Results 1  10
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19
Random matrices: The universality phenomenon for Wigner ensembles
, 2012
"... In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Centr ..."
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Cited by 16 (5 self)
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In this paper, we survey some recent progress on rigorously etablishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.
The Euclidean distance degree of an algebraic variety
, 2013
"... The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the EckartYoung Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest ..."
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Cited by 14 (2 self)
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The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the EckartYoung Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
Random matrices: Universality of local spectral statistics of nonHermitian matrices
, 2013
"... It is a classical result of Ginibre that the normalized bulk kpoint correlation functions of a complex n × n gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K∞(z, w): = 1pi e −z2/2−w2/2+zw in ..."
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Cited by 11 (1 self)
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It is a classical result of Ginibre that the normalized bulk kpoint correlation functions of a complex n × n gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K∞(z, w): = 1pi e −z2/2−w2/2+zw in the limit n→∞. In this paper we show that this asymptotic law is universal among all random n × n matrices Mn whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts, and whose moments match that of the complex gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex gaussian matrices in a small disk to these more general ensembles. These results are nonHermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the nonHermitian case, due to the instability of the spectrum for such ma
Practical Secrecy using Artificial Noise
"... Abstract—In this paper, we consider the use of artificial noise for secure communications. We propose the notion of practical secrecy as a new design criterion based on the behavior of the eavesdropper’s error probability PE, as the signaltonoise ratio goes to infinity. We then show that the pract ..."
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Cited by 10 (9 self)
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Abstract—In this paper, we consider the use of artificial noise for secure communications. We propose the notion of practical secrecy as a new design criterion based on the behavior of the eavesdropper’s error probability PE, as the signaltonoise ratio goes to infinity. We then show that the practical secrecy can be guaranteed by the randomly distributed artificial noise with specified power. We show that it is possible to achieve practical secrecy even when the eavesdropper can afford more antennas than the transmitter. Index Terms—Physical layer security, lattices, wiretap channel, artificial noise. I.
Betti numbers of random real hypersurfaces and determinants of random symmetric matrices. ArXiv eprints
, 2012
"... We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the re ..."
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Cited by 6 (4 self)
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We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from middimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the random real hypersurfaces.
Random matrices: Sharp concentration of eigenvalues
 DEPARTMENT OF MATHEMATICS, ZHEJIANG UNIVERSITY
, 2013
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Law of log determinant of sample covariance matrix and optimal estimation of differential entropy for highdimensional gaussian distributions. arXiv:1309.0482
, 2013
"... Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential e ..."
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Cited by 3 (1 self)
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Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential entropy and the logdeterminant of the covariance matrix. We first establish a central limit theorem for the log determinant of the sample covariance matrix in the high dimensional setting where the dimension p(n) can grow with the sample size n. An estimator of the differential entropy and the log determinant is then considered. Optimal rate of convergence is obtained. It is shown that in the case p(n)/n → 0 the estimator is asymptotically sharp minimax. The ultrahigh dimensional setting where p(n)> n is also discussed.
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices, Preprint arXiv:1210.5666
"... Abstract. We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large n for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed ..."
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Abstract. We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large n for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following TaoVu and Erdös, Yau, et al. and a LittlewoodPaley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the Hölder class C1/2+ in the case of matrices of Gaussian convolution type. We also give a variance bound which implies the CLT for test functions in the Sobolev space H1+ and C1− for general Wigner matrices satisfying moment conditions. Previous results on the CLT impose the existence and continuity of at least one classical derivative. 1.
Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble
"... The goal of this paper is to establish a relation between characteristic polynomials of N ×N GUE random matrices H as N → ∞, and Gaussian processes with logarithmic correlations. First, we introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian proces ..."
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The goal of this paper is to establish a relation between characteristic polynomials of N ×N GUE random matrices H as N → ∞, and Gaussian processes with logarithmic correlations. First, we introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN (z): = − log det(zI−H)  on mesoscopic scales as N →∞. By employing a Fourier integral representation, we show how this implies a continuous analogue of a result by Diaconis and Shahshahani [18]. On the macroscopic scale, DN (x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give