Propagation and organization in lattice random media

The authors show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, they consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (1) the state of each site is defined by its spin (up or down); (2) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (3) the state of the site is modified by the passage of the particle. They consider one-dimensional and triangular lattices, for which they give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular they prove that, in one dimension, the average propagation velocity is = 1/(3 {minus} 2q), with q the probability that a site has a spin {up{underscore}arrow}, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: = 1/8. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific reorganization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of their knowledge, presented here for the first time.