Multiscale analysis of the conductivity in the Lorentz mirrors model

We consider the mirrors model in $d$ dimensions on an infinite slab and with unit density. This is a deterministic dynamics in a random environment. We argue that the crossing probability of the slab goes like $κ/(κ+N)$ where $N$ is the width of the slab. We are able to compute $κ$ perturbatively by using a multiscale approach. The only small parameter involved in the expansion is the inverse of the size of the system. This approach rests on an inductive process and a closure assumption adapted to the mirrors model. For $d=3$, we propose the recursive relation for the conductivity $κ_n$ at scale $n$ : $κ_{n+1}=κ_n(1+\frac{κ_n}{2^{n}}α)$, up to $o(1/2^n)$ terms and with $α\simeq 0.0374$. This sequence has a finite limit.

💡 Research Summary

The paper investigates normal conductivity in the three‑dimensional mirrors model, a deterministic lattice Lorentz gas where each lattice site hosts a randomly oriented mirror that reflects a particle according to a reversible, no‑U‑turn rule. Despite the deterministic, non‑chaotic nature of the dynamics and the presence of infinitely many finite trapping loops, numerical simulations suggest that the model exhibits a finite, size‑independent conductivity.

The authors focus on the crossing probability C_N of a slab of width N (with transverse size taken large enough to neglect edge effects). The conductivity at scale N is defined as κ_N = N C_N / (1 − C_N). The main goal is to prove that κ_N converges to a finite constant κ as N → ∞ and to compute κ analytically.

A key methodological step is a multiscale decomposition: a slab of width 2^{n}+1 is split into two independent slabs of width 2^{n}. For a non‑backtracking random walk (the Markovian benchmark) independence yields an exact recursion for the crossing probability, leading to a constant conductivity κ̂ = d/(d − 1). In the mirrors model, however, trajectories that revisit the interface between the two halves are constrained by the deterministic compatibility of the mirrors, generating correlations that must be quantified.

To capture these correlations the authors introduce η_n(l), the deviation from independence for trajectories that cross the interface l − 1 times at scale 2^{n}, and define Δ_n as a weighted sum of differences η_n(l+1) − η_n(l). The crossing probability at the next scale satisfies

c_{n+1} = c_n^2