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Published in:
Asymptotic Freeness of Gaussian Random Matrices Read chapter Read first chapter

2017 | OriginalPaper | Chapter

4. Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices

Authors : James A. Mingo, Roland Speicher

Published in: Free Probability and Random Matrices

Publisher: Springer New York

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Abstract

After having developed the basic theory of freeness, we are now ready to have a more systematic look into the relation between freeness and random matrices. In Chapter 1, we showed the asymptotic freeness between independent Gaussian random matrices. This is only the tip of an iceberg. There are many more classes of random matrices which show asymptotic freeness. In particular, we will present such results for Wigner matrices, Haar unitary random matrices and treat also the relation between such ensembles and deterministic matrices. Furthermore, we will strengthen the considered form of freeness from the averaged version (which we considered in Chapter 1) to an almost sure one.

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previous chapter Free Harmonic Analysis
next chapter Fluctuations and Second Order Freeness
Literature
7.
G.W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118 (Cambridge University Press, Cambridge, 2010)
15.
Z. Bai, J.W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices. Springer Series in Statistics, 2nd edn. (Springer, New York, 2010)
57.
B. Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not. 2003(17), 953–982 (2003)MathSciNetCrossRefMATH
59.
B. Collins, P. Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)MathSciNetCrossRefMATH
66.
96.
F. Hiai, D. Petz, Asymptotic freeness almost everywhere for random matrices. Acta Sci. Math. (Szeged). 66(3–4), 809–834 (2000)
123.
V.A. Marchenko, L.A. Pastur, Distribution of eigenvalues for some sets of random matrices. Mat. Sbornik 114(4), 507–536 (1967)
129.
137.
A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335 (Cambridge University Press, Cambridge, 2006)
146.
N. Raj Rao, R. Speicher, Multiplication of free random variables and the S-transform: the case of vanishing mean. Electron. Commun. Probab. 12, 248–258 (2007)MathSciNetCrossRefMATH
160.
173.
178.
D. Voiculescu, Multiplication of certain non-commuting random variables. J. Oper. Theory 18(2), 223–235 (1987)MATH
180.
188.
D. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Not. 1998, 41–63 (1998)MathSciNetCrossRefMATH
203.
J. Wishart, The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A(1–2), 32–52 (1928)CrossRefMATH
206.
Y.Q. Yin, P.R. Krishnaiah, A limit theorem for the eigenvalues of product of two random matrices. J. Multivariate Anal. 13, 489–507 (1983)MathSciNetCrossRefMATH
Metadata
Title
Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices
Authors
James A. Mingo
Roland Speicher
Copyright Year
2017
Publisher
Springer New York
Book
Free Probability and Random Matrices

Print ISBN: 978-1-4939-6941-8

Electronic ISBN: 978-1-4939-6942-5

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4939-6942-5_4