Homogenization theory of random walks among deterministic conductances

Abstract

We study the asymptotic distribution of random walks on ${\mathbb{Z}}^d$ ($d\ge 1$) 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.

arXiv comment53 pages, version to appear in PTRF

References in corpus (20)

• Quenched invariance principle for simple random walk on percolation clusters
• Recent progress on the Random Conductance Model
• Quantitative stochastic homogenization and large-scale regularity
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• Quenched invariance principles for random walks with random conductances
• Disorder, entropy and harmonic functions
• Elliptic regularity and quantitative homogenization on percolation clusters
• Invariance principle for the random conductance model in a degenerate ergodic environment
• Quenched invariance principle for random walks with time-dependent ergodic degenerate weights
• Quenched invariance principle for random walks in balanced random environment
• Scaling limit for a class of gradient fields with nonconvex potentials
• Limit theory for random walks in degenerate time-dependent random environments
• Heat kernel estimates for random walks with degenerate weights
• Invariance Principle for the Random Conductance Model with dynamic bounded Conductances
• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights
• Gaussian estimates for spatially inhomogeneous random walks on ${\mathbf{Z}}^d$
• Quantitative version of the Kipnis-Varadhan theorem and Monte Carlo approximation of homogenized coefficients
• An invariance principle for one-dimensional random walks among dynamical random conductances
• Trapping in the random conductance model
• An invariance principle for one-dimensional random walks in degenerate dynamical random environments