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Diane Holcomb: Random Matrices via Differential Operators II

Tid: Ti 2017-11-28 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 2 we will focus on giving a proof of the soft edge limit with some detail (though not all). In particular one of the challenges is to first show the the limiting operator at the soft edge "makes sense," that is it is well defined and has pure point spectrum. We will split the time between discussing the limiting operator and give the convergence proof.