Random (element) matrix theory



When we talk about random matrices per default we mean matrices with independent distributions for the elements. More generally we might consider distributions over whole matrices, which are obviously also “random matrices” but not the ones addressed here.

The archetypal result in capitalised Random Matrix Theory is Wigner’s semicircle law, which gives the distribution of eigenvalues in growing symmetric square matrices with a certain element-wise real distribution. Results like that are what we are after. There is a lot more one would like to know in general of course; for example, eigenvectors of random matrices (O’Rourke, Vu, and Wang 2016), and different element-wise distributions and so on, and indeed those are all heavily researched.

I am a consumer not a constructor of these theorems, so this page will remain forever sparse.

There are many more fancy distributions now. I mostly encounter this though matrix concentration theorems, which use random matrix results to prove things. Especially interesting to me: random projections and orthonormal operators.

To read

References

Bordenave, Charles, and Djalil Chafaï. 2012. Around the Circular Law.” Probability Surveys 9 (0): 1–89.
Bose, Arup, Sourav Chatterjee, and Sreela Gangopadhyay. n.d. “Limiting Spectral Distributions of Large Dimensional Random Matrices,” 30.
Cheng, Xiuyuan, and Amit Singer. 2013. The Spectrum of Random Inner-Product Kernel Matrices.” Random Matrices: Theory and Applications 02 (04): 1350010.
Edelman, Alan, and N. Raj Rao. 2005. Random Matrix Theory.” Acta Numerica 14 (May): 233–97.
El Karoui, Noureddine. 2010. The Spectrum of Kernel Random Matrices.” The Annals of Statistics 38 (1).
Handel, Ramon van. 2017. Structured Random Matrices.” In Convexity and Concentration, edited by Eric Carlen, Mokshay Madiman, and Elisabeth M. Werner, 107–56. The IMA Volumes in Mathematics and Its Applications. New York, NY: Springer.
Koltchinskii, Vladimir, and Evarist Giné. 2000. Random Matrix Approximation of Spectra of Integral Operators.” Bernoulli 6 (1): 113–67.
Krbálek, Milan, and Petr Seba. 2000. The Statistical Properties of the City Transport in Cuernavaca (Mexico) and Random Matrix Ensembles.” Journal of Physics A: Mathematical and General 33 (26): L229–34.
Meckes, Elizabeth. 2021. The Eigenvalues of Random Matrices.” arXiv:2101.02928 [Math], January.
Meckes, Elizabeth S., and Mark W. Meckes. 2017. A Sharp Rate of Convergence for the Empirical Spectral Measure of a Random Unitary Matrix.” arXiv:1612.08100 [Math-Ph], March.
O’Rourke, Sean, Van Vu, and Ke Wang. 2016. Eigenvectors of Random Matrices: A Survey.” Journal of Combinatorial Theory, Series A, Fifty Years of the Journal of Combinatorial Theory, 144 (Supplement C): 361–442.
Ormerod, Paul, and Craig Mounfield. 2000. Random Matrix Theory and the Failure of Macro-Economic Forecasts.” Physica A: Statistical Mechanics and Its Applications 280 (3-4): 497–504.
Tropp, Joel A. 2015. An Introduction to Matrix Concentration Inequalities.
Vershynin, Roman. 2011. Introduction to the Non-Asymptotic Analysis of Random Matrices.” arXiv:1011.3027 [Cs, Math], November.
———. 2015. Estimation in High Dimensions: A Geometric Perspective.” In Sampling Theory, a Renaissance: Compressive Sensing and Other Developments, edited by Götz E. Pfander, 3–66. Applied and Numerical Harmonic Analysis. Cham: Springer International Publishing.
———. 2016. Four Lectures on Probabilistic Methods for Data Science,” December.
Wathen, Andrew J., and Shengxin Zhu. 2015. On Spectral Distribution of Kernel Matrices Related to Radial Basis Functions.” Numerical Algorithms 70 (4): 709–26.
Wigner, Eugene P. 1955. Characteristic Vectors of Bordered Matrices With Infinite Dimensions.” The Annals of Mathematics 62 (3): 548.

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