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Moment bounds for the corrector in stochastic homogenization of a percolation model. (English) Zbl 1326.39015
Summary: We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $$\mathbb{Z}^d$$, $$d>2$$. The model is obtained from the classical $$\{0,1\}$$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result of A. Gloria and F. Otto [Ann. Probab. 39, No. 3, 779–856 (2011; Zbl 1215.35025)], where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows subinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green’s function.

##### MSC:
 39A70 Difference operators 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K37 Processes in random environments 82C43 Time-dependent percolation in statistical mechanics 60F17 Functional limit theorems; invariance principles 60H25 Random operators and equations (aspects of stochastic analysis)
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