Walks in rigid environments

We consider a motion of the particle in a random configuration of scatterers on a lattice. There is a feedback of the particle on an environment which is characterized by its rigidity r. The type of a scatterer at a site of the lattice changes after the rth visit of the particle to this site. We show that in one-dimensional lattice the particle will eventually propagate in one direction with a random velocity if a rigidity is an odd number and it will oscillate (about origin) with an unbounded amplitude if r is an even number. The average velocity of propagation and the expected time until the first passage through a given site of the lattice are computed.