Estimates on the Probability of Outliers for Real Random Bargmann-Fock functions

In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there are no zeros in $[-r,r{]}^m\subset {\mathbb{R}}^m$ decays at least exponentially in terms of $r^m$. For this event we also prove a lower bound on the order of decay, which we do not expect to be sharp. Secondly, we compute the order of decay for the probability of families of events where the volume of the complex zero set is either much larger or much smaller then expected.