  Assuming certain conditions on the spectral measures of centered stationary
Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the
probability of the event that their zero count in an interval (resp., nodal
length in a square domain) is larger than $n$, where $n$ is much larger than
the expected value of the zero count in that interval (resp., nodal length in
that square domain), is exponentially small in $n$.
