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A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus

par Gloria, Antoine ;Nolen, James
Référence Communications on pure and applied mathematics, 69, 12, page (2304-2348)
Publication Publié, 2016-12

Article révisé par les pairs

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Résumé :

We study a random conductance problem on a d-dimensional discrete torus of size L > 0. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance AL of the network is a random variable, depending on L, that converges almost surely to the homogenized conductance Ahom. Our main result is a quantitative central limit theorem for this quantity as L → ∞. In particular, we prove there exists some σ > 0 such that dK (Ld/2 A – Ahom/ σ, g) ≲ L–d/2 logd L,where dK is the Kolmogorov distance and gis a standard normal variable. The main achievement of this contribution is the precise asymptotic description of the variance of AL.© 2015 Wiley Periodicals, Inc.