WIAS Preprint No. 1739, (2012)

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



Authors

  • Biskup, Marek
  • Salvi, Michele
  • Wolff, Tilman

2010 Mathematics Subject Classification

  • 37H10 60K37 60J60

Keywords

  • Random conductance model, second order discrete elliptic equations with random coefficients, homogenization theory

Abstract

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.

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