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

Figures (11)

• Figure 1: Original Lorentz mirror model in $d=2$ (periodic boundary conditions in the vertical direction). (a) A random choice of local pairings yields a collection of disjoint trajectories. (b) The induced perfect matching of the external edges (thick lines). There are two crossings.
• Figure 2: (a) The generation-$0$ block. (b) The construction of the generation-$n$ block from $2^d$ independent copies of the generation-$(n-1)$ block.
• Figure 3: Ratio $\bar{v}(L)/\bar{\mu}(L)$ of the sample-to-sample variance $\bar{v}(L)$ to the mean $\bar{\mu}(L)$ of the number of crossings, as a function of $L=2^n$ with $n=1,\ldots,7$. Error bars indicate $95\%$ confidence intervals. (a) Standard rule (all local pairings allowed). (b) Orthogonal rule (restricted local pairings; no edge is paired with its opposite). The horizontal line marks $2/3$.
• Figure SM.1: Equivalence between the mirror and tube representations in $d=2$. (a) A configuration of mirrors (empty sites, "/" mirrors, and "\\" mirrors) on the $L\times L$ square lattice with periodic boundary conditions vertically and external edges attached on the left and right. (b) Local correspondence between the three vertex types (empty, "/" mirror, and "\\" mirror) and the three pairings of the four incident edges. (c) The corresponding tube/trajectory picture obtained by translating each local mirror choice into a pairing of the four incident edges; the disjoint trajectories induce a perfect matching of the external edges.
• Figure SM.2: (3D Hierarchical model) Mean conductance $\mu_n$ in $d=3$ with initial crossing $A_0=2$. Left: $\log_2\mu_n$ versus iteration $n=\log_2 L_n$. The linear growth is consistent with the normal-transport scaling $\mu_n\propto L_n=2^n$. Right: As $n$ increases, the ratio $\mu_n/2^n$ converges to a constant (the conductivity).