Diffraction of stochastic point sets: Explicitly computable examples

TL;DR

This paper explores diffraction patterns of stochastic point processes with long-range order, providing explicit examples and a unified framework based on classical point process theory and Palm distributions.

Contribution

It introduces a unified approach to compute diffraction spectra of stochastic point sets with mixed types, emphasizing explicitly solvable cases and duality structures.

Findings

Explicitly computed diffraction spectra for specific stochastic point processes

Identification of duality structures analogous to Poisson summation

Development of a general framework for analyzing diffraction of aperiodic order

Abstract

Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.