1210.2371 (M. Biskup et al.)

A central limit theorem for the effective conductance: I. Linear
boundary data and small ellipticity contrasts    [PDF]

M. Biskup, M. Salvi, T. Wolff

We consider resistor networks on $\Z^d$ where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge 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 arbitrary ellipticity contrasts are to be addressed in a subsequent paper.

View original: http://arxiv.org/abs/1210.2371