Multiscale analysis of the conductivity in the Lorentz mirrors model

Abstract

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 $\kappa /(\kappa +N)$ where $N$ is the width of the slab. We are able to compute $\kappa$ 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 ${\kappa }_n$ at scale $n$ : ${\kappa }_{n+1}={\kappa }_n(1+\frac{{\kappa }_n}{2^n}\alpha )$, up to $o(1/2^n)$ terms and with $\alpha \simeq 0.0374$. This sequence has a finite limit.

Detailed explanations for the computations added