We study the statistics of Dirichlet eigenvalues of the random Schr "odinger operator @math , with @math the discrete Laplacian on @math and @math uniformly bounded independent random variables, on sets of the form @math for @math bounded, open and with a smooth boundary. If @math holds for some bounded and continuous @math , we show that, as @math , the @math -th eigenvalue converges to the @math -th Dirichlet eigenvalue of the homogenized operator @math , where @math is the continuum Dirichlet Laplacian on @math . Assuming further that @math for some positive and continuous @math , we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of @math against the product of squares of the corresponding eigenfunctions of @math .
To the best of our knowledge, the fluctuations of @math for independent and identically distributed conductances have not been studied yet. Notwithstanding, the analysis of a related effective conductance problem (Nolen @cite_8 , Rossignol @cite_18 , Biskup, Salvi and Wolff @cite_16 ) indicates that @math should be asymptotically normal with mean zero and variance that is a biquadratic expression in @math integrated over @math , where @math denotes a @math -th eigenfunction of the operator @math . A significant additional technical challenge of this problem is the need to employ the corrector method (this is what gives rise to the homogenized'' coefficients @math above).
@cite_18 @cite_16 @cite_8
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We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length n in Zd, when n goes to infinity. 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. We consider solutions of an elliptic partial differential equation in ( R ^d ) with a stationary, random conductivity coefficient that is also periodic with period (L ). Boundary conditions on a square domain of width (L ) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit (L ), this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size (L ) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincare inequality developed recently by Chatterjee.