The hole probability for Gaussian entire functions

Abstract

Consider the random entire function

$$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $$

, where the ϕ _n are independent standard complex Gaussian coefficients, and the a _n are positive constants, which satisfy

$$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $$

.

We study the probability P _H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a _n is logarithmically concave, we prove that

$$\log {P_H}(r) = - S(r) + o(S(r))$$

, where

$$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $$

, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.