Large Random Matrices and Determinantal Random Point Fields

University Of California-Davis, Davis CA

Investigators

Abstract

The principal investigator will work on several problems in random matrix theory and determinantal random point fields. The main emphasis of the research is on statistical properties of the eigenvalues of large random matrices, in particular on the universality conjecture. Building on the previous work on the largest eigenvalues of certain Wigner matrices he expects to extend his results to a wider class of Wigner matrices and prove similar results for sample covariance matrices. He also proposes to study universality in the bulk of the spectrum by using the renormalization group approach. Another foci of the project is concerned with determinantal random point fields. The goal is to find sufficiently general conditions for Central Limit Theorem type results for (rescaled) linear statistics and to study the ergodic properties of translation-invariant random point fields. The random matrix models that are proposed to study come from, or have applications in multivariate statistical analysis (principal component analysis), nuclear physics (statistics of energy levels of heavy nuclei), solid state physics (modelling transport properties of small metallic particles and quantum dots) and theoretical computer science (computational complexity, statistical analysis of errors and linear numerical algorithms). The importance of the field increases as many different areas of mathematics and physics including combinatorics, representation theory, operator algebras, number theory, integrable systems, quantum chaos, nuclear physics, statistical physics appear to have deep and fruitful connections to random matrices. Besides the various applications of the results indicated in the proposal the principal investigator believes that it is equally important to achieve a better understanding of some mathematical phenomena in random matrices, in particular, universality of local distribution of eigenvalues.