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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
G.W. Anderson, O. Zeitouni, A CLT for a band matrix model. Probab. Theory Relat. Fields 134(2), 283–338 (2006)
K.J. Dykema, Multilinear function series and transforms in free probability theory. Adv. Math. 208(1), 351–407 (2007)
W. Hachem, P. Loubaton, J. Najim, Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17(3), 875–930 (2007)
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 rnm086
D.S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, Foundations of Free Noncommutative Function Theory. Mathematical Surveys and Monographs, vol. 199 (American Mathematical Society, Providence, RI, 2014)
A. Nica, D. Shlyakhtenko, R. Speicher, Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Not. 2002(29), 1509–1538 (2002)
A. Nica, D. Shlyakhtenko, R. Speicher, R-cyclic families of matrices in free probability. J. Funct. Anal. 188(1), 227–271 (2002)
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)
D. Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Not. 1996(20), 1013–1025 (1996)
R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Am. Math. Soc. 627, 88 (1998)
A.M. Tulino, S. Verdú, Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory 1(1), 184 (2004)
D. Voiculescu, Operations on certain non-commutative operator-valued random variables. Recent advances in operator algebras (Orléans, 1992). Astérisque 232, 243–275 (1995)
D. Voiculescu, The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2000(2), 79–106 (2000)
D. Voiculescu, Free analysis questions. I. Duality transform for the coalgebra of ∂ X: B . Int. Math. Res. Not. 2004(16), 793–822 (2004)
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)
J.D. Williams, Analytic function theory for operator-valued free probability. J. Reine Angew. Math. (2013). Published online: 2015-01-20
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Mingo, J.A., Speicher, R. (2017). Operator-Valued Free Probability Theory and Block Random Matrices. In: Free Probability and Random Matrices. Fields Institute Monographs, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6942-5_9
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