Random Matrix Model for Free Meixner Laws

Lenczewski, Romuald

doi:10.1093/imrn/rnu041

Cited by 8 publications

(12 citation statements)

References 20 publications

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“…Nevertheless, this 'singular' case seems to be of importance since it leads to new random matrix models. For instance, it gives other types of asymptotic independence as those mentioned above as well as a new random matrix model for free Meixner laws and the associated asymptotic conditional freeness [24]. We also show that the study of non-Hermitian Gaussian random matrices (sometimes called the Ginibre Ensemble) can be reduced to the Hermitian case as in Voiculescu's paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 63%

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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Section: Introduction and Main Resultsmentioning

confidence: 63%

Limit distributions of random matrices

Lenczewski

2014

Advances in Mathematics

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“…In particular, we recover the expressions of the limiting spectral measures associated to the perturbations of the GUE(n) and the LUE nτ (n), which belong to the class of free Meixner laws. In the Gaussian setting, this was first observed in [25]. In the general case, this is a consequence of the local laws [24,23], as observed in [26].…”

Section: Introductionmentioning

confidence: 67%

Large deviations for spectral measures of some spiked matrices

Noiry,

Rouault

2020

Preprint

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“…This implies, in particular, that the asymptotic distribution of Mpu, nq under τ 1 pnq agrees with that of γpuq under Ψ 1 . The proof that the moments of γpuq are the moments of the free Meixner law associated with the parameters pα 1 puq, α 2 puq, β 1 puq, β 2 puqq is purely combinatorial and can be found in [9]. We also refer the reader to [9] for the proof of asymptotic conditional independence of the family of such matrices.…”

Section: Products Of Independent Gaussian Random Matricesmentioning

confidence: 99%

“…, d r , with block variances as additional parameters. In some cases, this enables us to construct simple random matrix models for certain families of probability measures, like the model for free Meixner laws [1,4] constructed in [9]. At the same time, studying such polynomials can lead to new results in free probability.…”

Section: Introductionmentioning

confidence: 99%

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Limit distributions of Gaussian block ensembles

Lenczewski

2015

Preprint

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It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial freeness, which is similar to freeness in free probability, but it also has some matricial features. Using this new concept of noncommutative independence, we described the asymptotics of blocks and symmetric blocks of certain classes of independent random matrices. In this paper, we present the main results obtained in this framework, concentrating on the ensembles of blocks of Gaussian random matrices.

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Random Matrix Model for Free Meixner Laws

Cited by 8 publications

References 20 publications

Limit distributions of random matrices

Limit distributions of random matrices

Large deviations for spectral measures of some spiked matrices

Limit distributions of Gaussian block ensembles

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