  We study the motion of a particle moving on a two-dimensional honeycomb
lattice, whose sites are randomly occupied by either right or left rotators,
which rotate the particle's velocity to its right or left, according to
deterministic rules. In the model we consider, the scatterers are each
initially oriented to the right with probability $p\in[0,1]$. This is done
independently, so that the initial configuration of scatterers, which forms the
medium through which the particle moves, are both independent and identically
distributed. For $p\in(0,1)$, we show that as the particle moves through the
lattice, it creates a number of reflecting structures. These structures
ultimately \emph{limit} the particle's motion, causing it to have a periodic
trajectory. As $p$ approaches either 0 or 1, and the medium becomes
increasingly homogenous, the particle's dynamics undergoes a discontinuous
transition from this self-limiting, periodic motion to a self-avoiding motion,
where the particle's trajectory, away from its initial position, is a
self-avoiding walk. Additionally, we show that the periodic dynamics observed
for $p\in(0,1)$ can persist, even if the initial configuration of scatterers
are not identically distributed. Furthermore, we show that if these
orientations are not chosen independently, this can drastically change the
particle's motion causing it to have a behavior that is nonperiodic.
