Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations

TL;DR

This paper explores the deep relationship between diffraction patterns and the underlying dynamics of discrete point processes, demonstrating that higher-order correlations uniquely determine the dynamics and establishing new theoretical results in the field.

Contribution

It proves that all higher correlations uniquely determine the dynamics of point processes and establishes a quantitative relation between autocorrelation, diffraction, and dual characters.

Findings

Higher correlations determine the dynamics of point processes.

A square-mean form of the Bombieri-Taylor conjecture is proved.

A quantitative relation between autocorrelation, diffraction, and epsilon dual characters is derived.

Abstract

The paper studies the relationship between diffraction and dynamics for uniformly discrete ergodic point processes in real spaces. This relationship takes the form of an isometric embedding of two L^2 spaces. Diffraction (or equivalently the 2-point correlations) usually cannot determine the dynamics entirely, but we prove that knowledge of all the higher correlations (2-point, 3-point, ...) does. A square-mean form of the Bombieri-Taylor conjecture is proved. A quantitative relation between autocorrelation, diffraction, and epsilon dual characters is derived. Most results of the paper are proved in the setting of multi-colour points and assignable scattering strengths.