Hyperbolic Geometry

We can use random matrix theory to solve problems in hyperbolic geometry, using random walks on graphs as an intermediary step.

Operators and Free Probability

Large chunks of random matrix theory was discovered by people studying operators (in particular, the $C^*$-algebra and von Neumann algebra cases), and free probability was a tool developed to do this.

Representation Theory

Random matrix theory can be used to study representation theory problems through Ramanujan graphs.