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Free Path Lengths in Quasicrystals

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A Correction to this article was published on 02 January 2020

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Abstract

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.

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Change history

  • 02 January 2020

    In order for Theorem 5.1 to be correct

  • 02 January 2020

    In order for Theorem 5.1 to be correct

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Correspondence to Jens Marklof.

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Communicated by L. Erdös

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement n. 291147. J.M. is furthermore supported by a Royal Society Wolfson Research Merit Award, and A.S. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.

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Marklof, J., Strömbergsson, A. Free Path Lengths in Quasicrystals. Commun. Math. Phys. 330, 723–755 (2014). https://doi.org/10.1007/s00220-014-2011-3

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