibility in the choice of the interval lengths. In Dirac notation, we set jhi ¼ ðh a ; h b Þ T , which satisfies jhð0Þi ¼ jvi with the right eigenvector jvi of the substitution matrix M from equation 2.…”

“…The convergence of this approximation is exponentially fast. We refer readers to Baake & Grimm (2019b) for further details and an extension of the cocycle approach to more general inflation systems, and to Baake & Grimm (2020) for a planar example.…”

“…As far as aperiodic structures are concerned, there are in fact a number of early, partly heuristic, results in the literature (Luck, 1993;Aubry et al, 1988;Godrè che & Luck, 1990). These have recently been reformulated and extended (Og uz et al, 2017(Og uz et al, , 2019 and rigorously established (Baake & Grimm, 2019a), using exact renormalization relations for primitive inflation rules (Baake, Frank et al, 2019;Baake & Gä hler, 2016;Mañ ibo, , 2019Baake, Gä hler & Mañ ibo, 2019;Baake et al, 2018); see also Fuchs et al (2019) for results for some planar aperiodic tilings.…”

“…Here, we review the result for variants of the onedimensional Fibonacci model sets considered above, where we now allow for changes of the windows. For a general discussion of this approach and more examples of systems with different types of diffraction, we refer readers to Baake & Grimm (2019a) and references therein.…”