Propagation and organization in lattice random media

We show that a signal can propagate in a particular direction through a mod el random medium regardless of the precise state of the medium. As a protot ype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the st ate of each site is defined by its spin (up or down); (ii) the particle arr iving at a site is scattered forward (backward) if the spin is up (down); ( iii) the stare of the site is modified by the passage of the particle, i.e. , the spin of the site where a scattering has taken place, flips (up arrow <=> down arrow). We consider one-dimensional and triangular lattices, for w hich we give a microscopic description of the dynamics, prove the propagati on of a particle through the scatterers, and compute analytically its stati stical properties. In particular we prove that, in one dimension, the avera ge propagation velocity is [c(q)] = 1/(3 - 2q), with q the probability that a site has a spin up arrow, and, in the triangular lattice, the average pr opagation velocity is independent of the scatterers distribution: [c] = 1/8 . In both cases, the origin of the propagation is a blocking mechanism, res tricting the motion of the particle in the direction opposite to the ultima te propagation direction, and there is a specific reorganization of the spi ns after the passage of the particle. A detailed mathematical analysis of t his phenomenon is, to the best of our knowledge, presented here for the fir st time.