Abstract
Consider the random entire function
, where the ϕ n are independent standard complex Gaussian coefficients, and the a n are positive constants, which satisfy
.
We study the probability P H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a n is logarithmically concave, we prove that
, where
, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.
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Research supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 171/07.
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Nishry, A. The hole probability for Gaussian entire functions. Isr. J. Math. 186, 197–220 (2011). https://doi.org/10.1007/s11856-011-0136-z
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DOI: https://doi.org/10.1007/s11856-011-0136-z