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The density of zeros of random power series with stationary complex Gaussian coefficients

Tomoyuki Shirai

TL;DR

This work analyzes zeros of random power series with stationary complex Gaussian coefficients by linking the radial density of zeros near the unit circle to the spectral measure of the coefficient process. Using the Edelman-Kostlan framework, it derives explicit asymptotics for the 1-point zero density $\rho_1(z)$ in three regimes determined by the local behavior of the spectral density $f_{\varphi}$ at $0$, and further refines these with precise asymptotics of Poisson-type integrals near the boundary. The paper also establishes a sharp connection between the spectral support and analytic continuation: the complement of the spectral support is the regular set across which continuation is possible, while degeneracies in the spectral density can reduce zero density and enable continuation across arcs. Collectively, the results quantify how coefficient dependence and spectral degeneracy shape zero distributions and boundary analyticity in random Gaussian power series, with explicit formulas for special cases such as 1-dependent coefficients.

Abstract

We study the zeros of random power series with stationary complex Gaussian coefficients, whose spectral measure is absolutely continuous. We analyze the precise asymptotic behavior of the radial density of zeros near the boundary of the circle of convergence. The dependence of the coefficients generally reduces the density of zeros compared with that of the hyperbolic Gaussian analytic function (the i.i.d. coefficients case), where the spectral density and its zeros plays a crucial role in this reduction. We also show the relationship between the support of the spectral measure and the analytic continuation at the boundary of the circle of convergence.

The density of zeros of random power series with stationary complex Gaussian coefficients

TL;DR

This work analyzes zeros of random power series with stationary complex Gaussian coefficients by linking the radial density of zeros near the unit circle to the spectral measure of the coefficient process. Using the Edelman-Kostlan framework, it derives explicit asymptotics for the 1-point zero density in three regimes determined by the local behavior of the spectral density at , and further refines these with precise asymptotics of Poisson-type integrals near the boundary. The paper also establishes a sharp connection between the spectral support and analytic continuation: the complement of the spectral support is the regular set across which continuation is possible, while degeneracies in the spectral density can reduce zero density and enable continuation across arcs. Collectively, the results quantify how coefficient dependence and spectral degeneracy shape zero distributions and boundary analyticity in random Gaussian power series, with explicit formulas for special cases such as 1-dependent coefficients.

Abstract

We study the zeros of random power series with stationary complex Gaussian coefficients, whose spectral measure is absolutely continuous. We analyze the precise asymptotic behavior of the radial density of zeros near the boundary of the circle of convergence. The dependence of the coefficients generally reduces the density of zeros compared with that of the hyperbolic Gaussian analytic function (the i.i.d. coefficients case), where the spectral density and its zeros plays a crucial role in this reduction. We also show the relationship between the support of the spectral measure and the analytic continuation at the boundary of the circle of convergence.
Paper Structure (11 sections, 9 theorems, 103 equations, 2 figures)

This paper contains 11 sections, 9 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Edelman-Kosltan's formula and spectral measures
  4. Precise asymptotics of Poisson integral and its relative
  5. Asymptotic expansion I
  6. Asymptotic expansion II
  7. Proof of Theorem \ref{['thm:main']}
  8. The case $f_{\varphi}(0)>0$
  9. The case $f_{\varphi}(0)=0$
  10. The case $f_{\varphi}(0)=f_{\varphi}"(0)=0$
  11. Analytic continuation

Key Result

Theorem 2.1

Let $F_{\varphi}(ds) = F(d(s+\varphi))$ on $(-\pi, \pi]$ modulo $2\pi$. Suppose $F_{\varphi}(ds) = f_{\varphi}(s)ds$ and $f_{\varphi}(s)$ is smooth at $s=0$. (i) When $f_{\varphi}(0) > 0$, (ii) When $f_{\varphi}(0) = 0$, (iii) When $f_{\varphi}(0)=f_{\varphi}"(0)=0$. Then,

Figures (2)

  • Figure 1: Left: The zeros of the approximate polynomial $X_{\mathrm{hyp}}^{(400)}(z) = \sum_{k=0}^{400} \xi_k z^k$ for $F(dt) = dt/(2 \pi)$, the i.i.d. case. Red points indicate zeros inside the unit disk, while blue points indicate zeros outside the unit disk. Right: The zeros of an approximate polynomial of $X_F(z)$ for $F(dt)=\mathbf{1}_{[-\pi/2,\pi/2]}(t)dt/\pi$.
  • Figure 2: The case $f(s) = I_{[-\pi/2,\pi/2]}(s)$ and $\varphi=3\pi/4$. From left, $f_{\varphi}(s)$, ${\widehat{f}}_{\varphi}(s)$, and $\check{f}_{\varphi}(s)$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Example 2.3: $1$-dependent case
  • Example 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • ...and 10 more