Scaling of Particle Trajectories on a Lattice I: Critical Behavior

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. For most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters. In addition to the two critical exponents $\tau=15/7$ and $d_{f}=7/4$ found before, the critical exponent $\sigma=3/7$, which is associated with the scaling function for trajectory size away from criticality, also appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity, at criticality. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class from that of percolation clusters. An argument for the scaling behavior found along the critical lines is presented.