LisMath Seminar 

13/05/2016, 16:00 — 17:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
David Garcia-Garcia, Universidade de Lisboa

Random Matrix Theory and Toeplitz Determinants

The first part of this talk will be an overview of some of the basic aspects of the theory of random matrices. This will include examples of the most common matrix ensembles, which are spaces of matrices whose entries are random variables. Focusing on an example of major importance, the Gaussian Unitary Ensemble, we will show how the probability distribution on these matrices is closely related to another probability distribution on their eigenvalues. We will also explore the relationship between random matrix theory and the theory of orthogonal polynomials. In the second part of the talk, to showcase the impact of random matrix theory on other fields, we will introduce Toeplitz determinants and discuss the Szeg Limit Theorem. If time allows it, we will comment some more general results in this direction.

Bibliography

1. P. Deift and D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics, 18 (2009). 
2. E. Basor, Toeplitz determinants, Fisher-Hartwig symbols, and random matrices, in Recent Perspectives in Random Matrix Theory and Number Theory, 309-336, Cambridge University Press (2005). 
3. P. Deift, A. Its and I. Krasovsky, Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results, Comm. Pure Appl. Math., 66, 13601438 (2013) [arXiv:1207.4990v3 [math.FA]]
4. D. Bump and P. Diaconis, Toeplitz Minors, J. Combin. Theory Ser. A, 97(2), 252-271 (2002).

See also

LisMath Seminar.pdf