The Statistical Distribution of the Zeros of Random Paraorthogonal Polynomials on the Unit Circle

TL;DR

This paper studies the zeros of random paraorthogonal polynomials on the unit circle, showing that as the degree grows, their local distribution resembles that of independent uniform points, indicating no local correlation.

Contribution

It proves that the zeros of these random paraorthogonal polynomials are asymptotically distributed like independent uniform points on the circle, revealing a Poisson-like local behavior.

Findings

Zeros are asymptotically Poisson distributed.

No local correlation between zeros for large degree.

Distribution matches that of independent uniform points.

Abstract

We consider polynomials on the unit circle defined by the recurrence relation \Phi_{k+1}(z) = z \Phi_{k} (z) - \bar{\alpha}_{k} \Phi_k^{*}(z) for k \geq 0 and \Phi_0=1. For each n we take \alpha_0, \alpha_1, ...,\alpha_{n-2} i.i.d. random variables distributed uniformly in a disk of radius r < 1 and \alpha_{n-1} another random variable independent of the previous ones and distributed uniformly on the unit circle. The previous recurrence relation gives a sequence of random paraorthogonal polynomials \{\Phi_n\}_{n \geq 0}. 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). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.