We will take a quick overview of Brownian motion and introduce the Dyson Brownian motion. The main goal is to follow Theorem. 3.1.16 in [1] to understand the behavior of the eigenvalues of random matrices.

 

P.S. there is some modification of Theorem 3.1.16, see Theorem 21 of [2].

 

References:

 

[1] §3.1. in Topics in random matrix theory by Terence Tao 

https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf

[2] https://terrytao.wordpress.com/2010/01/18/254a-notes-3b-brownian-motion-and-dyson-brownian-motion/