Diane Holcomb: Random matrices via differential operators I

Abstract: This is the first of a series of four talks. The talks will be the next four Tuesdays at 15:15 in F11.

The Gaussian unitary ensemble (GUE) is a random matrix model of an Hermitian matrix whose entries are maximally independent (up to symmetry constraints) with Gaussian distribution. This model was traditionally studied using its integrable structure. More recently Trotter, and later Dumitriu and Edelman gave a tridiagonal matrix whose eigenvalues have the same distribution as the GUE. It was observed by Edelman and Sutton that this tridiagonal matrix in a certain scale acted like a certain differential operator. This opened up a new area of research in random matrices using random differential operators to describe limits of random matrix ensembles. This short course will discuss some of the proofs of these operator limits as well as some of the advantages of treating random matrices using these methods. 

Week 1: This will be a colloquium style talk. We introduce the spectral and empirical measures of an n by n random matrix and discuss their limits in the n to infinity limit. We then discuss tridiagonalization arguments for the classic matrix models, and introduce the notion of a local limit. We will examine the way in which the tridiagonal matrix model in soft edge scaling acts like a differential operator on the appropriate space. If time permits we will discuss the advantages of describing these limits with differential operators.