Operator-Valued Free Multiplicative Convolution: Analytic Subordination Theory and Applications to Random Matrix Theory

Abstract

We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily numerically implementable. We present two concrete applications of our method: the product of two free operator-valued semicircular elements and the calculation of the distribution of dcd + d 2 cd 2 for scalar-valued c and d, which are free. Comparision between the solution obtained by our methods and simulations of random matrices shows excellent agreement. arXiv:1209.3508v1 [math.OA]

A general solution to (free) deterministic equivalents

“…This means that one only needs the individual B-valued Cauchy transforms of x, y (or good approximations of these) to obtain the B-valued Cauchy transform of x + y, and hence, its probability distribution. The operator-valued multiplicative convolution can also be numerically approximated (see [8]).…”

On the asymptotic distribution of block-modified random matrices

“…For our general numerical solution, where B is not commutative, the factorization property of the S-transform is implicitly being used in the subordination formulas (see [BSTV14]). …”

On the Capacity of Block Multiantenna Channels

“…Even though the qualitative facet of operator-valued free probability theory is still under development, recent techniques based on matricial fixed point equations have made it possible to numerically compute the asymptotic spectral distribution of a wide class of random matrix ensembles [23], [24]. A situation where these techniques are particularly useful is in the study of sums of products of independent block Gaussian matrices (see Section III-B).…”

“…With this information theoretic result and some free probability arguments, we prove an asymptotic capacity theorem that, in addition to reducing the optimization domain, does not depend on the dimension of the channel matrix. This theorem allows us to apply techniques already known by the information theory community [18], [19] as well as the subordination techniques recently introduced by Belinschi et al in [23], [24] to numerically compute the asymptotic capacity of the channels under consideration.…”