  We show that the fidelity decay between an initial eigenstate state evolved
under a unitary chaotic operator and the same eigenstate evolved under a
perturbed operator saturates well before the 1/N limit, where $N$ is the size
of the Hilbert space, expected for a generic initial state. We provide a
theoretical argument and numerical evidence that, for intermediate perturbation
strengths, the saturation level depends quadratically on the perturbation
strength.
