The Experts below are selected from a list of 24624 Experts worldwide ranked by ideXlab platform
Paul Bourgade - One of the best experts on this subject based on the ideXlab platform.
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maximum of the characteristic polynomial of random Unitary matrices
Communications in Mathematical Physics, 2017Co-Authors: Louispierre Arguin, David Belius, Paul BourgadeAbstract:It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a \({N\times N}\) random Unitary Matrix sampled from the Haar measure grows like \({CN/({\rm log} N)^{3/4}}\) for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range \({[N^{1 - \varepsilon},N^{1 + \varepsilon}]}\), for arbitrarily small \({\varepsilon}\).
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the characteristic polynomial of a random Unitary Matrix a probabilistic approach
arXiv: Probability, 2007Co-Authors: Paul Bourgade, C P Hughes, Ashkan Nikeghbali, Marc YorAbstract:In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random Unitary Matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith, using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random Unitary Matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.
Gordon W Semenoff - One of the best experts on this subject based on the ideXlab platform.
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large representation recurrences in large n random Unitary Matrix models
Journal of High Energy Physics, 2011Co-Authors: Joanna L Karczmarek, Gordon W SemenoffAbstract:In a random Unitary Matrix model at large N, we study the properties of the expectation value of the character of the Unitary Matrix in the rank k symmetric tensor representation. We address the problem of whether the standard semiclassical technique for solving the model in the large N limit can be applied when the representation is very large, with k of order N. We find that the eigenvalues do indeed localize on an extremum of the effective potential; however, for finite but sufficiently large k/N, it is not possible to replace the discrete eigenvalue density with a continuous one. Nonetheless, the expectation value of the character has a well-defined large N limit, and when the discreteness of the eigenvalues is properly accounted for, it shows an intriguing approximate periodicity as a function of k/N.
Thomas L Marzetta - One of the best experts on this subject based on the ideXlab platform.
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Multiple-antennas and isotropically random Unitary inputs: the received signal density in closed form
2002Co-Authors: Babak Hassibi, Thomas L Marzetta, Senior MemberAbstract:Abstract—An important open problem in multiple-antenna communications theory is to compute the capacity of a wireless link subject to flat Rayleigh block-fading, with no channel-state information (CSI) available either to the transmitter or to the receiver. The isotropically random (i.r.) Unitary Matrix—having orthonormal columns, and a probability density that is invariant to premultiplication by an independent Unitary Matrix—plays a central role in the calculation of capacity and in some special cases happens to be capacity-achieving. In this paper, we take an important step toward computing this capacity by obtaining, in closed form, the probability density of the received signal when transmitting i.r. Unitary matrices. The technique is based on analytically computing the expectation of an exponential quadratic function of an i.r. Unitary Matrix and makes use of a Fourier integral representation of the constituent Dirac delta functions in the underlying density. Our formula for the received signal density enables us to evaluate the mutual information for any case of interest, something that could previously only be done for single transmit and receive antennas. Numerical results show that at high signal-to-noise ratio (SNR), the mutual information is maximized for = min ( 2) transmit antennas, where is the number of receive antennas and is the length of the coherence interval, whereas at low SNR, the mutual information is maximized by allocating all transmit power to a single antenna. Index Terms—Isotropically random (i.r.) Unitary Matrix, mul-tiple antennas, Unitary space–time modulation (USTM), wireless communications. I
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multiple antennas and isotropically random Unitary inputs the received signal density in closed form
International Symposium on Information Theory, 2001Co-Authors: Babak Hassibi, Thomas L MarzettaAbstract:An important open problem in multiple-antenna communications theory is to compute the capacity of a wireless link subject to flat Rayleigh block-fading, with no channel-state information (CSI) available either to the transmitter or to the receiver. The isotropically random (i.r.) Unitary Matrix-having orthonormal columns, and a probability density that is invariant to premultiplication by an independent Unitary Matrix-plays a central role in the calculation of capacity and in some special cases happens to be capacity-achieving. We take an important step toward computing this capacity by obtaining, in closed form, the probability density of the received signal when transmitting i.r. Unitary matrices. The technique is based on analytically computing the expectation of an exponential quadratic function of an i.r. Unitary Matrix and makes use of a Fourier integral representation of the constituent Dirac delta functions in the underlying density. Our formula for the received signal density enables us to evaluate the mutual information for any case of interest, something that could previously only be done for single transmit and receive antennas. Numerical results show that at high signal-to-noise ratio (SNR), the mutual information is maximized for M=min(N, T/2) transmit antennas, where N is the number of receive antennas and T is the length of the coherence interval, whereas at low SNR, the mutual information is maximized by allocating all transmit power to a single antenna.
Neil Oconnell - One of the best experts on this subject based on the ideXlab platform.
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on the characteristic polynomial of a random Unitary Matrix
Communications in Mathematical Physics, 2001Co-Authors: C P Hughes, Jon P Keating, Neil OconnellAbstract:We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N Unitary Matrix, as N→∞. First we show that \(\), evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for \(\). For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
Hao Shen - One of the best experts on this subject based on the ideXlab platform.
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uniqueness analysis of non Unitary Matrix joint diagonalization
IEEE Transactions on Signal Processing, 2013Co-Authors: Martin Kleinsteuber, Hao ShenAbstract:Matrix Joint Diagonalization (MJD) is a powerful approach for solving the Blind Source Separation (BSS) problem. It relies on the construction of matrices which are diagonalized by the unknown demixing Matrix. Their joint diagonalizer serves as a correct estimate of this demixing Matrix only if it is uniquely determined. Thus, a critical question is under what conditions is a joint diagonalizer unique. In the present work we fully answer this question about the identifiability of MJD based BSS approaches and provide a general result on uniqueness conditions of Matrix joint diagonalization. It unifies all existing results which exploit the concepts of non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for complex BSS, which can be formulated in closed form in terms of an eigen and a singular value decomposition of two matrices.
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uniqueness analysis of non Unitary Matrix joint diagonalization
arXiv: Information Theory, 2011Co-Authors: Martin Kleinsteuber, Hao ShenAbstract:Matrix Joint Diagonalization (MJD) is a powerful approach for solving the Blind Source Separation (BSS) problem. It relies on the construction of matrices which are diagonalized by the unknown demixing Matrix. Their joint diagonalizer serves as a correct estimate of this demixing Matrix only if it is uniquely determined. Thus, a critical question is under what conditions a joint diagonalizer is unique. In the present work we fully answer this question about the identifiability of MJD based BSS approaches and provide a general result on uniqueness conditions of Matrix joint diagonalization. It unifies all existing results which exploit the concepts of non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for complex BSS, which can be formulated in a closed form in terms of an eigenvalue and a singular value decomposition of two matrices.
