A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus
Antoine Gloria, James H. Nolen · Communications on Pure and Applied Mathematics · 2015
Abstract 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 urn:x-wiley:0010-3640:media:cpa21614:cpa21614-math-0001 where d K is the Kolmogorov distance and 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.