  We consider difference equations in balanced, i.i.d. environments which are
not necessary elliptic. In this setting we prove a parabolic Harnack inequality
(PHI) for non-negative solutions to the discrete heat equation satisfying a
(rather mild) growth condition, and we identify the optimal Harnack constant
for the PHI. We show by way of an example that a growth condition is necessary
and that our growth condition is sharp. Along the way we also prove a parabolic
oscillation inequality and a (weak) quantitative homogenization result, which
we believe to be of independent interest.
