Disorder, entropy and harmonic functions

TL;DR

This paper investigates harmonic functions on random environments, especially on supercritical percolation clusters, establishing their finite-dimensionality, uniqueness of sublinear functions, and providing heat kernel bounds.

Contribution

It proves the finite-dimensionality of harmonic functions with linear growth on percolation clusters and introduces a new entropy-based method applicable to various environments.

Findings

Harmonic functions with linear growth form a (d+1)-dimensional space.

No nonconstant sublinear harmonic functions exist, ensuring corrector uniqueness.

Provides bounds on heat kernel derivatives, generalizing previous results.

Abstract

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.