PreprintCC BY 4.0arXiv
Rigidity of the Peres-Virág class
Abstract
Consider the hyperbolic Gaussian analytic function (GAF) F𝔻(z)=∑n=0∞ ξnzn, |z|<1, where the ξn are independent standard complex Gaussian variables. Peres-Virág theorem tells that the distribution of the zeros of F𝔻(z) is provided by a Determinantal Point Process (DPP). We prove that F𝔻(z) is the only GAF whose zeros are determinantal with a Hermitian kernel, up to multiplication by a zero-free holomorphic function and a one-to-one holomorphic change of variable, with a \ non-degeneracy condition imposed to exclude trivial cases. The proof identifies a geometric property of GAF's with (local) determinantal zeros: the metric defined by the first intensity must have constant curvature −4.
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BibTeX
@misc{abreu2026rigidity,
title = {{Rigidity of the Peres-Virág class}},
author = {Luís Daniel Abreu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03118},
eprint = {2611.03118},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03118},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rigidity of the Peres-Virág class
AU - Luís Daniel Abreu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03118
UR - https://arxiv.org/abs/2611.03118
AB - Consider the hyperbolic Gaussian analytic function (GAF) F_𝔻(z)=∑_n=0^∞ ξ_nz^n, |z|<1, where the ξ_n are independent standard complex Gaussian variables. Peres-Virág theorem tells that the distribution of the zeros of F_𝔻(z) is provided by a Determinantal Point Process (DPP). We prove that F_𝔻(z) is the only GAF whose zeros are determinantal with a Hermitian kernel, up to multiplication by a zero-free holomorphic function and a one-to-one holomorphic change of variable, with a \ non-degeneracy condition imposed to exclude trivial cases. The proof identifies a geometric property of GAF's with (local) determinantal zeros: the metric defined by the first intensity must have constant curvature −4.
ER -