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28
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... 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 rando ..."
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Cited by 30 (17 self)
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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 diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
Surprises in Diffuse Scattering
 Z. Kristallogr
, 2000
"... . Diffuse scattering is usually associated with some disorder in the analyzed material. Different kinds of disorder may produce different diffuse scattering  or not. In this letter, we demonstrate some aspects of the variety of diffuse scattering that occurs even in very simple examples, and how u ..."
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Cited by 21 (16 self)
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. Diffuse scattering is usually associated with some disorder in the analyzed material. Different kinds of disorder may produce different diffuse scattering  or not. In this letter, we demonstrate some aspects of the variety of diffuse scattering that occurs even in very simple examples, and how unawareness may lead astray. 1 Introduction For a long time, the diffuse part in diffraction spectra played only a minor role in crystallography. This was mainly due to experimental restrictions but also to a lack of theoretical studies. In this letter, we investigate some simple models with disorder, both deterministic and random, where the diffuse background of the diffraction must not be neglected in the structural analysis. We first start with an introduction to the language of mathematical diffraction theory, which is necessary to get a full understanding of the spectrum. The following two examples have exactly the same diffraction pattern albeit their disorder is of completely differen...
Deformation of Delone dynamical systems and pure point spectrum
 J. Fourier Anal. Appl
, 2005
"... Abstract. This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss ..."
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Cited by 18 (11 self)
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Abstract. This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under “equivariant” local perturbations and discuss various examples, including deformed model sets. A key step in the proof of stability consists in transforming the problem into a question on factors of dynamical systems. 1.
MultiComponent Model Sets And Invariant Densities
, 1998
"... this paper, we study selfsimilarities of multicomponent model sets. The main point may be simply summarized: whenever there is a selfsimilarity, there are also naturally related density functions. As in the case of ordinary model sets ..."
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Cited by 17 (6 self)
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this paper, we study selfsimilarities of multicomponent model sets. The main point may be simply summarized: whenever there is a selfsimilarity, there are also naturally related density functions. As in the case of ordinary model sets
Diffraction of stochastic point sets: Explicitly computable examples
 COMMUN. MATH. PHYS
, 2009
"... Stochastic point processes relevant to the theory of longrange aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory ..."
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Cited by 16 (11 self)
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Stochastic point processes relevant to the theory of longrange aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.
SelfSimilarities and Invariant Densities for Model Sets
, 1997
"... Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the selfsimilarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prov ..."
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Cited by 11 (6 self)
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Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the selfsimilarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous refinement operators, and Hutchinson measures.
Universal bounds on the selfaveraging of random diffraction measures
 PROBAB. THEORY RELAT. FIELDS 126, 29–50 (2003)
, 2003
"... ..."
Which distributions of matter diffract? Some answers
, 2002
"... This review revolves around the question which general distribution of scatterers (in a Euclidean space) results in a pure point diffraction spectrum. Firstly, we treat mathematical diffration theory and state conditions under which such a distribution has pure point diffraction. We explain how a cu ..."
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Cited by 10 (3 self)
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This review revolves around the question which general distribution of scatterers (in a Euclidean space) results in a pure point diffraction spectrum. Firstly, we treat mathematical diffration theory and state conditions under which such a distribution has pure point diffraction. We explain how a cut and project scheme naturally appears in this context and then turn our attention to the special situation of model sets and lattice substitution systems. As an example, we analyse the paperfolding sequence. In the last part, we summarize some aspects of stochastic point sets, with focus both on structure and diffraction.