# A remark on a surprising result by Bourgain in homogenization

## TL;DR

This paper discusses Bourgain's surprising detailed description of solutions to discrete elliptic equations with random coefficients, revealing insights that surpass typical expectations in stochastic homogenization and connecting to fluctuation theory.

## Contribution

It reformulates Bourgain's result to emphasize its significance in homogenization theory and proposes related conjectures for future research.

## Key findings

- Bourgain's result provides a detailed expectation description beyond standard accuracy.
- The reformulation highlights the relevance to fluctuation theory.
- Several conjectures are proposed to extend the understanding of homogenization.

## Abstract

In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on $\mathbb Z^d$ with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyond the expected accuracy suggested by recent results in quantitative stochastic homogenization. In this short article we reformulate Bourgain's result in a form that highlights its interest to the state-of-the-art in homogenization (and especially the theory of fluctuations), and we state several related conjectures.