## The Annals of Probability

### 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.

#### Article information

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

Dates
Revised: March 2014
First available in Project Euclid: 9 September 2015

https://projecteuclid.org/euclid.aop/1441792287

Digital Object Identifier
doi:10.1214/14-AOP934

Mathematical Reviews number (MathSciNet)
MR3395463

Zentralblatt MATH identifier
1337.60248

#### Citation

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. https://projecteuclid.org/euclid.aop/1441792287

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