Diane Holcomb: Random Matrices via Differential Operators IV

Abstract:

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.

In week 4 we will give a bulk process convergence for the circular ensemble. This limiting operator instead of being given by a second order differential operator is a first order Dirac operator. The soft and hard edge operators may also be put into this Dirac operator framework and will be discussed if time permits.