Semi-Quenched Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

TL;DR

This paper extends the invariance principle for the Lorentz gas from an averaged (annealed) setting to a nearly sure (quenched) setting, showing that the Lorentz trajectories behave diffusively for almost all environments.

Contribution

It introduces a method to upgrade the invariance principle for the Lorentz gas from annealed to quenched settings using synchronized coupling of trajectories.

Findings

Proves quenched invariance principle for Lorentz gas in R^3.

Demonstrates validity for almost all realizations of the environment.

Extends previous annealed results to a stronger quenched setting.

Abstract

By synchronously coupling multiple Lorentz trajectories exploring the same environment consisting of randomly placed scatterers in R^3 we upgrade the annealed invariance principle proved in [C. Lutsko, B. T\'oth, Commun. Math. Phys. 379 589-632 (2020)] to quenched setting (that is, valid for almost all realizations of the environment) along sufficiently fast extractor sequences.