Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

Details Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

2011-05-19

arxiv.org/abs/1105.3929v3

Mathematics

Probability

24 pages, 1 figure. Some corrections were made and presentation was improved

Scientific paper

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.

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