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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 $E_\\Gamma(H)>E_\\emptyset(H)$. 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