Hole Probability for Entire Functions Represented by Gaussian Taylor Series

Hole Probability for Entire Functions Represented by Gaussian Taylor Series - Alon Nishry

Alon Nishry
Tel Aviv University
October 9, 2012

We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions given by a Taylor series with i.i.d complex Gaussian random variables and arbitrary non-random coefficients. A 'hole' is the event where the function has no zeros in a disk of radius r.

We find exact asymptotics for the rate of decay of the hole probability for large values of r, outside a small (non-random) exceptional set.