Hole probability for entire functions represented by Gaussian Taylor series |
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Authors: | Alon Nishry |
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Institution: | 1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
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Abstract: | Consider the Gaussian entire functionf(z) = ?? n=0 ?? ?? n a n z n , where {?? n } is a sequence of independent and identically distributed standard complex Gaussians and {a n } is some sequence of non-negative coefficients, with a 0 > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability P H (r) that f(z) has no zeros in the disk {|z| < r}. We prove that log P H (r) = ?S(r) + o(S(r)), where S(r) = 2·?? n??0log+(a n r n ) as r tends to ?? outside a deterministic exceptional set of finite logarithmic measure. |
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