Limit distributions of random matrices

Abstract

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...

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

“…In the current paper, we consider an asymptotic distribution of the singular values of the matrix X 2 , where X is an elliptic random matrix. While singular value distributions of powers and products of random matrices with independent entries have been studied extensively (see [8,3,28,22,1,16,10,11,30], just to name a few), the recent results for elliptic random matrices ( [27,26,14]) concern mostly eigenvalue statistics. The obtained asymptotic distribution F is a new generalization of Fuss-Catalan distribution (see [20,5,21,17]).…”

Random Matrices, Continuous Circular Systems and the Triangular Operator

“…where u P U " rts, p, q P rrs and u 1 , u 2 are different copies of u. As we showed in [9], they can be realized as matrices of the form ζ p,q puq " cpp, q, uq b epp, qq,…”

“…The matricial circular operators studied in [10] live in a kind of matricially free Fock space. The original definition given in [7] was slightly generalized in [9,10], where we used the name of the 'matricially free Fock space of tracial type' which will be used below.…”

“…In our previous works we have studied the combinatorics of *-moments of various operators (creation, semicircular, circular, etc.) in the matricial (discrete) case [7,8,9,10]. It was based on the class of colored labeled noncrossing partitions (if only one label is used, we just have colored noncrossing partitions).…”

Random Matrix Model for Free Meixner Laws

“…In particular, we showed in [11,12] that the symmetric blocks of an ensemble of nˆn complex Hermitian Gaussian random matrices Y pu, nq with block-identical variances converge in moments under normailzed partial traces to the mixed moments of symmetrized Gaussian operators, namely T p,q pu, nq Ñ p ω p,q puq where u P U and 1 ď p ď q ď r, with U being an index set enumerating independent matrices. The operators p ω p,q puq are natural symmetrizations of square arrays pω p,q puqq of matricially free Gaussian operators playing the role of basic Gaussian operators.…”