Renormalization group and elliptic homogenization in high contrast

We prove a quantitative estimate for the homogenization length scale in terms of the ellipticity ratio $\Lambda/\lambda$ of the coefficient field. This upper bound applies to high-contrast elliptic equations exhibiting near-critical behavior. Specifically, we show, assuming a suitable decay of correlations, the length scale at which homogenization occurs is at most $\exp(C \log^2(1+\Lambda/\lambda))$. 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 derive an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach may be viewed 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.