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Characterization of the Anderson metalinsulator transport transition for non ergodic operators and application, arXiv:1110.4652, submitted y
"... Abstract. We study the Anderson metalinsulator transition for non ergodic random Schrdinger operators in both annealed and quenched regimes, based on a dynamical approach of localization, improving known results for ergodic operators into this more general setting. In the procedure, we reformulate ..."
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Abstract. We study the Anderson metalinsulator transition for non ergodic random Schrdinger operators in both annealed and quenched regimes, based on a dynamical approach of localization, improving known results for ergodic operators into this more general setting. In the procedure, we reformulate the Bootstrap Multiscale Analysis of Germinet and Klein to fit the non ergodic setting. We obtain uniform Wegner Estimates needed to perform this adapted Multiscale Analysis in the case of DeloneAnderson type potentials, that is, Anderson potentials modeling aperiodic solids, where the impurities lie on a Delone set rather than a lattice, yielding a break of ergodicity. As an application we study the Landau operator with a DeloneAnderson potential and show the existence of a mobility edge between regions of dynamical localization
Dynamical properties of almost repetitive Delone sets, Preprint arXiv:1210.2955
"... Abstract. We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almo ..."
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Abstract. We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations. 1. Point Sets and Dynamical Systems Non–periodic point sets, which still display some regularity, are interesting objects in discrete geometry. Such sets have been intensively studied in the context of uniformly discrete subsets P of Euclidean space M = Rd. A useful device in that situation is the hull of P, i.e., the orbit closure XP: = {tP  t ∈ T}, where T is a topological group such as Rd or E(d), the group of Euclidean motions, acting continuously on M from the left. Here the closure is taken with respect to a suitable topology on the space of uniformly discrete subsets of M. With the induced action of T on XP, the hull can be regarded as a topological dynamical system (XP, T). Regularity of P is then reflected in properties of its hull such as minimality or unique ergodicity. These properties may of course depend on the topology or on the group action. A frequently studied topology on the space of uniformly discrete point sets is well adapted to point sets arising from tilings [GrSh, Ru, RaWo]. This so–called local matching topology is generated by the metric dLM (P, P ′): = min
Percolation Hamiltonians
, 2010
"... There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathem ..."
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There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators.
DIFFRACTION OF RANDOM NOBLE MEANS WORDS
"... Abstract. In this paper, several aspects of the random noble means substitution are studied. Beyond important dynamical facets as the frequency of subwords and the computation of the topological entropy, the important issue of ergodicity is addressed. From the geometrical point of view, we outline a ..."
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Abstract. In this paper, several aspects of the random noble means substitution are studied. Beyond important dynamical facets as the frequency of subwords and the computation of the topological entropy, the important issue of ergodicity is addressed. From the geometrical point of view, we outline a suitable cut and project setting for associated point sets and present results for the spectral analysis of the diffraction measure. 1.
ERGODICITY AND DYNAMICAL LOCALIZATION FOR DELONE–ANDERSON OPERATORS
"... Abstract. We study the ergodic properties of DeloneAnderson operators, using the framework of randomly coloured Delone sets and Delone dynamical systems. In particular, we show the existence of the integrated density of states and, under some assumptions on the geometric complexity of the underly ..."
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Abstract. We study the ergodic properties of DeloneAnderson operators, using the framework of randomly coloured Delone sets and Delone dynamical systems. In particular, we show the existence of the integrated density of states and, under some assumptions on the geometric complexity of the underlying Delone sets, we obtain information on the almostsure spectrum of the family of random operators. We then exploit these results to study the Lifshitztail behaviour of the integrated density of states of a Delone–Anderson operator at the bottom of the spectrum. This is used as an input for the multiscale analysis to prove dynamical localization. We also estimate the size of the spectral region where dynamical localization occurs. 1.