2026/02/28 by Raphael Lefevere, Hal Tasaki
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #math-ph #math.MP #math.PR
17 pages, 14 figures, essential reference added in v2, minor changes in v3, v4, major revision in version in v5. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo
arxiv created 2026/08/04 · arxiv updated 2026/08/05
The Lorentz mirror model provides a clean setting to study macroscopic transport generated solely by quenched environmental randomness. We introduce a hierarchical version whose distribution of left--right crossings satisfies an exact recursion. In dimensions d≥3, we prove two-sided bounds that support normal transport: the mean conductance scales as (cross-section)/(length). A Gaussian closure, supported by numerics, predicts that the variance-to-mean ratio of the dimensionless conductance converges to the universal value 2/3 for all d≥2 (the ``2/3 law''). We provide numerical evidence for the 2/3 law in the original (non-hierarchical) Lorentz mirror model in d=3, and conjecture that it is a universal signature of normal transport induced by random current matching. In the marginal case d=2, our hierarchical recursion reproduces the known scaling of the mean and variance of conductance. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo