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An Isaac Newton Institute Workshop

Recent Perspectives in Random Matrix Theory and Number Theory

Toeplitz Determinants and Connections to Random Matrices

3rd April 2004

Author: Estelle Basor (California Polytechnic State University, San Luis Obispo)

Abstract

Lecture1: This lecture will contain an introduction of Toeplitz Operators, Hankel Operators, and finite Toeplitz matrices. We will give a simple proof of the Strong Szego Limit Theorem which describes an asymptotic formula for determinants of Toeplitz matrices with smooth symbols.

Lecture2: This lecture will show how to extend the Szego Limit Theorem to symbols with discontinuities, the class of Fisher-Hartwig symbols.

Lecture3: The final lecture will show the connection of the limit theorems to results about the distribution functions for random matrix ensembles. If time permits, Laguerre ensembles will be considered and also the analogues for Wiener-Hopf opeators.