Discrete Toeplitz Determinants and their Applications.

Abstract

In this dissertation, we consider the asymptotics of discrete Toeplitz determinants. We find a simple identity which express a discrete Toeplitz determinant as the product of a continuous Toeplitz determinant and a Fredholm determinant, whose kernel only depends on the continuous orthogonal polynomials and also the structure of the discrete nodes. This identity provides a way to find the asymptotics of discrete Toeplitz determinants by using the asymptotics of the corresponding continuous orthogonal polynomials.

We apply this method to the width of nonintersecting processes of several different types. When the starting points are the same as the ending points, the distribution of width can be represented in terms of a discrete Toeplitz determinant with real-valued symbol, and the asymptotics of the width can be obtained by directly using the identity mentioned above. We also consider one case when the starting points are different from the ending points. It turns out that the associated discrete Teoplitz determinant has a complex-valued symbol and one need further techniques to find the asymptotics of the Fredholm determinant in the identity. We develop such techniques in the later case and also obtain the asymptotics of the width.

The results of width lead us to an identity between the GUE Tracy-Widom distribution and the Airy process. We prove several similar identities.