Contemporary Mathematics Volume 237, 1999 Connections between random matrices and Szego limit theorems Estelle L. Basor ABSTRACT. The purpose of this paper is to describe the connection between asymptotic formulas for random matrices and Szego limit theorems. The pa- per will illustrate how the distribution formula for a random variable can be described asymptotically by the classical Szego theorems and then in turn how the ideas of random matrix theory allow one to find explicit expressions for Fredholm determinants. Szego type limit theorems for smooth symbols are in this way equivalent to certain distributions being normal. We will also give an example of a distribution computed via more general limit theorems that is not normal. 1. Introduction We will describe how the heuristic ideas behind the theory of random matrices give insight into finding certain expansions of integral operators and conversely to show how these expansions give immediate answers to problems in random matrix theory. We begin by outlining the situation for the standard Gaussian distribution defined on the space of N x N Hermitian matrices and later discuss the case of more general ensembles or distributions. A Gaussian distribution on the matrices is a distribution in which the real and imaginary parts of all the matrix components are independent (as much as they can be, given that the matrices are Hermitian) and Gaussian. Here we only state the results and refer the reader to [7] for general facts about random matrices or [1] for a detailed description of the computations involved. There is one fundamental property that connects the two fields. Suppose that we have a random variable of the form N h = L f(XiV2N /a.) i=l defined on the eigenvalues, Xl,.'" X N of the N x N Hermitian matrices. The function f must be a suitably smooth function. The factor v'2N rescales the eigenvalues ensuring that the mean spacing is bounded. The parameter a. will be 1991 Mathematics Subject Classification. 47 A35, 82B. Supported in part by NSF Grant DMS-9623278. 1 © 1999 American Mathematical Society
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