Hierarchical Lorentz Mirror Model: Normal Transport and a Universal $2\mathrm{/}3$ Mean--Variance Law

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 $d\ge 3$, we prove normal transport: the mean conductance scales as (cross-section)/(length) on all length scales. A Gaussian closure, supported by numerics, predicts that the variance-to-mean ratio of the conductance converges to the universal value $2\mathrm{/}3$ for all $d\ge 2$ (the ``$2\mathrm{/}3$ law’'). We provide numerical evidence for the $2\mathrm{/}3$ 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 conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo