Asymptotic Study of Random Unitary Ensembles with Singular Potentials

Description

Random matrix theory has important applications in many different areas ofmathematics and physics, such as number theory, combinatorics, statistics, string theory,condense matter physics, classical and quantum optics, etc.. Universality of eigenvaluespacings when the matrix size tends to infinity is one of the central problems in randommatrix theory. Universality means that the local eigenvalue statistics are the same formany different random matrix ensembles, or in another word, they do not rely on theexact probability distribution that is put on the matrices, but only on some generalcharacteristics of the ensemble. Classical universality results usually depend on theanalyticity of corresponding potentials of the ensemble. When analytic properties of thepotentials are changed, some new universality classes are expected to appear.In this project, we will study unitary ensembles whose potential possesses poles. Basedon the close relation between unitary ensembles and orthogonal polynomials, we willobtain the new limiting kernels from the uniform asymptotics of related orthogonalpolynomials. Note that when the potential has poles, the corresponding weight functionfor the orthogonal polynomials possesses essential singularities, as a consequence thepolynomials belong to the non-Szego class. We will apply the Deift-Zhou nonlinearsteepest descent method for Riemann-Hilbert problems to derive the uniformasymptotics of the polynomials. Near the singularities, new local parametrices will beconstructed. The transition properties of the limiting kernel will also be considered whenthe parameter associated with the poles varies. Some other important quantities, such asHankel determinants and Fredholm determinants will be studied, too.