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 .
The argument in @cite_17 is based on a perturbation expansion of the resolvent operator and an explicit representation of the leading-order correction to the eigenfunctions; cf. Remark . In order to control the remainder terms, one then needs that the Green function of the homogenized operator is square integrable, and this requires the restriction to @math . The method employed in the present article is different in it avoids having to deal with local perturbations altogether. Incidentally, as was recently shown by Gu and Mourrat @cite_33 , for the random elliptic operators (see Subsection below for a formulation) the limit laws of the local and global fluctuations to eigenfunctions are in fact not even the same.
@cite_33 @cite_17
1565615755 2962804622
We investigate the global fluctuations of solutions to elliptic equations with random coefficients in the discrete setting. In dimension @math and for i.i.d. coefficients, we show that after a suitable scaling, these fluctuations converge to a Gaussian field that locally resembles a (generalized) Gaussian free field. The paper begins with a heuristic derivation of the result, which can be read independently and was obtained jointly with Scott Armstrong. We consider the perturbation of elliptic pseudodifferential operators @math with more than square integrable Green's functions by random, rapidly varying, sufficiently mixing, potentials of the form @math . We analyze the source and spectral problems associated with such operators and show that the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as @math as explicit Gaussian processes. Such results may be seen as central limit corrections to homogenization (law of large numbers). Similar results are derived for more general elliptic equations with random coefficients in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions.