Electronic Communications in Probability

Rigidity for zero sets of Gaussian entire functions

Avner Kiro and Alon Nishry

Full-text: Open access

Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 30, 9 pp.

Dates
Received: 7 February 2019
Accepted: 22 April 2019
First available in Project Euclid: 5 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1559700466

Digital Object Identifier
doi:10.1214/19-ECP236

Subjects
Primary: 30Dxx: Entire and meromorphic functions, and related topics 60G55: Point processes 30B20: Random power series

Keywords
Gaussian entire functions point processes rigidity of linear statistics

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kiro, Avner; Nishry, Alon. Rigidity for zero sets of Gaussian entire functions. Electron. Commun. Probab. 24 (2019), paper no. 30, 9 pp. doi:10.1214/19-ECP236. https://projecteuclid.org/euclid.ecp/1559700466


Export citation

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.
  • [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.
  • [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.
  • [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.
  • [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.