Journal
COMMUNICATIONS IN MATHEMATICAL PHYSICS
Volume 328, Issue 2, Pages 701-731Publisher
SPRINGER
DOI: 10.1007/s00220-014-2024-y
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Funding
- NSF [DMS-1106850]
- NSA [H98230-11-1-0171]
- GACR [P201/11/1558]
- ESF
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1106850] Funding Source: National Science Foundation
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Given a resistor network on Z(d) with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is 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.
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