Multiscale analysis of the conductivity in the Lorentz mirrors model

2025/10/28 by Lefevere, Raphael
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2510.24091

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 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+\fracκ_n2ⁿα), up to o(1/2ⁿ) terms and with α≃ 0.0374. This sequence has a finite limit.

Citations

• Diffusivity of the Lorentz mirror walk in high dimensions
• Self-avoiding walks and polygons -- an overview

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