Hierarchical Lorentz Mirror Model: Normal Transport and a Universal $2/3$ Mean--Variance Law
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| Format: | Preprint |
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2026
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| _version_ | 1866915832750145536 |
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| author | Lefevere, Raphael Tasaki, Hal |
| author_facet | Lefevere, Raphael Tasaki, Hal |
| contents | 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 $d\geq 3$, 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 $2/3$ for all $d\geq 2$ (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 conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07988 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hierarchical Lorentz Mirror Model: Normal Transport and a Universal $2/3$ Mean--Variance Law Lefevere, Raphael Tasaki, Hal Statistical Mechanics Mathematical Physics Probability 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 $d\geq 3$, 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 $2/3$ for all $d\geq 2$ (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 conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo |
| title | Hierarchical Lorentz Mirror Model: Normal Transport and a Universal $2/3$ Mean--Variance Law |
| topic | Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2602.07988 |