A remark on a surprising result by Bourgain in homogenization

In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on Zd 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.