Random complex zeroes, I. Asymptotic normality

Asymptotic normality of linear statistics of zeros of random polynomials

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2nφ_n(z)}dz$ under suitable assumptions on the weight functions $φ_n$.

非紧复流形上随机全纯截面因子的渐近正态性

Asymptotic Normality of Divisors of Random Holomorphic Sections on Non-compact Complex Manifolds

本文研究非紧 Hermite 流形上随机全纯截面零因子的统计行为，证明相关光滑线性统计量满足中心极限定理。

On Zeroes of Random Polynomials and Applications to Unwinding

Let $μ$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim μ$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $μ$ is radial around the origin, then those solutions are also distributed according to $μ$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $μ$ and condense another part on curves. We…

Zeroes of random Reinhardt polynomials

For a Reinhardt domain $Ω$ with the smooth boundary in $\mathbb{C}^{m+1}$ and a positive smooth measure $μ$ on the boundary of $Ω$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $γ_{N}$ which is induced by $L^{2}(\partialΩ,dμ)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\in\partialΩ$. First of all we apply stationary phase method to the Boutet de Monvel-Sjöstrand theorem to get the asymptotic for the partial szegö kernel, $S_{N}(z,z)$, and then we compute the s…