Berry-Esseen Theorem and Quantitative homogenization for the Random Conductance Model with degenerate Conductances

We study the random conductance model on the lattice $\mathbb{Z}^d$, 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 $d\geq 3$ quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed $t^{-\frac 1 5+\varepsilon}$ for $d\geq 4$ and $t^{-\frac{1}{10}+\varepsilon}$ 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 $d\geq 3$ 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.