A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
For researchers in stochastic homogenization and statistical physics, this provides a rigorous CLT for a key quantity in disordered media, though it is incremental over existing results for the deterministic limit.
The paper proves a central limit theorem for the effective conductance in resistor networks on Z^d with i.i.d. conductances and small ellipticity contrast, showing that the scaled effective conductance converges to a deterministic limit with Gaussian fluctuations.
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.