Singularity, Universality, and Smoothness of Random Walks

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Random matrices play a central and fundamental role in various areas of science, including mathematical physics, data mining, random noise perturbation, combinatorics, statistics, and theoretical computer science. For example, the Wigner semicircle distribution, which arises as a limiting distribution of eigenvalues of random symmetric matrices in the large-size limit, was first observed in the study of nuclear physics. This research project is a rigorous mathematical study of the phenomena of singularity and universality in random matrices. This research is expected to lead to a more complete and deeper understanding of random matrices, with considerable impact on related areas of science. Technically speaking, two aspects of random systems will be studied: singularity and universality. This includes the analysis of the repulsion of Wigner matrix eigenvalues and of random polynomial roots, and specifically the study of certain Wegner-type estimates for matrix eigenvalues and, for random polynomial roots, the repulsion phenomenon under minimal conditions on the random coefficients. The goal under the universality aspect focuses on a central limit theorem-type result for the logarithmic determinant of Wigner matrices and certain equidistribution properties of the eigenvectors.