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Scaling limit of the step-reinforced and stochastic Levy--Lorentz model on weakly entangled integer lattice

Jiaming Chen

TL;DR

The paper analyzes a one-dimensional Lévy--Lorentz gas in a weakly entangled random medium where scatterer spacings can be correlated and the tracer may exhibit memory through step reinforcement. It proves quenched central limit theorems and functional CLTs for both discrete-time and continuous-time dynamics, and extends these results to a step-reinforced model with memory parameter $p$, revealing diffusive scaling with variance factors $(3-4p)^{-1}$ and identifying a critical regime at $p=3/4$. The authors develop a novel auxiliary-renewal framework to separate entanglement effects from medium randomness and employ martingale techniques to handle memory, culminating in an extended Skorokhod space topology to accommodate two-sided time indices. Collectively, the results advance understanding of scaling limits for random walks in correlated random media with memory, bridging quenched and functional perspectives and enabling rigorous Green-function-type limit theorems for reinforced transport models.

Abstract

This paper describes the stochastic Levy--Lorentz gas driven by general long-range reference random walk on correlated and entangled random medium. Further consideration has been laid on the stochastic reinforcement of the underlying random walk, where it now possesses memory. Central limit theorems are obtained in both cases.

Scaling limit of the step-reinforced and stochastic Levy--Lorentz model on weakly entangled integer lattice

TL;DR

The paper analyzes a one-dimensional Lévy--Lorentz gas in a weakly entangled random medium where scatterer spacings can be correlated and the tracer may exhibit memory through step reinforcement. It proves quenched central limit theorems and functional CLTs for both discrete-time and continuous-time dynamics, and extends these results to a step-reinforced model with memory parameter , revealing diffusive scaling with variance factors and identifying a critical regime at . The authors develop a novel auxiliary-renewal framework to separate entanglement effects from medium randomness and employ martingale techniques to handle memory, culminating in an extended Skorokhod space topology to accommodate two-sided time indices. Collectively, the results advance understanding of scaling limits for random walks in correlated random media with memory, bridging quenched and functional perspectives and enabling rigorous Green-function-type limit theorems for reinforced transport models.

Abstract

This paper describes the stochastic Levy--Lorentz gas driven by general long-range reference random walk on correlated and entangled random medium. Further consideration has been laid on the stochastic reinforcement of the underlying random walk, where it now possesses memory. Central limit theorems are obtained in both cases.
Paper Structure (11 sections, 14 theorems, 28 equations)

This paper contains 11 sections, 14 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Step-reinforced and stochastic Lévy--Lorentz gas
  3. Stochastic Lévy--Lorentz gas.
  4. Step-reinforced Lévy--Lorentz gas.
  5. Regular central limit theorem of Lévy--Lorentz gas
  6. Discrete-time scenario.
  7. Continuous-time scenario.
  8. Step-reinforced scenario.
  9. Functional central limit theorem and Skorokhod space
  10. Extension of Skorokhod space topology.
  11. Proof of the main results.

Key Result

Theorem 2.1

We have the following convergences in distribution, with the positive deterministic $\ell\coloneqq\lim_{n\to\infty}n^{-1}\omega_n$, $\mathbb{P}$-a.s., as well as the limiting identity for the continuous-time stochastic Lévy--Lorentz gas, where in general $\mathcal{N}(0,\sigma^2)$ denotes the centered Gaussian variable with variance $\sigma^2>0$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 6 more