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12
On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach, (extended version) arXiv:0710.4051. Julien Dumont was born in Flers
 and France Télécom Labs
, 1979
"... Abstract—In this paper, the capacityachieving input covariance matrices for coherent blockfading correlated multiple input multiple output (MIMO) Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the opti ..."
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Cited by 25 (13 self)
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Abstract—In this paper, the capacityachieving input covariance matrices for coherent blockfading correlated multiple input multiple output (MIMO) Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate, new results related to the accuracy of the approximation of the average mutual information are provided. Based on the accuracy of this approximation, an attractive optimization algorithm is proposed and analyzed. This algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information. Index Terms—Multiple input multiple output (MIMO) Rician channels, ergodic capacity, large random matrices, capacity achieving covariance matrices, iterative waterfilling. I.
Anomalous transport: A mathematical framework
 MR 99b:81046 162
, 1998
"... We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize ..."
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Cited by 15 (6 self)
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We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo’s formula for the conductivity and hence lead to anomalies in Drude’s formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner’s norbital model as well as the Anderson model in coherent potential approximation. 1
A new approach for capacity analysis of large dimensional multiantenna channels” submitted to IEEE trans. Inform. Theory, 2006. available at http ://arxiv.org/abs/cs.IT/0612076
"... This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorou ..."
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Cited by 12 (6 self)
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This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
A new approach for mutual information analysis of large dimensional multiantenna chennels
 4004, 2008. FOR CERTAIN STATISTICS OF GRAM RANDOM MATRICES 41
"... This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorou ..."
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Cited by 6 (5 self)
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This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorously established that the mutual information when properly centered and rescaled converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.
ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES
, 2000
"... We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the bandtype structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band w ..."
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Cited by 5 (0 self)
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We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the bandtype structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band width 1 ≪ b ≪ N, we derive explicit expressions for the first terms of 1/bexpansions of the average of the Green function N −1 Tr(H (N,b) −z) −1 and its correlation function as well. The expressions obtained show that there exist several scales of the universal forms of the spectral correlation function. These scales are determined by the rate of decrease of the function u(t). They coincide with those detected in theoretical physics for the localization length and densitydensity correlator in the bandtype random matrix ensembles. 1 Problem, motivation and results
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
THE CONDUCTIVITY MEASURE FOR THE ANDERSON MODEL
, 709
"... Dedicated to Leonid A. Pastur on the occasion of his 70th birthday Abstract. We study the acconductivity in linear response theory for the Anderson tightbinding model. We define the electrical acconductivity and calculate the linearresponse current at zero temperature for arbitrary Fermi energy. ..."
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Cited by 1 (0 self)
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Dedicated to Leonid A. Pastur on the occasion of his 70th birthday Abstract. We study the acconductivity in linear response theory for the Anderson tightbinding model. We define the electrical acconductivity and calculate the linearresponse current at zero temperature for arbitrary Fermi energy. In particular, the Fermi energy may lie in a spectral region where extended states are believed to exist. 1.
1 On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach
, 2008
"... In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are avai ..."
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In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed by numerical techniques. As the corresponding optimization algorithms are not very attractive, an approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to + ∞ at the same rate. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information, while being much more computationally attractive.
Contents
, 2008
"... We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using NonCommutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localize ..."
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We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using NonCommutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.