Asymptotics of the Hole Probability for Zeros of Random Entire Functions

Consider the random entire function $f(z)=\Sigma^{\infty}_{n=0}\phi_n \frac{z^n}{\sqrt{n!}},\ (\star)$ where the ϕn are independent and identically distributed (i.i.d.) standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability PH(r) that f has no zeros in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that $\log P_H(r)=-\frac {3e^2}{4}\cdot r^4+o(r^4)$ as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series. If ϕn are compactly supported random variables instead of Gaussians, we get a very different result: there exists r0 so that every random function of the form (★) must vanish in the disk {|z| < r0}.