Zeroes of random Reinhardt polynomials

TL;DR

This paper analyzes the distribution and correlation of zeros of random polynomials associated with Reinhardt domains in complex space, deriving their asymptotic behavior using advanced kernel and stationary phase techniques.

Contribution

It introduces a method to compute the scaling limit distribution and pair correlation functions of zeros for random Reinhardt polynomials in complex domains.

Findings

Asymptotic formulas for the partial Szegő kernel $S_N(z,z)$

Explicit scaling limit distribution function for zeros

Explicit scaling limit pair correlation function for zeros

Abstract

For a Reinhardt domain $\Omega$ with the smooth boundary in $\mathbb{C}^{m+1}$ and a positive smooth measure $\mu$ on the boundary of $\Omega$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $\gamma_{N}$ which is induced by $L^{2}(\partial\Omega,d\mu)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\in\partial\Omega$. First of all we apply stationary phase method to the Boutet de Monvel-Sj\"{o}strand theorem to get the asymptotic for the partial szeg\"{o} kernel, $S_{N}(z,z)$, and then we compute the scaling limit partial szeg\"{o} kernel in any direction in $\mathbb{C}^{m+1}$, then by using well-known Kac-Rice formula we compute scaling limit distribution function and scaling limit pair correlation function between zeros.