Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra

TL;DR

This paper demonstrates that for certain uniformly discrete point processes with pure point spectra, correlations up to the third order fully determine the system, especially when diffraction has no extinctions.

Contribution

It establishes that all essential physical information in such systems can be derived from a finite set of correlations, with bounds depending on spectral properties.

Findings

2 and 3 point correlations suffice when no diffraction extinctions occur

Correlation bounds are linked to the cycle structure of a related graph

Pure point diffractive systems have fully determined physical information from limited correlations

Abstract

The paper investigates how correlations can completely specify a uniformly discrete point process. The setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic. The first result is that all of the essential physical information in such a system is derivable from its $n$-point correlations, $n= 2, 3, >...$. If the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and Bragg spectra. In particular, if the diffraction has no extinctions, then the 2 and 3 point correlations contain all the relevant information.