Homogenization theory of random walks among deterministic conductances

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.