Berry-Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

We study the random conductance model on the lattice Zd, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension d≥3 quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed t−15+ε for d≥4 and t−110+ε for d=3. Additionally, in the uniformly elliptic case in low dimensions d=2,3 we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for d≥3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.