A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus

We study a random conductance problem on a d ‐dimensional discrete torus of size L  > 0. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance A L of the network is a random variable, depending on L , that converges almost surely to the homogenized conductance A hom . Our main result is a quantitative central limit theorem for this quantity as L  → ∞. In particular, we prove there exists some σ > 0 such that [ d K ( L d / 2A L − A homσ , G ) ≲ L − d / 2log d L , where d K is the Kolmogorov distance and G is a standard normal variable. The main achievement of this contribution is the precise asymptotic description of the variance of A L .© 2015 Wiley Periodicals, Inc.