Symmetric exclusion as a model of non-elliptic dynamical random conductances

TL;DR

This paper models a non-elliptic dynamical random conductance system using a symmetric exclusion process and proves laws of large numbers and central limit theorems for the associated random walk.

Contribution

It introduces a novel interpretation of symmetric exclusion as a non-elliptic conductance model and establishes fundamental probabilistic results for the driven random walk.

Findings

The random walk exhibits diffusive behavior in all dimensions.

Law of large numbers holds for the model.

Central limit theorem established for the walk.

Abstract

We consider a finite range symmetric exclusion process on the integer lattice in any dimension. We interpret it as a non-elliptic time-dependent random conductance model by setting conductances equal to one over the edges with end points occupied by particles of the exclusion process and to zero elsewhere. We prove a law of large number and a central limit theorem for the random walk driven by such a dynamical field of conductances by using the Kipnis-Varhadan martingale approximation. Unlike the tagged particle in the exclusion process, which is in some sense similar to this model, this random walk is diffusive even in the one-dimensional nearest-neighbor case.