Random point sets and their diffraction

Abstract

The diffraction of various random subsets of the integer lattice ℤ ^d , such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in ℝ ^d . We present several systems with an effective stochastic interaction that still allow for explicit calculations of the autocorrelation and the diffraction measure. We concentrate on one-dimensional examples for illustrative purposes, and briefly indicate possible generalisations to higher dimensions. In particular, we discuss the stationary Poisson process in ℝ ^d and the renewal process on the line. The latter permits a unified approach to a rather large class of one-dimensional structures, including random tilings. Moreover, we present some stationary point processes that are derived from the classical random matrix ensembles as introduced in the pioneering work of Dyson and Ginibre. Their reconsideration from the diffraction point of view improves the intuition on systems with randomness and mixed spectra.

Keywords:

• diffraction spectra
• stochastic point processes
• random matrix ensembles

Acknowledgements

MB would like to thank P. Forrester and O. Zeitouni for discussions on the Coulomb gas and the random matrix ensembles. This work was supported by the German Research Council (DFG), within the CRC 701.