Spectral homogenization vs. localization in the random conductance model

We study the asymptotic behavior of the top eigenvectors and eigenvalues of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq 2$) with zero Dirichlet conditions. Let the conductances $w$ be positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. Then we show that the spectrum of the Laplacian displays a sharp transition between a completely localized and a completely homogenized phase. A simple moment condition distinguishes between the two phases.

In the homogenized phase we can even generalize our results to stationary and ergodic conductances with additional jumps of arbitrary length. Here, our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincar\'e inequalities, Moser iteration and two-scale convergence.

The investigation of the homogenized phase is joint work with M. Slowik and M. Heida.