A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts

• Published in: Communications in mathematical physics Vol. 328; no. 2; pp. 701 - 731
• Main Authors: Biskup, M., Salvi, M., Wolff, T.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2014
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Meyers Estimate; Effective Conductance; Central Limit Theorem; Salvi; Green Function
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-014-2024-y

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.