Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories

TL;DR

This survey reviews the critical behavior of closed trajectories in 2D Lorentz lattice gases, emphasizing scale-free statistics, fractal geometry, and universality classes at special scatterer concentrations.

Contribution

It synthesizes numerical studies and theoretical insights on critical phenomena, scaling laws, and exponents in 2D lattice gas models with quenched disorder.

Findings

Identification of critical exponents τ=15/7, d_f=7/4, σ=3/7 in universality classes

Observation of scale-free statistics and fractal geometry at critical points

Alternative exponents found in partially occupied models

Abstract

Lorentz lattice gases (LLGs) are discrete-time transport models in which a point particle moves ballistically between lattice sites and is scattered by randomly placed, quenched local scatterers such as ``rotators'' or ``mirrors.'' Despite the elementary update rules, LLGs exhibit rich dynamical regimes: typically, trajectories close quickly and the distribution of loop lengths has exponential tails, but at special concentrations of scatterers one observes critical behavior with scale-free statistics and fractal geometry. This survey focuses on the critical behavior of closed trajectories in two-dimensional LLGs, starting from the numerical study of Cao and Cohen, and its relation to percolation-hull scaling and kinetic hull-generating walks. We highlight the scaling hypothesis for loop-length distributions, the emergence of critical exponents $\tau=15/7$, $d_f=7/4$, and $\sigma=3/7$ in several universality classes, and the appearance of alternative exponents in partially occupied models.