  We study harmonic functions on random environments with particular emphasis
on the case of the infinite cluster of supercritical percolation on
$\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at
most linearly is $(d+1)$-dimensional almost surely. Further, there are no
nonconstant sublinear harmonic functions (thus implying the uniqueness of the
corrector). A main ingredient of the proof is a quantitative, annealed version
of the Avez entropy argument. This also provides bounds on the derivative of
the heat kernel, simplifying and generalizing existing results. The argument
applies to many different environments; even reversibility is not necessary.
