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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Naomi Dvora Feldheim

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  1. Naomi Dvora Feldheim
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Correspondence to Naomi Dvora Feldheim.

Additional information

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.

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Feldheim, N.D. Variance of the number of zeroes of shift-invariant Gaussian analytic functions. Isr. J. Math. 227, 753–792 (2018). https://doi.org/10.1007/s11856-018-1737-6

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  • DOI: https://doi.org/10.1007/s11856-018-1737-6

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