Uniformity Transition for Ray Intensities in Random Media

We analyse a model for the intensity of distribution for rays
propagating without absorption in a random medium. The random medium
is modelled as a dynamical map. After N iterations, the intensity is modelled as
a sum of M contributions from different trajectories, each of which is
a product of N independent identically distributed random variables, representing
successive focussing or de-focussing events. The number of ray
trajectories reaching a given point is assumed to proliferate
exponentially: M=A^N, for some constant A.
We investigate the probability distribution of the sum.
We find a phase transition as
parameters of the model are varied. There is a phase where the
fluctuations are suppressed as N increases, and a phase where
theintensity has large fluctuations, for which we provide a large deviation
analysis.