Disorder, entropy and harmonic functions

Itai Benjamini, Hugo Duminil-Copin, Gady Kozma, Ariel Yadin

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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.

Original languageEnglish
Pages (from-to)2332-2373
Number of pages42
JournalAnnals of Probability
Issue number5
StatePublished - 1 Jan 2015


  • Anomalous diffusion
  • Avez
  • Corrector
  • Entropy
  • Harmonic functions
  • Iic
  • Kaimanovich-vershik
  • Percolation
  • Planar map
  • Random walk in random environment
  • Stationary graphs
  • Uipq

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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