We prove an elliptic Harnack inequality at large scale on the lattice for nonnegative solutions of a difference equation with balanced i.i.d. coefficients which are not necessarily elliptic. We also identify the optimal constant in the Harnack inequality. Our proof relies on a quantitative homogenization result of the corresponding invariance principle to Brownian motion and on percolation estimates. As a corollary of our main theorem, we derive an almost optimal Hölder estimate.
We thank David Criens for sharing with us his thesis  which helped us fix some errors in earlier versions.
We thank two anonymous referees for their thoughtful comments and suggestions which improved the presentation of our work.
"An elliptic Harnack inequality for difference equations with random balanced coefficients." Ann. Probab. 50 (3) 835 - 873, May 2022. https://doi.org/10.1214/21-AOP1544