Uniformity Transition for Ray Intensities in Random Media

TL;DR

This paper studies the distribution of ray intensities in a random medium, revealing a phase transition between regimes of suppressed and large fluctuations, using a model based on dynamical maps and large deviation theory.

Contribution

It introduces a new model for ray intensity distribution in random media and analyzes the phase transition in fluctuation behavior.

Findings

Identifies a phase transition in intensity fluctuations.

Provides a large deviation analysis for the large fluctuation phase.

Shows exponential proliferation of ray trajectories affects intensity distribution.

Abstract

This paper analyses 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 $S$ of ${\cal N}$ contributions from different trajectories, each of which is a product of $N$ independent identically distributed random variables $x_k$, representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: ${\cal N}=\Lambda^N$, for some $\Lambda>1$. We investigate the probability distribution of $S$. We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of $S$ are suppressed as $N\to \infty$, and a phase where the $S$ has large fluctuations, for which we provide a large deviation analysis.