  We consider random walks among random conductances on $\mathbb{Z}^2$ and
establish precise asymptotics for the associated potential kernel and the
Green's function of the walk killed upon exiting balls. The result is proven
for random walks on i.i.d. supercritical percolation clusters among ergodic
degenerate conductances satisfying a moment condition. We also provide a
similar result for the time-dynamic random conductance model. As an application
we present a scaling limit for the variances in the Ginzburg-Landau $\nabla
\phi$-interface model.
