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A guide to mathematical quasicrystals
, 1999
"... The discovery of alloys with longrange orientational order and sharp diffraction images of noncrystallographic symmetry [65, 35] has initiated an intensive investigation of the possible structures and physical properties of such systems. Although there were various precursors, both theoretically a ..."
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Cited by 37 (13 self)
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The discovery of alloys with longrange orientational order and sharp diffraction images of noncrystallographic symmetry [65, 35] has initiated an intensive investigation of the possible structures and physical properties of such systems. Although there were various precursors, both theoretically and experimentally
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.
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 29 (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.
Repetitive Delone sets and quasicrystals
, 1999
"... This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) ..."
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Cited by 27 (0 self)
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This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) of radius T being finite for all T. A Delone set X of finite type is repetitive if there is a function MX(T) such that every closed ball of radius MX(T)+T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R naction. There is a lower bound for MX(T) in terms of NX(T), namely MX(T) ≥ c(NX(T)) 1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T) purely in terms of NX(T). The complexity of a repetitive Delone set of finite type is measured by the growth rate of its repetitivity function MX(T). For example, the function MX(T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T) = O(T) as T → ∞ and is densely repetitive if MX(T) = O(NX(T)) 1/n as T → ∞. We show that linearly repetitive sets
The Planar Dimer Model With Boundary: A Survey.
 CRM Proceedings and Lecture Notes
, 1998
"... this paper we would like to give a short survey of some of these new results. ..."
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Cited by 21 (1 self)
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this paper we would like to give a short survey of some of these new results.
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
"... Abstract. Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity ..."
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Cited by 13 (8 self)
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Abstract. Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.
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.