The propagation of a signal in a disordered or a periodic structure is considered in a simple model that contains, as limiting cases, both the propagation of a classical persistent random walker AND the coherent propagation of a wave in the presence of elastic
scatterers of mesoscopic physics. A complete statistical analysis of the problem is performed by using a loop-expansion of all the possible paths in a one-dimensional system;
the statistics of the distribution of the length of the paths is presented and the transition from the ballistic to the diffusive regime is discussed.
A combination of "strip-techniques" allows to generalize this analysis to higher order dimensional systems, for different boundary conditions, and to obtain reflection and transmission coefficients. In particular results for propagation in small N x M structures will be presented, showing the presence of band gaps. The possible application to simple imaging problems will be discussed The different roles of the ''randomness'' in the underlying statistical model and the disorder of the medium and their interaction will be discussed, both analytically and through numerical simulations.