  We prove a quantitative estimate on the homogenization length scale in terms
of the ellipticity ratio $\Lambda/\lambda$ of the coefficient field. This upper
bound is applicable to high contrast elliptic equations demonstrating
near-critical behavior. Specifically, we show that, given a suitable decay of
correlation, the length scale at which homogenization is observed is at most
$\exp(C \log^3(1+\Lambda/\lambda))$. The proof introduces the new concept of
coarse-grained ellipticity, which measures the effective ellipticity ratio of
the equation -- and thus the strength of the disorder -- after integrating out
smaller scales. By a direct analytic argument, we obtain an approximate
differential inequality for this coarse-grained ellipticity as a function of
the length scale. This approach can be interpreted as a rigorous
renormalization group argument, and provides a quantitative framework for
homogenization that can be iteratively applied across an arbitrary number of
length scales.
