Results 1  10
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19
Weighted Dirac combs with pure point diffraction
, 2002
"... A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a ..."
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Cited by 39 (26 self)
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A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.
Dynamical Systems on Translation Bounded Measures: PURE POINT DYNAMICAL AND DIFFRACTION SPECTRA
, 2003
"... Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure ..."
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Cited by 38 (23 self)
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Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
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 26 (16 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.
Diffractive point sets with entropy
"... Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of ..."
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Cited by 25 (16 self)
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Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in welldefined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.
Uniform existence of the integrated density of states for . . . Z^d
"... We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that n ..."
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Cited by 14 (8 self)
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We give an overview and extension of recent results on ergodic random Schrödinger operators for models on Zd. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed ShubinPastur type trace formula. The set of points of increase of the IDS supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.
Diffraction of Weighted Lattice Subsets
"... A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, a ..."
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Cited by 11 (11 self)
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A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diraction measure is periodic, with the dual lattice as lattice of periods. This statement remains true in the setting of a locally compact Abelian group that is also compact.
Model sets: a survey
, 1999
"... This article surveys the mathematics of the cut and project method as applied to point sets, called here model sets. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems. Dedicated to the memory of Richard (Dick) Sl ..."
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Cited by 7 (1 self)
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This article surveys the mathematics of the cut and project method as applied to point sets, called here model sets. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems. Dedicated to the memory of Richard (Dick) Slansky The spirit of the universe is subtle and informs all life. Things live and die and change their forms, without knowing the root from which they come. Abundantly it multiplies; eternally it stands by itself. The greatest reaches of space do not leave its confines, and the smallest down of a bird in autumn awaits its power to assume form. — Chuang Tzu (tr. Lin Yutang) 1
RANDOM COLOURINGS OF APERIODIC GRAPHS: ERGODIC AND SPECTRAL PROPERTIES
, 709
"... Abstract. We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic measures, and prove an ergodic theorem. For covari ..."
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Cited by 6 (1 self)
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Abstract. We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic measures, and prove an ergodic theorem. For covariant operators of finite range defined on those graphs, we show the existence and selfaveraging of the integrated density of states, as well as the nonrandomness of the spectrum. Our main result establishes Lifshits tails at the lower spectral edge of the graph Laplacian on bond percolation subgraphs, for sufficiently small probabilities. Among other assumptions, its proof requires exponential decay of the clustersize distribution for percolation on rather general graphs. 1.
Which distributions of matter diffract? Some answers
 Quasicrystals, Structure and Physical properties
, 2003
"... Diffraction experiments are important to determine the structure of a solid, even more so with the refined methods available today. Recent applications include aperiodic systems as well as systems with disorder. Kinematic diffraction can be described and understood in terms of Fourier analysis. The ..."
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Cited by 6 (3 self)
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Diffraction experiments are important to determine the structure of a solid, even more so with the refined methods available today. Recent applications include aperiodic systems as well as systems with disorder. Kinematic diffraction can be described and understood in terms of Fourier analysis. The