Inhomogeneous Markov Approach to Percolation Theory Based Propagation in Random Media

In recent years it has been shown that ray propagation in two-dimensional (2D) uniform random percolation lattices is well approximated by stochastic processes. Two of the probabilistic interpretations can be solved exactly but impose some additional restrictions on the ray dynamics: the 1999 model by G. Franceschetti demands that ray reflections inside the lattice are independent and identically distributed (IID); and the 2006 model by A. Martini assumes that for the analysis of vertical ray penetration reflections on vertical lattice cell faces can be neglected. In this paper we present a 2D generalization of the 1999 model that does not rely on the IID assumption. It is based on a set of coupled iteration formulas for right stochastic matrices that describe a time-inhomogeneous Markov process. Upon expanding the space of transition states, the iterative process can be recast in the projection of a time-homogeneous Markov chain. Thus the Perron-Frobenius theorem can be applied to analyze the stationary states of the process, which leads to a simple analytical expression for the probability of a ray ensemble reaching a specified lattice level after infinitely many reflections. We discuss different boundary conditions at the absorbing barriers of the ray propagation process and show that (1) by imposing the IID assumption, the 1999 model by G. Francescetti can be recovered; and (2) a one-dimensional reduction of the approach exactly leads to the 2006 model by A. Martini The presented stochastic approach provides a framework in which basic assumptions of probabilistic interpretations of percolation theory based ray propagation can be studied in some detail. It also allows for the consistent derivation of the stationary probability states which are needed to apply the approach to a wider range of phenomena such as internal-source ray propagation in both uniform and nonuniform random media.