Absence of singular continuous diffraction for discrete multi-component particle models

TL;DR

This paper proves that for certain discrete multi-component particle models with a Gibbs measure, the diffraction pattern lacks a singular continuous component, consisting only of pure point and absolutely continuous parts.

Contribution

It establishes the absence of singular continuous diffraction in a broad class of aperiodic particle models with finitely many types of particles.

Findings

Diffraction measure exists almost surely.

Diffraction measure has only pure point and absolutely continuous parts.

No singular continuous component is present.

Abstract

Particle models with finitely many types of particles are considered, both on $\mathbb{Z}^d$ and on discrete point sets of finite local complexity. Such sets include many standard examples of aperiodic order such as model sets or certain substitution systems. The particle gas is defined by an interaction potential and a corresponding Gibbs measure. Under some reasonable conditions on the underlying point set and the potential, we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density. In particular, no singular continuous part is present.