  We consider discrete Schr\"odinger operators of the form $H=-\Delta +V$ on
$\ell^2(\Z^d)$, where $\Delta$ is the discrete Laplacian and $V$ is a bounded
potential. Given $\Gamma \subset \Z^d$, the $\Gamma$-trimming of $H$ is the
restriction of $H$ to $\ell^2(\Z^d\setminus\Gamma)$, denoted by $H_\Gamma$. We
investigate the dependence of the ground state energy $E_\Gamma(H)=\inf \sigma
(H_\Gamma)$ on $\Gamma$. We show that for relatively dense proper subsets
$\Gamma$ of $\Z^d$ we always have $E_\Gamma(H)>E_\emptyset(H)$. We use this
lifting of the ground state energy to establish Wegner estimates and
localization at the bottom of the spectrum for $\Gamma$-trimmed Anderson
models, i.e., Anderson models with the random potential supported by the set
$\Gamma$
