Spectrum of the product of independent random Gaussian matrices

Burda, Z.; Janik, Romuald A.; Waclaw, Bartłomiej

doi:10.1103/physreve.81.041132

Cited by 108 publications

(190 citation statements)

References 36 publications

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“…Figure 1a) confronts our prediction with the numerical calculations. Recently, (23) was confirmed by independent calculation using diagrammatic methods [34]. • n=2, κ=1/3…”

Section: Examplesmentioning

confidence: 75%

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Belinschi¹,

Nowak²,

Speicher³

et al. 2017

J. Phys. A: Math. Theor.

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We extend the so-called "single ring theorem" [1], also known as the Haagerup-Larsen theorem [2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.

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“…Figure 1a) confronts our prediction with the numerical calculations. Recently, (23) was confirmed by independent calculation using diagrammatic methods [34]. • n=2, κ=1/3…”

Section: Examplesmentioning

confidence: 75%

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Belinschi¹,

Nowak²,

Speicher³

et al. 2017

J. Phys. A: Math. Theor.

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“…In the case when Bpnq is a power of a square random matrix, then it has recently been shown by Alexeev et al [1] that the limit moments are given by Fuss-Catalan numbers (products of independent random matrices have also been studied recently [2,9,10]). The distributions defined by the Fuss-Catalan numbers were explicitly determined by Penson and 9…”

Section: Wishart Matrices and Related Productsmentioning

confidence: 99%

“…The usual settings for Wishart matrices are slightly different since different normalizations are used [10,19]. For instance, if BpNq P GRMpmpNq, N, 1{Nq is a Gaussian random matrix of dimension mpNqˆN and 1{N is the variance of each entry, where N P N, it is assumed that t " lim N Ñ8 mpNq N and then one computes the limit distribution of B˚pNqBpNq under normalized trace trpNq composed with classical expectation.…”

Section: Wishart Matrices and Related Productsmentioning

confidence: 99%

Limit distributions of random matrices

Lenczewski

2014

Advances in Mathematics

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Abstract. We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type. Introduction and main resultsOne of the most important features of free probability is its close relation to random matrices. It has been shown by Voiculescu [33] that Hermitian random matrices with independent Gaussian entries are asymptotically free. This result has been generalized by Dykema to non-Gaussian random matrices [12] and has been widely used by many authors in their studies of asymptotic distributions of random matrices. It shows that there is a concept of noncommutative independence, called freeness, which is fundamental to the study of large random matrices and puts the classical result of Wigner [36] on the semicircle law as the limit distribution of certain symmetric random matrices in an entirely new perspective.In particular, if we are given an ensemble of independent Hermitian nˆn random matrices tY pu, nq : u P Uu whose entries are suitably normalized and independent complex Gaussian random variables for each natural n, then lim nÑ8 τ pnqpY pu 1 , nq . . . Y pu m , nqq " Φpωpu 1 q . . . ωpu m qq for any u 1 , . . . , u m P U, where tωpuq : u P Uu is a semicircular family of free Gaussian operators living in the free Fock space with the vacuum state Φ and τ pnq is the normalized trace composed with classical expectation called the trace in the sequel. This realization of the limit distribution gives a fundamental relation between random matrices and operator algebras.2010 Mathematics Subject Classification: 46L54, 15B52, 60F99 Key words and phrases: free probability, freeness, matricial freeness, random matrix, asymptotic freeness, matricially free Gaussian operatorsThe basic original random matrix model studied by Voiculescu corresponds to independent complex Gaussian variables, where the entries Y i,j pu, nq of each matrix Y pu, nq satisfy the Hermiticity condition, have mean zero, and the variances of real-valued diagonal Gaussian variables Y j,j pu, nq are equal to 1{n, whereas those of the real and imaginary parts of the off-diagonal (complex-valued) Gaussian variables Y i,j pu, nq are equal to 1{2n. If we rel...

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“…With tools from free probability and diagrammatic expansions, one may find the limiting global eigenvalue distributions as in [8,15,16,33]. It turns out that, as in the theory of a single random matrix, the various limits exhibit a rich and interesting mathematical structure, which also show a large degree of universality, see e.g.…”

Section: Products Of Ginibre Random Matricesmentioning

confidence: 99%

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

261

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Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

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Spectrum of the product of independent random Gaussian matrices

Cited by 108 publications

References 36 publications

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Limit distributions of random matrices

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

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