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Diane Holcomb: Random Matrices via Differential Operators III

Tid: Ti 2017-12-05 kl 15.15

Plats: F11, KTH

Medverkande: Diane Holcomb, KTH

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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 3 we will focus on giving a proof of the hard edge limit. This proof has a different flavor than the soft edge because it works by considering inverses and showing convergence of an integral kernel. If time permits we will discuss some results that it is possible to prove about the hard edge process.