  Diffusive random walks feature the surprising property that the average
length of all possible random trajectories that enter and exit a finite domain
is determined solely by the domain boundary. Changes in the diffusion constant
or the mean-free path, that characterize the diffusion process, leave the mean
path length unchanged. Here we demonstrate experimentally that this result can
be transferred to the scattering of waves, even when wave interference leads to
marked deviations from a diffusion process. Using a versatile microwave setup,
we establish the mean path length invariance for the crossover to Anderson
localization and for the case of a band gap in a photonic crystal. We obtain
these results on the mean path length solely based on a transmission matrix
measurement through a novel procedure that turns out to be more robust to
absorption and incomplete measurement in the localized regime as compared to an
assessment based on the full scattering matrix.
