Abstract
Recently more and more disciplines of science and engineering have found Random Matrix Theory valuable. Some disciplines use the limiting densities to indicate the cutoff between “noise” and “signal.” Other disciplines are finding eigenvalue repulsions a compelling model of reality. This survey introduces both the theory behind these applications and matlab experiments allowing a reader immediate access to the ideas.
Note to our readers: This survey paper is in large part a precursor to a book on Random Matrix Theory that will be forthcoming. We reserve the right to reuse materials in the book.
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Notes
- 1.
Strictly speaking, the random variables should be written x ij (n).
- 2.
If the x ij (n) are identically distributed, a very common assumption, then it is sufficient to require finite moments.
References
J.T. Albrecht, C.P. Chan, and A. Edelman. Sturm sequences and random eigenvalue distributions. Foundations of Computational Mathematics, 9(4):461–483, 2009.
G.W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118, 2010.
Z.D. Bai. Convergence rate of expected spectral distributions of large random matrices. Part II. Sample covariance matrices. Annals of Probability, 21:649–672, 1993.
Z.D. Bai. Convergence rate of expected spectral distributions of large random matrices. Part I. Wigner matrices. The Annals of Probability, 625–648, 1993.
Z. Bai and J. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edn. Science Press, Beijing, 2010.
O.E. Barndorff-Nielsen and S. Thorbjørnsen. Lévy laws in free probability. Proceedings of the National Academy of Sciences, 99(26):16568, 2002.
R. Couillet and M. Debbah. Random Matrix Methods for Wireless Communications. Cambridge University Press, Cambridge, 2011.
V. Dahirel, K. Shekhar, F. Pereyra, T. Miura, M. Artyomov, S. Talsania, T.M. Allen, M. Altfeld, M. Carrington, D.J. Irvine, et al. Coordinate linkage of hiv evolution reveals regions of immunological vulnerability. Proceedings of the National Academy of Sciences, 108(28):11530, 2011.
I. Dumitriu. Eigenvalue statistics for beta-ensembles. Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 2003.
A. Edelman and N.R. Rao. Random matrix theory. Acta Numerica, 14(233–297):139, 2005.
A. Edelman. The random matrix technique of ghosts and shadows. Markov Processes and Related Fields, 16(4):783–790, 2010.
P.J. Forrester. Log-Gases and Random Matrices, vol. 34. Princeton University Press, Princeton, 2010.
V.L. Girko. Circular law. Teoriya Veroyatnostei I Ee Primeneniya, 29:669–679, 1984.
V.A. Marčenko and L.A. Pastur. Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik, 1:457, 1967.
M.L. Mehta. Random Matrices, vol. 142. Academic Press, San Diego, 2004.
F. Mezzadri and N.C. Snaith. Recent Perspectives in Random Matrix Theory and Number Theory. Cambridge University Press, Cambridge, 2005.
R.J. Muirhead. Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1982.
A. Nica and R. Speicher. Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, New York, 2006.
D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica, 17(4):1617, 2007.
S.M. Popoff, G. Lerosey, R. Carminati, M. Fink, A.C. Boccara, and S. Gigan. Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media. Physical Review Letters, 104(10):100601, 2010.
P. Šeba. Parking and the visual perception of space. Journal of Statistical Mechanics: Theory and Experiment, 2009:L10002, 2009.
B.D. Sutton. The stochastic operator approach to random matrix theory. Ph.D. thesis, Massachusetts Institute of Technology, 2005.
K.A. Takeuchi and M. Sano. Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Physical review letters, 104(23):230601, 2010.
E.P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62:548–564, 1955.
Acknowledgements
The first author was supported in part by DMS 1035400 and DMS 1016125.
We acknowledge many of our colleagues and friends, too numerous to mention here, whose work has formed the basis of Random Matrix Theory. We particularly thank Raj Rao Nadakuditi for always bringing the latest applications to our attention.
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Edelman, A., Wang, Y. (2013). Random Matrix Theory and Its Innovative Applications. In: Melnik, R., Kotsireas, I. (eds) Advances in Applied Mathematics, Modeling, and Computational Science. Fields Institute Communications, vol 66. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5389-5_5
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