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Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

Yun-Heng Du

Abstract

We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_β(z)=\sum_{n=0}^{\infty}\frac{ξ_{n}}{\sqrt{Γ\bigl(\frac{2}β(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^β}$ on $\mathbb{C}$, where $β>0$. Under the condition that $F_β(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $μ_{0}^β$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/β}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $β=2$), who first revealed this striking conditional convergence and the emergence of a hole. Our analysis extends their phenomenon to the entire family of power-exponential weights.

Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

Abstract

We establish the \emph{hole phenomenon} for the Gaussian analytic function associated with the power-exponential weight on , where . Under the condition that has no zeros in , the scaled zero counting measure converges to a limiting measure vaguely in distribution. This limit exhibits a \emph{forbidden region} which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case ), who first revealed this striking conditional convergence and the emergence of a hole. Our analysis extends their phenomenon to the entire family of power-exponential weights.
Paper Structure (5 sections, 19 theorems, 169 equations)

This paper contains 5 sections, 19 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Upper Bounds of the Hole Probabilities
  4. The Lower Bounds of Hole Probabilities
  5. Proof of the Main Results

Key Result

Theorem 1.1

Let $\phi$ be a smooth function on $\mathbb{C}$ with compact support, define As $r \to \infty$, one has the asymptotic formula Here, where $m_{\mathbb{S}^1}$ denotes the normalized uniform measure on the unit circle $\mathbb{S}^1 = \{ |z|=1 \}$, and $\widehat{m}^{\beta}$ is a measure on $\mathbb{C}$ whose expression in polar coordinates is

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Lemma 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2: gn
  • Lemma 2.3
  • proof
  • ...and 20 more