Zeros of random orthogonal polynomials on the unit circle

Abstract

We study the statistical distribution of the zeros of some classes of random orthogonal polynomials on the unit circle. For each n we take the random Verblunsky coefficients alpha_0, alpha_1,...,alpha_{n-2} to be independent identically distributed random variables uniformly distributed in a disk of radius r < 1 and alpha_{n-1} to be another random variable independent of the previous ones and distributed uniformly on the unit circle. These coefficients define a sequence of random paraorthogonal polynomials Phi_n. For any n, the zeros of Phi_n are n random points on the unit circle. We prove that, for any point p on the unit circle, the distribution of the zeros of Phi_n in intervals of size O(1/n) near p is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). Therefore, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials. The same result holds when we take alpha_0, alpha_1,...,alpha_{n-2} to be independent identically distributed random variables uniformly distributed in a circle of radius r < 1 and alpha_{n-1} to be another random variable independent of the previous ones and distributed uniformly on the unit circle.