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17
Delocalization in Random Polymer Models
 Commun. Math. Phys
"... A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer ..."
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Cited by 26 (4 self)
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A random polymer model is a onedimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have purepoint spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.
Fractal dimensions and the phenomenon of intermittency in quantum dynamics
 Duke Math. J
"... We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions D ± () µψ 1/ ..."
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Cited by 18 (9 self)
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We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions D ± () µψ 1/(1 + p/d) of the measure µψ (where d is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension. 1.
Linear Response Theory for Magnetic Schrödinger Operators in Disordered Media
, 2004
"... We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators whe ..."
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Cited by 16 (9 self)
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We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the wellknown KuboStreda formula for the quantum Hall conductivity at zero temperature.
Generalized Fractal Dimensions: Equivalences and Basic Properties
, 2000
"... Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q & ..."
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Cited by 14 (7 self)
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Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > 0. We then use these relations to prove some regularity properties for (q); we also provide some estimates for these functions (in particular estimates on their behaviour at 1), as well as for the dimensions corresponding to convolution of two measures. We nally present some calculations for speci c examples. 1
Intermittent Lower Bound on Quantum Diffusion
, 1999
"... An intermittent lower bound on quantum diffusion is proven in presence of a multifractal spectral measure. ..."
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Cited by 9 (2 self)
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An intermittent lower bound on quantum diffusion is proven in presence of a multifractal spectral measure.
Lower Bounds on Wave Packet Propagation By Packing Dimensions of Spectral Measures
, 1999
"... We prove that, for any quantum evolution in ` 2 (Z D ), there exist arbitrarily long time scales on which the qth moment of the position operator increases at least as fast as a power of time given by q=D times the packing dimension of the spectral measure. Packing dimensions of measures and the ..."
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Cited by 9 (0 self)
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We prove that, for any quantum evolution in ` 2 (Z D ), there exist arbitrarily long time scales on which the qth moment of the position operator increases at least as fast as a power of time given by q=D times the packing dimension of the spectral measure. Packing dimensions of measures and their connections to scaling exponents and boxcounting dimensions are also discussed.
Upper bounds for quantum dynamics governed by Jacobi matrices with selfsimilar spectra
, 1998
"... We study a class of onesided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of spectral measures, and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds invol ..."
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Cited by 8 (5 self)
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We study a class of onesided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of spectral measures, and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds involve the Hausdorff dimension of the spectral measure, multiplied by a correction calculated from the dynamical entropy, the density of states, and the capacity of the support. For Julia matrices, the correction disappears and the growth is ruled by the fractal dimension.
Phaseaveraged transport for quasiperiodic Hamiltonians
 Commun. Math. Phys
"... For a class of discrete quasiperiodic Schrödinger operators defined by covariant representations of the rotation algebra, a lower bound on phaseaveraged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on ..."
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Cited by 3 (0 self)
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For a class of discrete quasiperiodic Schrödinger operators defined by covariant representations of the rotation algebra, a lower bound on phaseaveraged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almostMathieu) operator. As a byproduct, a new solution of the frame problem associated with WeylHeisenbergGabor lattices of coherent states is given. 1
Physical Applications of Freeness
"... Introduction The mathematical concept `freeness' was introduced by Voiculescu about 15 years ago in order to get some insight into the structure of the von Neumann algebras of the free groups. The starting idea of Voiculescu was to separate the notion of freeness from that concrete problem and ..."
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Cited by 3 (0 self)
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Introduction The mathematical concept `freeness' was introduced by Voiculescu about 15 years ago in order to get some insight into the structure of the von Neumann algebras of the free groups. The starting idea of Voiculescu was to separate the notion of freeness from that concrete problem and to consider it on its own sake. One leading principle of his investigations was to consider freeness as a noncommutative analogue to the classical notion `independence of random variables' This approach was very successful. Gradually it became clear that freeness has indeed a very interesting structure of its own and that there exist a lot of links to other fields. One of the main links of freeness to an apriori unrelated context is the description of random matrices via freeness. This has opened the possibility to use freeness as an exact mathematical concept for the description of physical models and approximations which build (explicitely or implicitly) on random matrices. One model of