Many dimensions plus linear algebra plus probability equals Random Matrix Theory.

Tends to pop up certain random linear operators, leading to some elegant results where linear algebra is (“the whole modern world”) and herefore trendy.

I mostly encounter this though matrix concentration, which can apparently leverage random matrix results to prove things.

The most famous result is Wigner’s semicircle law, which gives the distribution of eigenvalues in growing symmetric square matrices with a certain element-wise real distribution and finite second moments.

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.

## To read

- Kenneth Tay gives a the 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)

Edelman, Alan, and N. Raj Rao. 2005. “Random Matrix Theory.” *Acta Numerica* 14 (May): 233–97. https://doi.org/10.1017/S0962492904000236.

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–L234. https://doi.org/10.1088/0305-4470/33/26/102.

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. https://doi.org/10.1016/S0378-4371(00)00075-3.

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. https://doi.org/10.1016/j.jcta.2016.06.008.

Tropp, Joel A. 2015. “An Introduction to Matrix Concentration Inequalities,” January. http://arxiv.org/abs/1501.01571.