Homogenization theory of random walks among deterministic conductances

Abstract: We study asymptotic laws of random walks on $\mathbb Z^d$ ($d\ge1$) in deterministic reversible environments defined by an assignment of a positive and finite conductance to each edge of $\mathbb Z^d$. The walk jumps across an edge with probability proportional to its conductance. We identify a deterministic set of conductance configurations for which an Invariance Principle (i.e., convergence in law to Brownian motion under diffusive scaling of space and time) provably holds. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs are based on martingale approximations whose control relies on 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 proofs of "deterministic homogenization" in other motions in disordered media.