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Transport Regimes in Random Walks in Random Environments

Mihir Metkar, Neha Sah, Zoey Zhou

TL;DR

Transport Regimes in Random Walks in Random Environments surveys how quenched disorder shapes transport in random media across discrete-time, continuous-time, and reversible models. It organizes theory around transport observables, one-dimensional potential representations, and higher-dimensional approaches via environment-from-the-particle, correctors, regeneration, and large deviations. The work highlights mechanisms such as barriers/valleys in 1D, corrector-martingale decompositions in reversible media, and regeneration-based ballisticity in non-reversible settings, while integrating correlation methods, rare-event techniques, and numerical diagnostics. The synthesis provides a practical framework for diagnosing regimes, estimating effective diffusivity, and guiding inference in disordered transport and related materials.

Abstract

Random walks in random environments (RWRE) model transport in quenched disorder, incorporating spatial heterogeneity, trapping, random drift, and random geometry. This paper summarizes discrete and continuous time formulations, identifies principal transport regimes through quantitative observables (velocity, diffusivity, mean-square displacement, first-passage, large deviations, aging), and reviews core methods in one dimension (potential/valley mechanisms) and in higher dimensions (environment-seen-from-the-particle, correctors/homogenization, regeneration and ballisticity criteria).

Transport Regimes in Random Walks in Random Environments

TL;DR

Transport Regimes in Random Walks in Random Environments surveys how quenched disorder shapes transport in random media across discrete-time, continuous-time, and reversible models. It organizes theory around transport observables, one-dimensional potential representations, and higher-dimensional approaches via environment-from-the-particle, correctors, regeneration, and large deviations. The work highlights mechanisms such as barriers/valleys in 1D, corrector-martingale decompositions in reversible media, and regeneration-based ballisticity in non-reversible settings, while integrating correlation methods, rare-event techniques, and numerical diagnostics. The synthesis provides a practical framework for diagnosing regimes, estimating effective diffusivity, and guiding inference in disordered transport and related materials.

Abstract

Random walks in random environments (RWRE) model transport in quenched disorder, incorporating spatial heterogeneity, trapping, random drift, and random geometry. This paper summarizes discrete and continuous time formulations, identifies principal transport regimes through quantitative observables (velocity, diffusivity, mean-square displacement, first-passage, large deviations, aging), and reviews core methods in one dimension (potential/valley mechanisms) and in higher dimensions (environment-seen-from-the-particle, correctors/homogenization, regeneration and ballisticity criteria).
Paper Structure (42 sections, 2 theorems, 36 equations, 1 figure)

This paper contains 42 sections, 2 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quenched vs. annealed laws.
  3. Discrete-time nearest-neighbor RWRE on $\mathbb{Z}^d$.
  4. Continuous-time models and random waiting times.
  5. Reversible random conductance model (RCM).
  6. Environment viewed from the particle.
  7. Transport observables and "statistics".
  8. Fractional kinetics benchmark (physics).
  9. One-dimensional RWRE: potential representation, explicit formulas, and sharp regimes
  10. Model and potential.
  11. Direction of transience (criterion).
  12. Ballistic vs. sub-ballistic transience.
  13. Exact gambler's-ruin formula.
  14. Recurrent strong-disorder regime and logarithmic displacement.
  15. Sub-ballistic transience and stable indices.
  16. ...and 27 more sections

Key Result

Theorem 2.1

If $\mathbb{E}[\xi_0]=0$, the walk is recurrent. If $\mathbb{E}[\xi_0]<0$, then $X_n\to+\infty$$\mathbb{P}_0$-a.s.; if $\mathbb{E}[\xi_0]>0$, then $X_n\to-\infty$$\mathbb{P}_0$-a.s.

Figures (1)

  • Figure 1: Simulated trajectories (blue lines) of RWRE under 3 different regimes.

Theorems & Definitions (2)

  • Theorem 2.1: Solomon
  • Proposition 2.2: Speed formula