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

Abstract: The Lorentz mirror model provides a clean setting to study macroscopic transport generated solely by quenched environmental randomness. We introduce a hierarchical version that admits an exact recursion for the distribution of left--right crossings, and prove normal transport: the mean conductance scales as (cross-section)/(length) for all length scales if $d\ge3$. A Gaussian approximation, supported by numerics, predicts that, in the marginal case $d=2$, this scaling acquires a logarithmic correction and that the variance-to-mean ratio of conductance converges to the universal value $2/3$ (the ``$2/3$ law'') for all $d\ge2$. We conjecture that both effects persist beyond the hierarchical setting. We finally provide numerical evidence for the $2/3$ law in the original Lorentz mirror model in $d=3$, and interpret it as a universal signature of normal transport induced by random current matching. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo