Spectral representation of correlation functions for zeros of Gaussian power series with stationary coefficients

TL;DR

The paper develops a spectral framework for Gaussian analytic functions (GAFs) with stationary Gaussian coefficients, showing that the zeros are governed by the spectral measure $F$ through a covarianc kernel $K(z,w)=\int s(z,t)\overline{s(w,t)}\,F(dt)$. It derives an Edelman–Kostlan 1-point density in terms of $K$ and $F$, and proves a general integral representation for the $n$-point correlation function, enabling explicit analysis in several examples. Through three detailed cases, including the i.i.d. (hyperbolic) GAF, periodic-atomic spectral measures, and long-range correlated spectra, the work shows how the zeros can form determinantal point processes with Bergman-type kernels or exhibit arc-supported boundary behavior, illustrating the central role of the spectral measure. The convergence results connect weak limits of spectral measures to convergence of both the GAFs and their zero sets, providing a robust framework for understanding zeros under spectral perturbations and for validating approximate models. Collectively, the results offer a spectral-origin lens on zero statistics of GAFs and furnish tools for analyzing higher-order correlations and related phenomena in random analytic settings.

Abstract

We analyze Gaussian analytic functions (GAFs) defined as power series with coefficients modeled by discrete stationary Gaussian processes, utilizing their spectral measures. We revisit some limit theorems for random analytic functions and examine some examples of GAFs through numerical computations. Furthermore, we provide an integral representation of the n-point correlation functions of the zero sets of GAFs in terms of the spectral measures of the underlying coefficient Gaussian processes.

Figures (4)

• Figure 1: Left: The zeros of the approximate polynomial $X_{\mathrm{hyp}}^{(100)}(z) = \sum_{k=0}^{100} \xi_k z^k$ for $F(dt) = dt/(2 \pi)$. Red points indicate zeros inside the unit disk, while blue points indicate zeros outside the unit disk. Right: The zeros of $X_F(z)$ for $F(dt)=\mathbf{1}_{[-\pi/2,\pi/2]}(t)dt/\pi$. The zeros in the right half are distributed randomly, similar to those in the left panel, while the zeros in the left half are neatly aligned near the unit circle. The latter corresponds to the zeros that might disappear in the limit as $N \to \infty$, which is consistent with the fact that the density of the zeros on the left is $O(1)$.
• Figure 2: The graph of $g_r(\theta) = \pi(1-r^2)^2 \rho_1(r e^{i\theta})$ for $r=1/4, 1/2, 3/4, 1-$. The common intersection point of all the graphs is $1-(2/\pi)^2 \approx 0.594715$ at $\theta=\pm \pi/2$.
• Figure 3: The zeros of $X_F^{(N)}(z)$ were computed for $N =200$. The red points indicate zeros inside the unit disk, while the blue points indicate zeros outside the unit disk. The patterns of red and blue points seem to random.
• Figure 4: The zeros of $X_F^{(100)}(z)$ are computed $10,000$ times. The left (resp. right) panel shows the number of zeros inside the unit disk in the left-half (resp. right-half) plane.

Theorems & Definitions (18)

• Theorem 2.1: Edelman-Kostlan
• Proposition 2.2
• Proposition 2.3
• Corollary 2.4
• proof
• Proposition 4.1
• Lemma 4.2
• proof
• Remark 4.3
• Lemma 4.4