  We study the asymptotic behavior of the principal eigenvector and eigenvalue
of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq
2$) with zero Dirichlet condition. We assume that the conductances $w$ are
positive i.i.d. random variables, which fulfill certain regularity assumptions
near zero. If $\gamma=\sup \{ q\geq 0\colon \mathbb{E} [w^{-q}]<\infty \}<1/4$,
then we show that for almost every environment the principal Dirichlet
eigenvector asymptotically concentrates in a single site and the corresponding
eigenvalue scales subdiffusively. The threshold $\gamma_{\rm c} = 1/4$ is
sharp. Indeed, other recent results imply that for $\gamma>1/4$ the top of the
Dirichlet spectrum homogenizes. 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, and path arguments.
