  Waves propagating through a weakly scattering random medium show a pronounced
branching of the flow accompanied by the formation of freak waves, i.e.,
extremely intense waves. Theory predicts that this strong fluctuation regime is
accompanied by its own fundamental length scale of transport in random media,
parametrically different from the mean free path or the localization length. We
show numerically how the scintillation index can be used to assess the scaling
behavior of the branching length. We report the experimental observation of
this scaling using microwave transport experiments in quasi-two-dimensional
resonators with randomly distributed weak scatterers. Remarkably, the scaling
range extends much further than expected from random caustics statistics.
