Mathematical Sciences: Application of Operator Theory to Random Matrices and Random Variables

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ABSTRACT Proposal: DMS-9623278 PI: Basor The main objects of study in the theory of random matrices are certain Fredholm determinants. These arise to describe the level spacing distribution functions for random matrix models. Other random variables also have distribution function given by Fredholm determinants. The primary focus of this research is to describe the distribution function of random variables for different matrix models. In the classical case the operators are convolution operators, but for other models the operators are generalizations of the convolution type. The techniques already developed to study the classical Szego limit theorems will be generalized for use in the more complicated cases. If the random variable is of a certain discontinuous type, then the operators have singular symbols. The known theory for discontinuous symbols will be extended to apply in the more general setting. This will involve extensions of the Fisher-Hartwig conjecture. Many physical systems possess such complicated behavior that exact predictions become impossible. However, it is sometimes still possible to describe the average or statistical properties of the systems. For example in slow nuclear reactions it is important to understand the average properties of the energy levels of a compound nucleus. Quantum mechanics tells us that the energy levels of the system are simulated by the eigenvalues of a large Hermitian matrix. A matrix is a square array of numbers and the eigenvalues are a set of fundamental numbers associated to the matrix. The purpose of the proposed research is to study the statistical properties of the eigenvalues for different classes of randomly generated matriaes. The theory of random matrices also occurs in other areas of mathematics and physics, such as the distribution of the zeros of the Riemann zeta function, the study of disordered conductors and chaotic systems.