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On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

A. Pajor, L. Pastur (2009)

Studia Mathematica

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix H ₙ ( 0 ) and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of H ₙ ( 0 ) and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

On the reduction of a random basis

Ali Akhavi, Jean-François Marckert, Alain Rouault (2009)

ESAIM: Probability and Statistics

For p ≤ n, let b1(n),...,bp(n) be independent random vectors in ℝ n with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If b ^ 1 ( n ) , ... , b ^ p ( n ) is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios...

On the singular values of random matrices

Shahar Mendelson, Grigoris Paouris (2014)

Journal of the European Mathematical Society

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on ℝ n that is supported in a relatively small ball and linear functionals are uniformly bounded in L p for some p > 8 , in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c n / N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure...

Orbit measures, random matrix theory and interlaced determinantal processes

Manon Defosseux (2010)

Annales de l'I.H.P. Probabilités et statistiques

A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal....

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio Auffinger, Gérard Ben Arous, Sandrine Péché (2009)

Annales de l'I.H.P. Probabilités et statistiques

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (Electron. Commun. Probab.9 (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

Polynomial Riccati equations with algebraic solutions

Henryk Żołądek (2002)

Banach Center Publications

We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.

Product of exponentials and spectral radius of random k-circulants

Arup Bose, Rajat Subhra Hazra, Koushik Saha (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling...

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