Disorder, entropy and harmonic functions

Citation
Benjamini, Itai et al., Disorder, entropy and harmonic functions, Annals of probability (Online) , 43(5), 2015, pp. 2332-2373
ISSN journal
2168894X
Volume
43
Issue
5
Year of publication
2015
Pages
2332 - 2373
Database
ACNP
SICI code
Abstract
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.