A quantitative central limit theorem for the effective conductance on the discrete torus

Antoine Gloria 1, 2 James Nolen 3
2 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : 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, and the main result is a quantitative central limit theorem for this quantity as L→∞. In terms of scalings we prove that this nonlinear nonlocal function AL essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of AL.
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https://hal.archives-ouvertes.fr/hal-01093352
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Submitted on : Wednesday, December 10, 2014 - 3:29:01 PM
Last modification on : Tuesday, August 20, 2019 - 5:12:02 PM

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Antoine Gloria, James Nolen. A quantitative central limit theorem for the effective conductance on the discrete torus. Communications on Pure and Applied Mathematics, Wiley, 2016, 69 (12), pp.2304--2348. ⟨10.1002/cpa.21614⟩. ⟨hal-01093352⟩

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