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

Abstract

We study a family of random Taylor series

$$\begin{aligned} F(z) = \sum _{n\ge 0} \zeta _n a_n z^n \end{aligned}$$

with radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients \((\zeta _n)\); 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 \(\{|z|\leqslant r\}\) as \(r\uparrow 1\). Our bounds take different forms according to whether the non-random coefficients \((a_n)\) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients \((a_n)\) display power-law behavior.

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Notes

  1. 1.

    Recently, Royen [17] proved the full version of the Gaussian correlation inequality, see also the paper by Latała and Matlak [9].

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Acknowledgements

The authors thank Alexander Borichev, Fedor Nazarov, and the referee for several useful suggestions.

Author information

Correspondence to Jeremiah Buckley.

Additional information

Jeremiah Buckley: Supported by ISF Grants 1048/11 and 166/11, by ERC Grant 335141 and by the Raymond and Beverly Sackler Post-Doctoral Scholarship 2013–14.

Ron Peled: Supported by ISF Grants 1048/11 and 861/15 and by IRG Grant SPTRF.

Mikhail Sodin: Supported by ISF Grants 166/11 and 382/15.

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Buckley, J., Nishry, A., Peled, R. et al. Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence. Probab. Theory Relat. Fields 171, 377–430 (2018). https://doi.org/10.1007/s00440-017-0782-0

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Mathematics Subject Classification

  • 30B20
  • 30C15
  • 60G55