Variance of the number of zeroes of shift-invariant Gaussian analytic functions

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Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [−T,T] × [a, b] is asymptotically between cT and CT2, with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.

Original languageEnglish
Pages (from-to)753-792
Number of pages40
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - 1 Aug 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.


∗ Research supported by the Science Foundation of the Israel Academy of Sciences and Humanities, grant 166/11; by the United States–Israel Binational Science Foundation, grant 2012037; and by a National Science Foundation postdoctoral fellowship grant. Received May 23, 2016 and in revised form September 8, 2017

FundersFunder number
National Science Foundation
United States-Israel Binational Science Foundation2012037
Israel Academy of Sciences and Humanities166/11


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