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 theorem...