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Author: Salvi, Michele × End date: 2012 × Start date: 2010 × DDC: 510 × Has files: Yes ×

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    A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Biskup, Marek; Salvi, Michele; Wolff, Tilman
    We consider resistor networks on Zd 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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    Large deviations for the local times of a random walk among random conductances
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) König, Wolfgang; Salvi, Michele; Wolff, Tilman
    We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.