Zeros of random linear combinations of entire functions with complex Gaussian coefficients

TL;DR

This paper analyzes the distribution of zeros in random linear combinations of entire functions with Gaussian coefficients, providing explicit formulas and limits, especially for orthogonal polynomials on the real line.

Contribution

It derives explicit zero distribution formulas for random entire function combinations and explores their limits for orthogonal polynomials with Szeg\

Findings

Explicit intensity function for zeros in any Jordan region.

Simplified intensity formula for orthogonal polynomials via Christoffel-Darboux.

Limit of zero distribution for Szeg\

Abstract

We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ in any Jordan region $\Omega \subset \mathbb C$. The basis functions $f_j$ are entire functions 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 $n$. Our main application is to polynomials orthogonal on the real line. 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 Szeg\H{o} weights.