A sharp transition in zero overcrowding and undercrowding probabilities for stationary Gaussian processes

Denote by N_F(T) the number of zeros in the interval [0, T] of a real stationary Gaussian process F whose spectral measure is supported on [−A, −B] ∪ [B, A], with 0 ≤ B < A. We study linear deviations events for N_F(T), namely η-overcrowding events ({NF(T)>ηT}forη>ENF(1)) and η-undercrowding events ({NF(T)<ηT}forη<ENF(1)). We show that, as T tends to infinity, η-overcrowding probability undergoes a transition from exponential decay to Gaussian decay at η=Aπ, while η-undercrowding probability undergoes the reverse transition at η=Bπ.