A Functional Central Limit Theorem for Window Counts of Hardy--Szegő Zeros
Abstract
The Hardy--Szegő zero process, investigated in the disk by Peres and Virág through the independent identically distributed Gaussian analytic function, is a canonical conformally invariant determinantal point process. This paper studies its upper half-plane realization. Although conformally equivalent to the disk model, this realization has its own natural geometry: real-translation invariance turns the process into a stationary object along the boundary and makes long horizontal windows the natural observables. For every admissible height window, we prove a Donsker-type functional central limit theorem for the centered zero counts in expanding horizontal windows, with an explicit intensity and variance depending on the height window. The proof is based on factorial cumulants and Brillinger mixing. The main technical input is a family of all-order integrability estimates for reduced cumulant densities, obtained by exploiting the determinantal cycle structure before integrating over the height variables. As further consequences, we derive an explicit covariance density, an asymptotic variance formula, and a macroscopic Gaussian white-noise limit for linear statistics.
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BibTeX
@misc{ai2026functional,
title = {{A Functional Central Limit Theorem for Window Counts of Hardy--Szegő Zeros}},
author = {Qian Ai and Feng Guo and Qi Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06583},
eprint = {2609.06583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06583},
}
RIS
TY - UNPB
TI - A Functional Central Limit Theorem for Window Counts of Hardy--Szegő Zeros
AU - Qian Ai
AU - Feng Guo
AU - Qi Zhou
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06583
UR - https://arxiv.org/abs/2609.06583
AB - The Hardy--Szegő zero process, investigated in the disk by Peres and Virág through the independent identically distributed Gaussian analytic function, is a canonical conformally invariant determinantal point process. This paper studies its upper half-plane realization. Although conformally equivalent to the disk model, this realization has its own natural geometry: real-translation invariance turns the process into a stationary object along the boundary and makes long horizontal windows the natural observables. For every admissible height window, we prove a Donsker-type functional central limit theorem for the centered zero counts in expanding horizontal windows, with an explicit intensity and variance depending on the height window. The proof is based on factorial cumulants and Brillinger mixing. The main technical input is a family of all-order integrability estimates for reduced cumulant densities, obtained by exploiting the determinantal cycle structure before integrating over the height variables. As further consequences, we derive an explicit covariance density, an asymptotic variance formula, and a macroscopic Gaussian white-noise limit for linear statistics.
ER -