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arXiv · 2210.02151

Hyperuniformity and non-hyperuniformity of quasicrystals

Michael Björklund·Tobias Hartnick
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Abstract

We develop a general framework to study hyperuniformity of various mathematical models of quasicrystals. Using this framework we provide examples of non-hyperuniform quasicrystals which unlike previous examples are not limit-quasiperiodic. Some of these examples are even anti-hyperuniform or have a positive asymptotic number variance. On the other hand we establish hyperuniformity for a large class of mathematical quasicrystals in Euclidean spaces of arbitrary dimension. For certain models of quasicrystals we moreover establish that hyperuniformity holds for a generic choice of the underlying parameters. For quasicrystals arising from the cut-and-project method we conclude that their hyperuniformity depends on subtle diophantine properties of the underlying lattice and window and is by no means automatic.

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Michael Björklund, Tobias Hartnick. 2022-10-05. Hyperuniformity and non-hyperuniformity of quasicrystals. https://arxiv.org/abs/2210.02151

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