Hole probability for entire functions represented by Gaussian Taylor series

Abstract

Consider the Gaussian entire functionf(z) = Σ ^∞ _n=0 ζ _n a _n z ⁿ, where {ζ _n } is a sequence of independent and identically distributed standard complex Gaussians and {a _n } is some sequence of non-negative coefficients, with a ₀ > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability P _H (r) that f(z) has no zeros in the disk {|z| < r}. We prove that log P _H (r) = −S(r) + o(S(r)), where S(r) = 2·Σ _n≥0log⁺(a _n r ⁿ) as r tends to ∞ outside a deterministic exceptional set of finite logarithmic measure.