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.
- 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)