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Homogenization theory of random walks among deterministic conductances

Marek Biskup

TL;DR

The paper develops a de-randomized homogenization framework to establish an Invariance Principle for random walks on Z^d in deterministic reversible edge conductances. By constructing a spatial averaging mechanism that yields a stochastic surrogate $\mathbb{P}_\omega$ and leveraging heat-kernel bounds, the authors prove that a large, translation-invariant class of conductance configurations $\\Omega^*$ exhibits diffusive scaling to Brownian motion, provided mild moment conditions hold (on $p,q>1$ with $1/p+1/q<2/d$ for $d\ge2$). The main contribution is identifying $\\Omega^*$—stable under translations and zero-density perturbations—and proving an IIP for $\\omega\\in\\Omega^*\cap\\Omega_{p,q}$, with the limit covariance given by a variational formula tied to a homogenized Dirichlet energy. This framework unifies and extends stochastic homogenization results to fully deterministic environments and offers a blueprint for de-randomized treatments of other disordered systems, including zero-density perturbations and subdomain or time-dependent variants. The work combines martingale approximations, a Conversion Lemma based on heat-kernel bounds, and Dirichlet-energy homogenization to achieve the CLT, providing both qualitative criteria and a pathway toward quantitative extensions and broader models.

Abstract

We study the asymptotic distribution of random walks on $\mathbb Z^d$ ($d\ge1$) in deterministic reversible environments defined by an assignment of a positive conductance to each edge of $\mathbb Z^d$. We identify a deterministic set of conductance configurations for which the walk obeys an Invariance Principle; i.e., converges in law to a non-degenerate Brownian motion under diffusive scaling of space and time. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs rely on martingale approximations whose main step is the conversion of averages in time and physical space under the deterministic environment to those in a suitable stochastic counterpart. Our study sets up a framework for "de-randomized homogenization" of other motions in disordered media.

Homogenization theory of random walks among deterministic conductances

TL;DR

The paper develops a de-randomized homogenization framework to establish an Invariance Principle for random walks on Z^d in deterministic reversible edge conductances. By constructing a spatial averaging mechanism that yields a stochastic surrogate and leveraging heat-kernel bounds, the authors prove that a large, translation-invariant class of conductance configurations exhibits diffusive scaling to Brownian motion, provided mild moment conditions hold (on with for ). The main contribution is identifying —stable under translations and zero-density perturbations—and proving an IIP for , with the limit covariance given by a variational formula tied to a homogenized Dirichlet energy. This framework unifies and extends stochastic homogenization results to fully deterministic environments and offers a blueprint for de-randomized treatments of other disordered systems, including zero-density perturbations and subdomain or time-dependent variants. The work combines martingale approximations, a Conversion Lemma based on heat-kernel bounds, and Dirichlet-energy homogenization to achieve the CLT, providing both qualitative criteria and a pathway toward quantitative extensions and broader models.

Abstract

We study the asymptotic distribution of random walks on () in deterministic reversible environments defined by an assignment of a positive conductance to each edge of . We identify a deterministic set of conductance configurations for which the walk obeys an Invariance Principle; i.e., converges in law to a non-degenerate Brownian motion under diffusive scaling of space and time. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs rely on martingale approximations whose main step is the conversion of averages in time and physical space under the deterministic environment to those in a suitable stochastic counterpart. Our study sets up a framework for "de-randomized homogenization" of other motions in disordered media.
Paper Structure (30 sections, 26 theorems, 250 equations)

This paper contains 30 sections, 26 theorems, 250 equations.

Table of Contents

  1. Introduction and results
  2. Main question and outlook
  3. Framework for de-randomized homogenization
  4. Main result
  5. Connections and proof ideas
  6. Earlier work
  7. Key obstacles
  8. Proof outline
  9. Organization
  10. Preliminaries
  11. Stochastic framework
  12. Heat-kernel bounds and the Conversion lemma
  13. Exit time estimate and tightness
  14. Ergodicity of Markov chain on environments
  15. Averaging for the random walk
  16. ...and 15 more sections

Key Result

Proposition 1.3

Write $C_{\text{\rm b}}(\Omega)$ for the set of bounded $f\in C(\Omega)$ and let $\mathcal{F}$ be the product $\sigma$-algebra on $\Omega$. Define the set of "tight" conductance configurations by Then for each averaging $\omega\in\Omega'$, the limit E:1.1 exists for all $f\in C_{\text{\rm b}}(\Omega)$ and there exists a unique probability measure $\mathbb P_\omega$ on $(\Omega,\mathcal{F})$ such

Theorems & Definitions (37)

  • Definition 1.1: Individual Invariance Principle
  • Definition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4: Conversion lemma
  • ...and 27 more