The zeros of Gaussian random holomorphic functions on ${\text{C}}^n$, and hole probability

We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over ${\text{C}}^n$. 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.