Self-Limiting Trajectories of a Particle Moving Deterministically in a Random Medium

Abstract

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.