Comparison of quenched and annealed invariance principles for random conductance model

TL;DR

This paper demonstrates that in certain ergodic conductance environments, the weak invariance principle can hold while the quenched invariance principle fails, highlighting differences in their applicability.

Contribution

It provides the first example of an environment where the annealed invariance principle holds but the quenched invariance principle does not.

Findings

Weak invariance principle holds in some ergodic environments.

Quenched invariance principle can fail in the same environments.

Highlights differences between annealed and quenched invariance principles.

Abstract

We show that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold.