DISORDER, ENTROPY AND HARMONIC FUNCTIONS

• 成果类型: Article
• 署名作者: Benjamini, Itai; Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel
• 署名单位: Weizmann Institute of Science; University of Geneva; Ben-Gurion University of the Negev
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP934
• 发表日期: 2015
• 页码: 2332-2373
• 关键词: quenched invariance-principle; parabolic harnack inequality; reversible markov-processes; incipient infinite cluster; random-walks; Stochastic Homogenization; volume growth; limit-theorem; local limit; percolation

摘要

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