# Random (element) matrix theory

November 10, 2014 — October 10, 2019

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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 , 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.

## 2 References

Bordenave, and Chafaï. 2012. Probability Surveys.
Bose, Chatterjee, and Gangopadhyay. n.d. “Limiting Spectral Distributions of Large Dimensional Random Matrices.”
Cheng, and Singer. 2013. Random Matrices: Theory and Applications.
Edelman. 1989.
Edelman, and Rao. 2005. Acta Numerica.
El Karoui. 2010. The Annals of Statistics.
Koltchinskii, and Giné. 2000. Bernoulli.
Krbálek, and Seba. 2000. Journal of Physics A: Mathematical and General.
Meckes, Elizabeth. 2021. arXiv:2101.02928 [Math].
Meckes, Elizabeth S., and Meckes. 2017. arXiv:1612.08100 [Math-Ph].
O’Rourke, Vu, and Wang. 2016. Journal of Combinatorial Theory, Series A, Fifty Years of the Journal of Combinatorial Theory,.
Ormerod, and Mounfield. 2000. Physica A: Statistical Mechanics and Its Applications.
Scarpazza. 2003.
Tropp. 2015. An Introduction to Matrix Concentration Inequalities.
van Handel. 2017. In Convexity and Concentration. The IMA Volumes in Mathematics and Its Applications.
Vershynin. 2011. arXiv:1011.3027 [Cs, Math].
———. 2015. In Sampling Theory, a Renaissance: Compressive Sensing and Other Developments. Applied and Numerical Harmonic Analysis.
Wathen, and Zhu. 2015. Numerical Algorithms.
Wigner. 1955. The Annals of Mathematics.