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 -