Scaling of Particle Trajectories on a Lattice II: The Critical Region

TL;DR

This paper investigates the scaling behavior of closed particle trajectories on lattices near criticality, revealing Gaussian scaling functions and stretched exponential dependencies, with new exponents identified for specific models.

Contribution

It introduces detailed numerical analysis of trajectory scaling functions near critical points, discovering Gaussian forms and new exponents, challenging previous assumptions of exponential dependence.

Findings

Scaling function $f(x)$ is symmetric double Gaussian with $\sigma \,\approx 3/7$

Trajectory length distribution follows a stretched exponential $e^{-S^{6/7}}$

New exponent $\sigma' \,= 1.6 \,\pm 0.3$ for rotator model on square lattice

Abstract

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice in the critical region are investigated. We study numerically two scaling functions: $f(x)$ related to the trajectory length distribution $n_S$ and $h(x)$ related to the trajectory size $R_S$ (gyration radius) as introduced by Stauffer for the percolation problem, where $S$ is the length of a closed trajectory. The scaling function $f(x)$ is in most cases found to be symmetric double Gaussians with the same characteristic size exponent $\sigma=0.43\approx 3/7$ as was found at criticality. In contrast to previous assumptions of an exponential dependence of $n_S$ on $S$, the Gaussian functions lead to a stretched exponential dependence of $n_S$ on $S$, $n_S\sim e^{-S^{6/7}}$. However, for the rotator model on the partially occupied square lattice, an alternative scaling function near criticality is found, leading to a new exponent $\sigma '=1.6\pm0.3$ and a super exponential dependence of $n_S$ on $S$. The appearance of the same exponent $\sigma=3/7$ describing the behavior at and near the critical point is discussed. Our numerical simulations show that $h(x)$ is essentially a constant, which depends on the type of lattice and on the concentration of the scatterers.