Disorder, entropy and harmonic functions

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.

Published at http://dx.doi.org/10.1214/14-AOP934 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Random walks on supercritical percolation clusters
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• Invariance principle for the random conductance model with unbounded conductances
• Anomalous heat-kernel decay for random walk among bounded random conductances
• The quenched invariance principle for random walks in random environments admitting a bounded cycle representation
• On one property of distances in the infinite random quadrangulation
• Stochastic homogenization of horospheric tree products

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• Elliptic regularity and quantitative homogenization on percolation clusters
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• Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla \varphi$ Interface Model
• Annealed estimates on the Green function
• Efficient First-Order Contextual Bandits: Prediction, Allocation, and Triangular Discrimination
• Optimal corrector estimates on percolation clusters
• Minimal growth harmonic functions on lamplighter groups
• A functional analysis proof of Gromov's polynomial growth theorem
• Supercritical loop percolation on ${\mathbb{Z}}^d$ for $d\ge 3$
• First-order expansion of homogenized coefficients under Bernoulli perturbations
• Diffusive estimates for random walks on stationary random graphs of polynomial growth
• A Liouville Property for Isotropic Diffusions in Random Environment
• A Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space with Homogeneous Neumann Boundary Data
• Homogenization via sprinkling
• Homogenization theory of random walks among deterministic conductances
• Ancient caloric functions on graphs with unbounded Laplacians