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Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

Research output: Contribution to journalArticlepeer-review

Jeremiah Anthony Buckley, Alon Nishry, Ron Peled, Mikhail Sodin

Original languageEnglish
Pages (from-to)377-430
JournalPROBABILITY THEORY AND RELATED FIELDS
Volume171
Issue number1-2
Early online date20 May 2017
DOIs
Accepted/In press6 Apr 2017
E-pub ahead of print20 May 2017
PublishedJun 2018

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Abstract

We study a family of random Taylor series
F(z)=∑n≥0ζnanzn
with radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients (ζn) ; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that F does not vanish in the disk {|z|⩽r} as r↑1 . Our bounds take different forms according to whether the non-random coefficients (an) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients (an) display power-law behavior.

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