Scaling of particle trajectories on a lattice

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. On both lattices, 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. At criticality, in addition to the two critical exponents tau = 15/7 and d(f) = 7/4 found before, the critical exponent sigma = 3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach Infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters. Nea, criticality, in the critical region, two scaling functions were determined numerically: f(x), related to the trajectory length (S) distribution n(S), and h(x), related to the trajectory size R-S (gyration radius) distribution, respectively. The scaling function f(x:) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent sigma = 0.43 approximate to 3/7 as at criticality, leading to a stretched exponential dependence of n(S) on S, n(S) similar to exp(-S-6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent sigma' = 1.6 +/- 0.3 and a superexponential dependence of n(S) on S. h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent sigma = 3/7 at and near a critical point is discussed.