Consider the random entire function
$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $
, where the
? 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.