I will present a simple but curious observation on the zeros of centered stationary Gaussian processes (SGP) on $\mathbb{R}$. The object of interest is
$N_T$ which is the number of zeros in the interval $[0,T]$. We restrict our attention to SGP with compactly supported spectral measure $\mu$. Let $A > 0$ be the smallest number such that supp$(\mu) \subseteq [-A, A]$.

Our primary interest is in the probability of overcrowding (resp. under crowding) event, which is the event that there is an excess (resp. deficit) of zeros in $[0,T]$ compared to the expected number, which is proportional to $T$. Comparing a couple of known results, we observe that there is a change in the behaviour of the probability $\p(N_T \geq \eta T)$, as $\eta$ varies. We show that there is indeed a \textit{sharp transition}. That is, this probability is at least of the order of $\exp(-C_{\eta}T)$ for small $\eta$, and at most of order $\exp(-c_{\eta}T^2)$ for large $\eta$. We identify the critical $\eta$ where this transition happens to be $\eta_c = A/\pi$. We also prove a similar result for under crowding probability when supp$(\mu)$ has a gap at the origin.

This talk is based on a joint work with Naomi Feldheim $\&$ Ohad Feldheim.