%0 Journal Article
%T A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
%A Marek Biskup
%A Michele Salvi
%A Tilman Wolff
%J Mathematics
%D 2012
%I arXiv
%R 10.1007/s00220-014-2024-y
%X Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.
%U http://arxiv.org/abs/1210.2371v3