  We consider a class of Gaussian random holomorphic functions, whose expected
zero set is uniformly distributed over $\C^n $. This class is unique (up to
multiplication by a non zero holomorphic function), and is closely related to a
Gaussian field over a Hilbert space of holomorphic functions on the reduced
Heisenberg group. For a fixed random function of this class, we show that the
probability that there are no zeros in a ball of large radius, is less than
$e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to
this result we also compute probability estimates for the event that a random
function's unintegrated counting function deviates significantly from its mean.
