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Homogenization theory of random walks in degenerate random environment

Sebastian Andres

TL;DR

This review surveys homogenization results for the random conductance model (RCM) in both i.i.d. and ergodic degenerate environments, focusing on quenched functional central limit theorems, quenched local limit theorems, and Gaussian heat kernel estimates under moment conditions. It explains the corrector method and the environment seen from the particle, establishing sublinearity of the corrector and proving quenched invariance principles for random walks with both variable- and constant-speed dynamics, with covariance operators $\Sigma_X^2$ and $\Sigma_Y^2$. The work extends these results to time-dynamic conductances and to models with long-range jumps, outlining the necessary moment and regularity conditions and highlighting the differences between annealed and quenched frameworks. Overall, the article connects probabilistic homogenization of random walks in degenerate media to broader stochastic homogenization theory, clarifying when diffusion-like behaviour emerges and when trapping or non-Gaussian effects persist.

Abstract

Recent progress on the understanding of the Random Conductance Model is reviewed. A particular emphasis is on homogenization results such as functional central limit theorems, local limit theorems and heat kernel estimates for almost every realization of the environment, established for random walks among stationary ergodic conductances that are possibly unbounded but satisfy certain moment conditions.

Homogenization theory of random walks in degenerate random environment

TL;DR

This review surveys homogenization results for the random conductance model (RCM) in both i.i.d. and ergodic degenerate environments, focusing on quenched functional central limit theorems, quenched local limit theorems, and Gaussian heat kernel estimates under moment conditions. It explains the corrector method and the environment seen from the particle, establishing sublinearity of the corrector and proving quenched invariance principles for random walks with both variable- and constant-speed dynamics, with covariance operators and . The work extends these results to time-dynamic conductances and to models with long-range jumps, outlining the necessary moment and regularity conditions and highlighting the differences between annealed and quenched frameworks. Overall, the article connects probabilistic homogenization of random walks in degenerate media to broader stochastic homogenization theory, clarifying when diffusion-like behaviour emerges and when trapping or non-Gaussian effects persist.

Abstract

Recent progress on the understanding of the Random Conductance Model is reviewed. A particular emphasis is on homogenization results such as functional central limit theorems, local limit theorems and heat kernel estimates for almost every realization of the environment, established for random walks among stationary ergodic conductances that are possibly unbounded but satisfy certain moment conditions.

Paper Structure

This paper contains 14 sections, 16 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The model
  4. Questions and aims
  5. The i.i.d. case
  6. Quenched invariance principle and other scaling limits
  7. Heat kernel behaviour
  8. Random conductance models in ergodic environments
  9. Quenched invariance principle
  10. A detailed overview of the proof
  11. Local limit theorems
  12. Heat kernel estimates
  13. Random conductance models with time-dynamic conductances
  14. Random conductance models with long-range jumps

Key Result

Theorem 2.1

Let $d \ge 2$ and suppose that $(\omega(e), e\in E_d)$ are i.i.d., $\omega(e)\geq 0$$\mathop{\mathrm{\mathbb{P}}}\nolimits$-a.s. and $\mathop{\mathrm{\mathbb{P}}}\nolimits( \omega(e) >0 ) > p_c$.

Figures (2)

  • Figure 1: Simulations provided by courtesy of Martin Slowik. (a) $\omega(e)\in[0,1]$ i.i.d. with $\mathbb{P}[\omega(e) \leq t] \sim t$; (b) $\omega(x,y)=\lambda(x) \vee \lambda(y)$, $\lambda(x)\in [0,1]$ i.i.d. with $\mathbb{P}[\lambda(x) \leq t] \sim t$; (c) $\omega(x,y)=\lambda(x) \wedge \lambda(y)$, $\lambda(x)\in[0,1]$ i.i.d. with $\mathbb{P}[\lambda(x) \leq t] \sim t$;
  • Figure :

Theorems & Definitions (34)

  • Theorem 2.1: ABDH13
  • Definition 3.2
  • Theorem 3.3: QFCLT
  • proof
  • Remark 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • proof
  • ...and 24 more