Matricially free random variables

Advances in Mathematics

2014

Self Cite

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...

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The definitions of boolean independence, monotone independence and s-freeness can be found in [22]. Let us remark that the notion of s-freeness is related to the subordination property for the free additive convolution [7,34].…”

Section: Asymptotic Monotone Independence and S-freenessmentioning

confidence: 99%

Limit distributions of random matrices

Advances in Mathematics

2014

Self Cite

“…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).…”

Section: Mixed *-Momentsmentioning

confidence: 99%

“…At the same time r is the number of summands in the direct sum decomposition of our Fock space (with r vacuum vectors). The color of each block is related in a matricial way to the color of its nearest outer block [7], as shown in Fig. 1.…”

Section: Mixed *-Momentsmentioning

confidence: 99%

Random Matrices, Continuous Circular Systems and the Triangular Operator

2019

cosa

Self Cite

Using suitably defined continuous analogs of the matricial circular systems and the direct integral of Hilbert spaces H " ş ' Γ Hpγqdγ, we study the operators living in H which give the asymptotic joint *-distributions of complex independent Gaussian random matrices with not necessarily equal variances of the entries. These operators are decomposed in terms of continuous circular systems tζpx, y; uq : x, y P r0, 1s, u P Uu acting between the fibers of H, the continuous analogs of matricial circular systems obtained when the Gaussian entries are block-identically distributed. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of the formula for the moments of T˚T , where T is a triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.2010 Mathematics Subject Classification: 46L54, 60B20, 47C15

“…where we use the matricial two-index notation of [10,11]. This notation is often helpful (and will be used when we refer to the results of these papers) since the second index shows onto which basis vectors the operators act non-trivially (it must match the first index of the basis vector).…”

Section: Operatorial Realizationmentioning

confidence: 99%

Random Matrix Model for Free Meixner Laws

International Mathematics Research Notices

2014

Abstract. Applying the concept of matricial freeness which generalizes freeness in free probability we have recently studied asymptotic joint distributions of symmetric blocks of Gaussian random matrices (Gaussian Symmetric Block Ensemble). This approach gives a block refinement of the fundamental result of Voiculescu on asymptotic freeness of independent Gaussian random matrices. In this paper, we show that this framework is natural for constructing a random matrix model for free Meixner laws. We also demonstrate that the ensemble of independent matrices of this type is asymptotically conditionally free with respect to the pair of partial traces.