Diffraction of random tilings: some rigorous results

TL;DR

This paper rigorously analyzes the diffraction patterns of various stochastic point sets and random tilings, providing explicit spectra and a measure-theoretic framework for understanding their diffraction components.

Contribution

It derives the diffraction spectra for 1D and planar random tilings, and summarizes the measure-theoretic approach to diffraction theory.

Findings

Diffraction spectrum of 1D random tilings derived.

Diffraction from planar random tilings based on dimer models analyzed.

Outline of measure-theoretic approach to diffraction spectrum decomposition.

Abstract

The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diffraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diffraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.