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, eigen*vectors* 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

- Kenneth Tay’s shortest explanation of the semicircular law.
- Bibliography and history at Scholarpedia
- Anderson, Guionnet and Zeitounni’s course
- Tao’s course and blog posts
- Djalil Chafaï, Around the circular law: an update, an update and conversation started from
*Around the circular law.*(Bordenave and Chafaï 2012)

## References

*Random Matrices: Theory and Applications*02 (04): 1350010.

*Acta Numerica*14 (May): 233–97.

*The Annals of Statistics*38 (1).

*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.

*Bernoulli*6 (1): 113–67.

*Journal of Physics A: Mathematical and General*33 (26): L229–34.

*arXiv:2101.02928 [Math]*, January.

*arXiv:1612.08100 [Math-Ph]*, March.

*Journal of Combinatorial Theory, Series A*, Fifty Years of the Journal of Combinatorial Theory, 144 (Supplement C): 361–442.

*Physica A: Statistical Mechanics and Its Applications*280 (3-4): 497–504.

*An Introduction to Matrix Concentration Inequalities*.

*arXiv:1011.3027 [Cs, Math]*, November.

*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.

*Numerical Algorithms*70 (4): 709–26.

*The Annals of Mathematics*62 (3): 548.

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