A quantitative central limit theorem for the effective conductance on the discrete torus
Résumé
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 AL of the network is a random variable, depending on L, and the main result is a quantitative central limit theorem for this quantity as L→∞. In terms of scalings we prove that this nonlinear nonlocal function AL essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of AL.