Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

Abstract

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.

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

References in corpus (3)

• Exact statistics of complex zeros for Gaussian random polynomials with real coefficients
• Condensation of the roots of real random polynomials on the real axis
• Density of Complex Zeros of a System of Real Random Polynomials

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