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arXiv · 2512.23129

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

Tianyi Zhou
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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.

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Tianyi Zhou. 2025-12-29. Survey on Lattice Gas Models on 2D Lattices: Critical Behavior of Closed Trajectories. https://arxiv.org/abs/2512.23129

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