## Electronic Communications in Probability

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

### Rigidity for zero sets of Gaussian entire functions

#### 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