Free Probability and Random Matrices pp 225-247 | Cite as

# Operator-Valued Free Probability Theory and Block Random Matrices

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## Abstract

Gaussian random matrices fit quite well into the framework of free probability theory, asymptotically they are semi-circular elements, and they have also nice freeness properties with other (e.g. non-random) matrices. Gaussian random matrices are used as input in many basic models in many different mathematical, physical, or engineering areas. Free probability theory provides then useful tools for the calculation of the asymptotic eigenvalue distribution for such models. However, in many situations, Gaussian random matrices are only the first approximation to the considered phenomena, and one would also like to consider more general kinds of such random matrices. Such generalizations often do not fit into the framework of our usual free probability theory. However, there exists an extension, operator-valued free probability theory, which still shares the basic properties of free probability but is much more powerful because of its wider domain of applicability. In this chapter, we will first motivate the operator-valued version of a semi-circular element and then present the general operator-valued theory. Here we will mainly work on a formal level; the analytic description of the theory, as well as its powerful consequences, will be dealt with in the following chapter.

## Keywords

Operator-valued Free Probability Theory Gaussian Random Matrices Limiting Eigenvalue Distribution Semicircular Element Circular Family## References

- 6.G.W. Anderson, O. Zeitouni, A CLT for a band matrix model. Probab. Theory Relat. Fields
**134**(2), 283–338 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 68.K.J. Dykema, Multilinear function series and transforms in free probability theory. Adv. Math.
**208**(1), 351–407 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 90.W. Hachem, P. Loubaton, J. Najim, Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab.
**17**(3), 875–930 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 94.J.W. Helton, R. Rashidi Far, R. Speicher, Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints. Int. Math. Res. Not. IMRN
**2007**(22), 15 (2007). Art. ID rnm086Google Scholar - 107.D.S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov,
*Foundations of Free Noncommutative Function Theory*. Mathematical Surveys and Monographs, vol. 199 (American Mathematical Society, Providence, RI, 2014)Google Scholar - 139.A. Nica, D. Shlyakhtenko, R. Speicher, Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Not.
**2002**(29), 1509–1538 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 140.A. Nica, D. Shlyakhtenko, R. Speicher,
*R*-cyclic families of matrices in free probability. J. Funct. Anal.**188**(1), 227–271 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 147.R. Rashidi Far, T. Oraby, W. Bryc, R. Speicher, On slow-fading MIMO systems with nonseparable correlation. IEEE Trans. Inf. Theory
**54**(2), 544–553 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 155.D. Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Not.
**1996**(20), 1013–1025 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 163.R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Am. Math. Soc.
**627**, 88 (1998)MathSciNetzbMATHGoogle Scholar - 174.A.M. Tulino, S. Verdú, Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory
**1**(1), 184 (2004)Google Scholar - 184.D. Voiculescu, Operations on certain non-commutative operator-valued random variables. Recent advances in operator algebras (Orléans, 1992). Astérisque
**232**, 243–275 (1995)Google Scholar - 190.D. Voiculescu, The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not.
**2000**(2), 79–106 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 193.D. Voiculescu, Free analysis questions. I. Duality transform for the coalgebra of
*∂*_{X: B}. Int. Math. Res. Not.**2004**(16), 793–822 (2004)Google Scholar - 195.D.-V. Voiculescu, Free analysis questions II: the Grassmannian completion and the series expansions at the origin. Journal für die reine und angewandte Mathematik (Crelles J.)
**2010**(645), 155–236 (2010)Google Scholar - 202.J.D. Williams, Analytic function theory for operator-valued free probability. J. Reine Angew. Math. (2013). Published online: 2015-01-20Google Scholar