Quantitative estimates for high-contrast random media

TL;DR

This paper provides optimal bounds for regularity and growth estimates in the homogenization of elliptic equations with random holes, advancing error analysis in stochastic homogenization and applications to double-porosity models.

Contribution

It introduces new bounds for the regularity radius and corrector growth in high-contrast random media with holes, under specific size and separation assumptions.

Findings

Derived optimal bounds for the regularity radius r_*

Established suboptimal growth estimates for the corrector

Enhanced error analysis in stochastic homogenization

Abstract

This paper studies quantitative homogenization of elliptic equations with random, uniformly elliptic coefficients that vanish in a union of random holes. Assuming an upper bound on the size of the holes and a separation condition between them, we derive optimal bounds for the regularity radius $r_*$ and suboptimal growth estimates for the corrector. These results are key ingredients for error analysis in stochastic homogenization and serve as crucial input for recent developments in the double-porosity model, such as those by Bonhomme, Duerinckx, and Gloria (arXiv:2502.02847).