RUI: Applications of Operator Theory to Random Matrix Theory

点击翻译按钮获取中文摘要

DMS-9970879ABSTRACTAsymptotic properties of determinants for different classes of operators that depend on a parameter have been important in several branches of physics and mathematics. In random matrix theory such determinants yield formulas that compute many fundamental statistical properties. For example, the distribution function for a linear statistic, (or a certain kind of random variable) in the classical Gaussian Unitary Ensemble is described by a Fredholm determinant of a convolution operator. Other ensembles of random matrices and other statistical objects give rise to more complicated operators. In the orthogonal and symplectic ensembles, matrix-valued symbols of operators naturally occur, while non-smooth random variables lead to operators with singular symbols. In all the above cases the asymptotic expansions are generalizations of the Strong Szego Limit Theorem to various classes of operators. Many physical systems possess such complicated behavior that exact predictions become impossible. Two billiard balls that are set into motion close together on an irregularly shaped billiard table may have very different paths. The energy level of a particle of a compound nucleus in a slow nuclear reaction also has complicated unpredictable behavior.Random matrix theory provides mathematical models that allow a simulation of the energy levels of the particle or the energies of the billiard balls.One of the goals of the subject is to understand the statistical behaviorof the energy levels. Here is an example. Suppose we add all the energies together. Does the distribution of the sum of the energies change depending on our initial assumptions about our models? For a large class of models, the answer is no. The distribution of the sum of energies, after appropriate normalization is bell-shaped. These results agree with experimentally acquired data and illustrate the universality of the theories.