  We 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, we
consider a point particle moving on a one-dimensional lattice whose sites are
occupied by scatterers with the following properties: (i) the state of each
site is defined by its spin (up or down); (ii) the particle arriving at a site
is scattered forward (backward) if the spin is up (down); (iii) the state 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 ($\uparrow \Leftrightarrow \downarrow
$). We consider one dimensional and triangular lattices, for which we give a
microscopic description of the dynamics, prove the propagation of a particle
through the scatterers, and compute analytically its statistical properties. In
particular we prove that, in one dimension, the average propagation velocity is
$<c(q)> = 1/(3-2q)$, with $q$ the probability that a site has a spin
$\uparrow$, and, in the triangular lattice, the average propagation velocity is
independent of the scatterers distribution: $<c> = 1/8$. 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 re-organization of the spins after the passage of the
particle. A detailed mathematical analysis of this phenomenon is, to the best
of our knowledge, presented here for the first time.
