Hole Probability for Entire Functions represented by Gaussian Taylor Series (1105.5734v2)

Abstract: We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions in Taylor series form 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 disc of radius r. We find exact asymptotics for the rate of decay of the hole probability for large values of r, outside a small exceptional set. The exceptional set depends only on the non-random coefficients. We assume no regularity conditions on the non-random coefficients.