  A fundamental insight in the theory of diffusive random walks is that the
mean length of trajectories traversing a finite open system is independent of
the details of the diffusion process. Instead, the mean trajectory length
depends only on the system's boundary geometry and is thus unaffected by the
value of the mean free path. Here we show that this result is rooted on a much
deeper level than that of a random walk, which allows us to extend the reach of
this universal invariance property beyond the diffusion approximation.
Specifically, we demonstrate that an equivalent invariance relation also holds
for the scattering of waves in resonant structures as well as in ballistic,
chaotic or in Anderson localized systems. Our work unifies a number of specific
observations made in quite diverse fields of science ranging from the movement
of ants to nuclear scattering theory. Potential experimental realizations using
light fields in disordered media are discussed.
