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September 2015 Disorder, entropy and harmonic functions
Itai Benjamini, Hugo Duminil-Copin, Gady Kozma, Ariel Yadin
Ann. Probab. 43(5): 2332-2373 (September 2015). DOI: 10.1214/14-AOP934


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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Itai Benjamini. Hugo Duminil-Copin. Gady Kozma. Ariel Yadin. "Disorder, entropy and harmonic functions." Ann. Probab. 43 (5) 2332 - 2373, September 2015.


Received: 1 May 2013; Revised: 1 March 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1337.60248
MathSciNet: MR3395463
Digital Object Identifier: 10.1214/14-AOP934

Primary: 60K37
Secondary: 20P05, 31A05, 37A35, 60B15, 60J10, 82B43

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • September 2015
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