The Annals of Probability

Disorder, entropy and harmonic functions

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

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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.

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Ann. Probab., Volume 43, Number 5 (2015), 2332-2373.

Received: May 2013
Revised: March 2014
First available in Project Euclid: 9 September 2015

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 31A05: Harmonic, subharmonic, superharmonic functions 82B43: Percolation [See also 60K35] 37A35: Entropy and other invariants, isomorphism, classification 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 20P05: Probabilistic methods in group theory [See also 60Bxx]

Harmonic functions percolation random walk in random environment stationary graphs entropy Avez Kaimanovich–Vershik corrector IIC UIPQ planar map anomalous diffusion


Benjamini, Itai; Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel. Disorder, entropy and harmonic functions. Ann. Probab. 43 (2015), no. 5, 2332--2373. doi:10.1214/14-AOP934.

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