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Abstract: The diffraction of various random subsets of the integer lattice$\mathbb{Z}^{d}$, such as the coin tossing and related systems, are wellunderstood. Here, we go one important step beyond and consider random pointsets in $\mathbb{R}^{d}$. We present several systems with an effectivestochastic interaction that still allow for explicit calculations of theautocorrelation and the diffraction measure. We concentrate on one-dimensionalexamples for illustrative purposes, and briefly indicate possiblegeneralisations to higher dimensions.In particular, we discuss the stationary Poisson process in $\mathbb{R}^{d}$and the renewal process on the line. The latter permits a unified approach to arather large class of one-dimensional structures, including random tilings.Moreover, we present some stationary point processes that are derived from theclassical random matrix ensembles as introduced in the pioneering work of Dysonand Ginibre. Their re-consideration from the diffraction point of view improvesthe intuition on systems with randomness and mixed spectra.



Autor: Michael Baake Bielefeld, Holger Koesters Bielefeld

Fuente: https://arxiv.org/







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