Random point sets and their diffraction

TL;DR

This paper explores the diffraction properties of various random point sets in Euclidean space, extending understanding from lattice subsets to more complex stochastic systems, with explicit calculations and examples in one dimension.

Contribution

It introduces new explicit diffraction calculations for random point sets with stochastic interactions, including Poisson, renewal, and matrix-derived processes, and discusses potential higher-dimensional generalizations.

Findings

Explicit autocorrelation and diffraction measures for several random point processes.

Demonstration of diffraction properties for processes derived from random matrix ensembles.

Unified approach to one-dimensional structures, including random tilings and renewal processes.

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.