*almost sure* bounds on the increments of a certain martingale, so that the fluctuations of this martingale can be strongly controlled by Azuma’s inequality. An assumption which allows for long-range correlations would not have this property. On a more philosophical level, our strategy is quantitative and therefore requires a quantitative ergodic assumption– but it turns out that the finite range of dependence condition happens to be the only assumption under which it is known how to prove quantitative results. Indeed, it is completely open to obtain quantitative results for the homogenization of even first-order Hamilton-Jacobi equations under any assumption that allows for long-range correlations. Even obtaining a convergence rate for the shape theorem in first-passage percolation is well-known to be open when the edges do not satisfy a finite range of dependence!

In the case of the forced mean curvature equation  with periodic coefficients, a lot of attention has been given to the existence of *plane-like* solutions, which began with the work of [@CD]. In our setting, these are solutions, for a given $\mu>0$ of the stationary problem $$-\operatorname{tr}\left( A(Du,x) D^2u\right) + H(Du,x) = \mu \quad \mbox{in}  \ {\mathbb{R}^d}$$ whose graphs stay within a bounded distance from an affine function. 