RUI: Determinant Identities, Szego Type Limit Theorems, and Connections to Random Matrices

California Polytechnic State University Foundation, San Luis Obispo CA

Investigators

Abstract

Determinants of Toeplitz matrices have arisen in many branches of mathematics and physics. For example, they describe the spin correlation between two sites in the classical Ising model of a two-dimensional magnet. More recently, they have been used to describe statistical properties of random matrices, whose eigenvalues model complicated systems. The asymptotic behavior of determinants of Toeplitz matrices is described using the Strong Szego-Limit Theorem and its generalizations. Recently, this theorem was improved in the sense that a new identity was found for the determinant that allows one to find very good estimates for the error in the Szego expansion of the determinant. The major purpose of this project is to extend this identity to other classes of matrices and operators. This has applications such as finding the distributions of linear statistics in random matrix theory, as well as finding the level spacing of the eigenvalues. Many physical systems possess such complicated behavior that exact predictions become impossible, so instead average properties of these systems are studied. For example, the energy level of a particle of a compound nucleus in a slow nuclear reaction has complicated unpredictable behavior. Random matrix theory provides mathematical models that allow a simulation of the energy levels of the particle. One of the tools that is used to study the statistical behavior of the random matrices and thus of the energy levels, is a determinant of a Toeplitz matrix. A determinant is a number that yields important information about a square array of numbers. These Toeplitz determinants occur in many branches of applied mathematics. One classical use was to study the properties of models of two-dimensional (or very thin) magnets. Determinants are often hard to compute. However, there is a result, called the Strong Szego Limit Theorem, which yields an estimate for such a determinant. Recently, this estimate was improved so that error terms could be calculated more easily. One major goal of the project is to extend these results to other classes of determinants. This will yield information about the energy levels of complicated systems in other types of models, not covered by the Toeplitz case.