Markov Approach to Percolation Theory Based Propagation in Random Media

For line-of-sight links in random media or urban areas, propagation may be approximated through sequential reflections of an optical ray in a two-dimensional medium of disordered lossless scatterers. Franceschetti approximated such percolation-based optical-ray propagation by a Markov process with two absorbing barriers, provided numerical solutions for the probability of a ray passing through the percolation lattice and solved-both approximately and exactly-a corresponding problem based on the theory of martingales. In this paper we solve exactly the Markov-theoretical formulation of the problem and prove that both the Markov and martingale approaches are equivalent. Our proof is an application of the Perron-Frobenius theory which provides an elegant framework for the study of the asymptotic behavior of stochastic processes. We demonstrate that for a wide range of vacancies and incident angles the exact solution of the Markov-theoretical formulation performs significantly better than the commonly used Wald approximation in the martingale approach. This has a number of implications on the accuracy of the model, especially for low density propagation media.