Abstract |
: |
Toeplitz and Hankel determinants arise in many different areas of mathematics, such as statistical mechanics and random matrix theory. In Chapter 1, we review some historical aspects and known results on these topics. In Chapter 2 and Chapter 3, we study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol is supported on the whole unit circle, has two jump discontinuities and tends to zero at a certain rate on an arc of the unit circle. In Chapter 2, we find asymptotics for such Toeplitz matrices if the rate is sufficiently fast. This generalizes a result proved by Widom, which was known only for symbols supported on an arc of the unit circle. In Chapter 3, we find asymptotics in the slower regime, which interpolates between a symbol supported on an arc and a symbol with Fisher-Hartwig singularities. This allows us to compute various probabilistic quantities in the thinned and conditional Circular Unitary Ensemble. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. In Chapter 4, we study the distribution of the ratio probability between the smallest and second smallest eigenvalue in the Laguerre Unitary Ensemble. We express this distribution as an integral of a Hankel determinant. The limiting distribution as the size of the matrices tends to infinity is found in terms of a function related to special solutions of a system of ODEs which can be expressed in terms of a Riemann-Hilbert problem. |