A quantitative CLT 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 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.

Joint work with J. Nolen