Induced vs spontaneous breakdown of $S$-matrix unitarity: probability of no return in quantum chaotic and disordered systems

We investigate systematically sample-to sample fluctuations of the probability $\tau$ of no return into a given entrance channel for wave scattering from disordered systems. For zero-dimensional («quantum chaotic») and quasi one-dimensional systems with broken time-reversal invariance we derive explicit formulas for the distribution of $\tau$, and investigate particular cases. Finally, relating $\tau$ to violation of $S$-matrix unitarity induced by internal dissipation, we use the same quantity to identify the Anderson delocalisation transition as the phenomenon of spontaneous breakdown of $S$-matrix unitarity.