Asymptotics of the hole probability for zeros of random entire functions

Alon Nishry*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Consider the random entire function f(z) = Σ n=0 Φn zn/n (*) where the φn are independent and identically distributed (i.i.d.) standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability PH(r) that f has no zeros in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that log P H(r) = -3e2/4•r4 + o(r4) as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series. If φn are compactly supported random variables instead of Gaussians, we get a very different result: there exists r0 so that every random function of the form (*) must vanish in the disk {|z| < r0}.

Original languageEnglish
Pages (from-to)2925-2946
Number of pages22
JournalInternational Mathematics Research Notices
Issue number15
StatePublished - 2010


FundersFunder number
Israel Science Foundation of the Israel Academy of Sciences and Humanities171/07


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