Random (element) matrix theory

When we talk about random matrices, by 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 through matrix concentration theorems, which use random matrix results to prove things. Especially interesting to me: random projections and orthonormal operators.