Rigidity for zero sets of Gaussian entire functions.

*(English)*Zbl 1418.30002Summary: In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane.

We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is “fully rigid”. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.

We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is “fully rigid”. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.

##### MSC:

30B20 | Random power series in one complex variable |

30D20 | Entire functions of one complex variable (general theory) |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

PDF
BibTeX
XML
Cite

\textit{A. Kiro} and \textit{A. Nishry}, Electron. Commun. Probab. 24, Paper No. 30, 9 p. (2019; Zbl 1418.30002)

**OpenURL**

##### References:

[1] | S. Ghosh and M. Krishnapur, Rigidity hierarchy in random point fields: random polynomials and determinantal processes, arXiv:1510.08814 (2015). |

[2] | S. Ghosh and J. L. Lebowitz, Generalized stealthy hyperuniform processes: maximal rigidity and the bounded holes conjecture, Comm. Math. Phys. 363 (2018), no. 1, 97–110. · Zbl 1401.60096 |

[3] | S. Ghosh and Y. Peres, Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues, Duke Math. J. 166 (2017), no. 10, 1789–1858. · Zbl 1405.60067 |

[4] | A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008, Translated from the 1970 Russian original by Mikhail Ostrovskii, With an appendix by Alexandre Eremenko and James K. Langley. |

[5] | W. K. Hayman, A generalisation of Stirling’s formula, J. Reine Angew. Math. 196 (1956), 67–95. · Zbl 0072.06901 |

[6] | J. Ben Hough, M. Krishnapur, Yuval Peres, and Bálint Virág, Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009. |

[7] | A. Kiro and M. Sodin, On functions \(K\) and \(E\) generated by a sequence of moments, Expo. Math. 35 (2017), no. 4, 443–477. |

[8] | F. Nazarov and M. Sodin, Random complex zeroes and random nodal lines, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1450–1484. · Zbl 1296.30005 |

[9] | R. P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. · Zbl 0928.05001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.