We introduce and study a new type of convolution of probability measures, denoted μ --ν and called the s-free additive convolution, which is defined by the subordination functions associated with the free additive convolution. We derive an alternating decomposition of μ --ν for compactly supported μ and ν, using another convolution called orthogonal additive convolution. This decomposition leads to two types of 'complete' alternating decompositions of the free additive convolution μ ν. More importantly, we develop an operatorial approach to the subordination property and introduce the associated notion of s-free independence. Moreover, we establish relations between convolutions associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, our result leads to natural decompositions of the free product of rooted graphs.

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...

We study the asymptotics of sums of matricially free random variables called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are `asymptotically matricially free' whereas the corresponding symmetric random blocks are `asymptotically symmetrically matricially free', where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, lower-block-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.Comment: 33 pages, 2 figure

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