Disorder, entropy and harmonic functions

We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on Zd. 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.