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 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.

漫射随机游走具有令人惊讶的特性，即进入和离开有限域的所有可能随机轨迹的平均长度仅由域边界确定。扩散常数或无均值路径的变化（表征扩散过程）使平均路径长度保持不变。在这里，我们通过实验证明，即使波干扰导致与扩散过程的明显偏离，该结果也可以转移到波的散射中。使用通用的微波设置，我们建立了到安德森定位的交叉以及光子晶体中带隙情况的平均路径长度不变性。