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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 \(d\ge 3\) quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed \(t^{-\frac{1}{5}+\varepsilon }\) for \(d\ge 4\) 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 \(d\ge 3\) 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.

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