Asymptotic properties of random matrices and pseudomatrices

Lenczewski, Romuald

doi:10.1016/j.aim.2011.07.011

Cited by 15 publications

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“…Let us remark that similar distributions for one HGRM were derived in our previous paper [23]. Here, we generalize this result in several important directions:…”

Section: Introduction and Main Resultsmentioning

confidence: 61%

“…Finally, we take this opportunity to issue an erratum to the definition of the symmetrically matricially free array of units given in [23]. The new definition entails certain changes which are discussed in the Appendix.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Each space M q is similar to the matricially free-boolean Fock space introduced in [23] except that its tensor products do not have to end with diagonal free Fock spaces and that we use a family of arrays of Hilbert spaces indexed by u P U. Thus, M is equipped with a family of (vacuum) unit vectors.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Of course, the quantities N q , n q , D q depend on n, but this is suppressed in our notation. At the same time, we decompose each semicircular Gaussian operator in terms of the corresponding rˆr array of matricially free Gaussian operators [23], namely ωpuq " r ÿ p,q"1 ω p,q puq, living in the matricially free Fock space of tracial type, similar to the matricially free Fock space introduced in [23], in which we distinguish a family of states tΨ 1 , . .…”

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confidence: 99%

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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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“…Let us remark that similar distributions for one HGRM were derived in our previous paper [23]. Here, we generalize this result in several important directions:…”

Section: Introduction and Main Resultsmentioning

confidence: 61%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Limit distributions of random matrices

Lenczewski

2014

Advances in Mathematics

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“…In addition, the random matrix result in [2], once again dealing with some "block-modified" Wishart matrices, and once again leading to certain free combinations of free Poisson laws, is waiting as well to be generalized. There are probably some connections as well with the work of Lenczewski in [5]. We intend to come back to these algebraic questions, and to perform as well a systematic study of the resulting asymptotic measures, in a forthcoming paper [3].…”

Section: Discussionmentioning

confidence: 99%

Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

Banica

Nechita

2012

J Theor Probab

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Abstract. We study the partial transposition W Γ = (id ⊗ t)W ∈ M dn (C) of a Wishart matrix W ∈ M dn (C) of parameters (dn, dm). Our main result is that, with d → ∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n ± 1)/2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line. IntroductionThe partial transposition of aΓ obtained by transposing each of the n × n blocks of W . That is, we let W Γ = (id ⊗ t)W , where id is the identity of M d (C), and t is the transposition of M n (C). The partial transposition operation can be defined using coordinates, as follows. A particular decompositionThe entries of a matrix W ∈ M d (C) ⊗ M n (C) can be indexed by four indices, i, j ∈ {1, . . . , d} identifying the block of A the matrix entry belongs to, and a, b ∈ {1, . . . , n} fixing the position of the matrix entry inside the block. Then, the partial transposed matrix W Γ has coordinates W Γ ia,jb = W ib,ja . Motivated by questions in quantum information theory, the partial transposition operation for Wishart matrices was studied by Aubrun [1], who showed that, for certain special values of the parameters, the empirical spectral distribution of W Γ converges in moments to a non-centered semicircular distribution. In this paper we discuss the general case, our main result being as follows.Theorem A. Let W be a complex Wishart matrix of parameters (dn, dm). Then, with d → ∞, the empirical spectral distribution of mW Γ converges in moments to a free difference of free Poisson distributions of respective parameters m(n ± 1)/2.Observe that if one applies the transposition map on the first factor of the tensor product M d (C)⊗M n (C) instead of the second one, the spectral distribution of W Γ remains2000 Mathematics Subject Classification. 60B20 (46L54, 81P45).

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Matricially free random variables

Lenczewski

2010

Journal of Functional Analysis

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We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a generalization of both freeness and monotone independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial generalizations of semicirle laws.Comment: 38 pages, 4 figure

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Asymptotic properties of random matrices and pseudomatrices

Cited by 15 publications

References 14 publications

Limit distributions of random matrices

Limit distributions of random matrices

Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

Matricially free random variables

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