Higher-order sum rules for zeros of the planar Gaussian entire function
Abstract
We establish marginal sum rules for the truncated correlation functions of zeros of the planar Gaussian entire function (GEF). For each fixed integer k≥2, the integral in one coordinate converges absolutely and equals −(k−1) times the truncated correlation of order k−1. This answers the marginal-identity question posed by Krishnapur and Yogeshwaran for the planar GEF. We also prove a criterion for simple point processes under local moment and absolute integrability assumptions. Suppose there is a sequence of compactly supported cutoffs between zero and one that are eventually equal to one on each compact set. If the variances of their linear statistics tend to zero, the marginal identities hold almost everywhere. Strong clustering and variance bounds for smooth linear statistics verify these assumptions for GEF zeros. The proof uses mixed cumulants and the partition formula relating ordinary and factorial cumulants.
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BibTeX
@misc{zhu2026higherorder,
title = {{Higher-order sum rules for zeros of the planar Gaussian entire function}},
author = {Xiaolin Zhu and Feng Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23697},
eprint = {2610.23697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23697},
}
RIS
TY - UNPB
TI - Higher-order sum rules for zeros of the planar Gaussian entire function
AU - Xiaolin Zhu
AU - Feng Guo
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23697
UR - https://arxiv.org/abs/2610.23697
AB - We establish marginal sum rules for the truncated correlation functions of zeros of the planar Gaussian entire function (GEF). For each fixed integer k≥2, the integral in one coordinate converges absolutely and equals −(k−1) times the truncated correlation of order k−1. This answers the marginal-identity question posed by Krishnapur and Yogeshwaran for the planar GEF. We also prove a criterion for simple point processes under local moment and absolute integrability assumptions. Suppose there is a sequence of compactly supported cutoffs between zero and one that are eventually equal to one on each compact set. If the variances of their linear statistics tend to zero, the marginal identities hold almost everywhere. Strong clustering and variance bounds for smooth linear statistics verify these assumptions for GEF zeros. The proof uses mixed cumulants and the partition formula relating ordinary and factorial cumulants.
ER -