Asymptotics of The Hole Probability for Zeros of Random Entire Functions

Details Asymptotics of The Hole Probability for Zeros of Random Entire Functions Asymptotics of The Hole Probability for Zeros of Random Entire Functions

2009-03-28

arxiv.org/abs/0903.4970v1

Mathematics

Complex Variables

21 pages

Scientific paper

10.1093/imrn/rnp229

We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the event where the function has no zeros in a disc of radius r. We show that the logarithm of the probability of the hole event decays asymptotically like -3/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole probability with other types of random coefficients.

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