TY - JOUR

T1 - Rigidity for zero sets of gaussian entire functions

AU - Kiro, Avner

AU - Nishry, Alon

N1 - Publisher Copyright:
© 2019, Institute of Mathematical Statistics. All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Gaussian entire functions

KW - Point processes

KW - Rigidity of linear statistics

UR - http://www.scopus.com/inward/record.url?scp=85068789742&partnerID=8YFLogxK

U2 - 10.1214/19-ECP236

DO - 10.1214/19-ECP236

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AN - SCOPUS:85068789742

SN - 1083-589X

VL - 24

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

M1 - 30

ER -