Zeros of random linear combinations of OPUC with complex Gaussian coefficients

Abstract

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n{\eta }_j{\varphi }_j(z),$$ in any Jordan region $\Omega \subset \mathbb{C}$. The basis functions ${\varphi }_j$ are orthogonal polynomials on the unit circle (OPUC) that are real-valued on the real line, and ${\eta }_0,\dots ,{\eta }_n$ are complex-valued iid Gaussian random variables. We derive an explicit intensity function for the number of zeros of $P_n$ in $\Omega$ for each fixed $n$. Using the Christoffel-Darboux formula, the intensity function takes a very simple shape. Moreover, we give the limiting value of the intensity function when the orthogonal polynomials are associated to Szegő weights.

This article relies heavily on the results of section 2 from arXiv:1605.06836 to give the analogues of the applications in sections 3 and 4 of arXiv:1605.06836 for OPUC