Fluctuations in random complex zeroes: Asymptotic normality revisited

Details Fluctuations in random complex zeroes: Asymptotic normality revisited Fluctuations in random complex zeroes: Asymptotic normality revisited

2010-03-22

arxiv.org/abs/1003.4251v1

Mathematics

Probability

30 pages

Scientific paper

By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant with respect to isometries of the complex plane. Extending the previous results of Sodin and Tsirelson, we compute the variance of linear statistics of random complex zeroes, and find close to optimal conditions on a test-function that yield asymptotic normality of fluctuations of the corresponding linear statistics. We also provide examples of test-functions with abnormal fluctuations of linear statistics.

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