A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts

TL;DR

This paper proves a central limit theorem for the effective conductance in resistor networks with i.i.d. conductances and small ellipticity contrast, showing fluctuations around the deterministic limit follow a normal distribution.

Contribution

It establishes a CLT for effective conductance with linear boundary conditions and small ellipticity contrast, extending previous results to a probabilistic fluctuation analysis.

Findings

Effective conductance scaled by volume converges to a deterministic limit.

Under small ellipticity contrast, fluctuations are normally distributed.

The proof uses the corrector method, Martingale CLT, and Meyers estimate.

Abstract

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.