The complex zeros of random orthogonal polynomials

TL;DR

This paper derives explicit formulas for the density and distribution of complex zeros of random orthogonal polynomials on the unit circle and disk, using advanced complex analysis and probability techniques.

Contribution

It introduces new explicit formulas for the zero distributions of random orthogonal polynomials and explores their asymptotic behavior.

Findings

Explicit formulas for zero density and mean distribution

Asymptotic analysis of zero distributions

Application of complex analysis and probability methods

Abstract

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for the density and the mean distribution of complex zeros of random polynomials spanned by orthogonal polynomials on the unit circle and on the unit disk. We then inquire into the consequences of their asymptotical evaluations.