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

Bálint Tóth

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. Tóth, 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.

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

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. Tóth, 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.
Paper Structure (8 sections, 9 theorems, 84 equations)

This paper contains 8 sections, 9 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Construction and Quenched Coupling
  3. Prologue to the coupling
  4. Synchronous Lorentz trajectories
  5. Quenched coupling with independent Markovian flight processes
  6. Control of tightness of the coupling
  7. Proof of Theorem \ref{['thm:sqip']}
  8. Triangular array of processes

Key Result

Theorem 1

(lutsko-toth-2020 Theorem 1) Let $d=3$, $\varepsilon\to0$, $r_\varepsilon=\varepsilon^{d/(d-1)}$ and $T_\varepsilon\to\infty$ be such that Let $t\mapsto X_\varepsilon(t)$ be the sequence of Lorentz trajectories among the spherical scatterers of radius $r_\varepsilon$ centred at the points $\varpi_\varepsilon$ cf. varpieps, and with deterministic initial velocities $v_\varepsilon\in S^{d-1}$. For

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 4 more