Random point sets and their diffraction

Abstract

The diffraction of various random subsets of the integer lattice ${\mathbb{Z}}^d$, such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in ${\mathbb{R}}^d$. 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 ${\mathbb{R}}^d$ 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 re-consideration from the diffraction point of view improves the intuition on systems with randomness and mixed spectra.

9 pages, 2 figures; talk presented at ICQ 11 (Sapporo)

References in corpus (12)

• Matrix Models for Beta Ensembles
• Determinantal random point fields
• Continuum limits of random matrices and the Brownian carousel
• Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
• Characterizations of model sets by dynamical systems
• Homometric model sets and window covariograms
• Deformation of Delone dynamical systems and pure point diffraction
• Can Kinematic Diffraction Distinguish Order from Disorder?
• Diffraction of stochastic point sets: Explicitly computable examples
• Homometric Point Sets and Inverse Problems
• The singular continuous diffraction measure of the Thue-Morse chain
• Planar dynamical systems with pure Lebesgue diffraction spectrum

Cited by in corpus (4)

• Kinematic Diffraction from a Mathematical Viewpoint
• Scaling of diffraction intensities near the origin: Some rigorous results
• Territorial behaviour of buzzards versus random matrix spacing distributions
• Diffraction Theory of Point Processes: Systems with Clumping and Repulsion