Rigidity for zero sets of gaussian entire functions

Avner Kiro, Alon Nishry

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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.

Original languageEnglish
Article number30
JournalElectronic Communications in Probability
StatePublished - 2019


FundersFunder number
Horizon 2020 Framework Programme692616
European Research Council
Israel Science Foundation382/15, 1903/18


    • Gaussian entire functions
    • Point processes
    • Rigidity of linear statistics


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