  We study local regularity properties of linear, non-uniformly parabolic
finite-difference operators in divergence form related to the random
conductance model on $\mathbb Z^d$. In particular, we provide an oscillation
decay assuming only certain summability properties of the conductances and
their inverse, thus improving recent results in that direction. As an
application, we provide a local limit theorem for the random walk in a random
degenerate and unbounded environment.
