Spectral properties of the random conductance model

Charge and exciton transport in disordered media plays an essential role in modern technologies. Classical examples are amorphous and organic semiconductors where the disorder can give rise to localized electron states. These localized electrons effectively behave like discrete particles hopping between discrete sites in an inhomogeneous environment. A popular model for such a hopping process is the random walk among random conductances where the jump rate between any two sites is the same in both directions. The long-time behavior of such a random walk, when it is killed at the boundary of a large box, is intimately linked to the first eigenvectors and eigenvalues of its generator with zero Dirichlet boundary condition. This follows directly from the spectral decomposition of the associated heat equation. In this thesis, we study these first eigenvectors and eigenvalues when the underlying lattice is Z^d. In addition to the spectrum, we study the homogenization properties of the corresponding Poisson equation. Regarding the spectrum, we find that in dimensions d≥2 and for independent and identically distributed positive conductances, there is a sharp transition between a completely localized and a completely homogenized regime. This transition hinges on the exponent q=sup{r≥0: E[ω^(-r)]<∞} where E[ω^(-r)] is the inverse rth moment of the conductance ω. If q<1/4, then we show that for almost every environment the first Dirichlet eigenvectors asymptotically concentrate in a single site and the corresponding eigenvalues scale subdiffusively. We further prove weak convergence of the rescaled eigenvalues to non-degenerate random variables. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, path arguments and the Bauer-Fike theorem. On the other hand, if q>1/4, then we show that the properly rescaled first eigenvectors and eigenvalues converge almost-surely to the first eigenvectors and eigenvalues of a homogenized operator. For this result it is sufficient to assume stationary and ergodic conductances, which are positive between nearest neighbors. Apart from that, we further allow unbounded-range connections. In this general case we need a stronger integrability condition on the lower tail of the conductances, which coincides with a well-known necessary condition for the validity of a local central limit theorem for the random walk among random conductances. The main result on the way to spectral homogenization is the homogenization of the corresponding Poisson equation. More precisely, we prove two-scale convergence of the solutions and their gradients. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.

Author dc:creator

• Flegel, Franziska

Advisor dc:contributor.advisor

• König, Wolfgang

Licence dc:rights.uri

• https://creativecommons.org/licenses/by/4.0/

Language dc:language.iso

en

Identifier URI

http://dx.doi.org/10.14279/depositonce-7572

OAI identifier oai:identifier

oai:depositonce.tu-berlin.de:11303/8426