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Ostilli","submitted_at":"2015-12-27T20:14:51Z","abstract_excerpt":"Given a $M\\times M$ Hermitian matrix $\\mathcal{H}$ with possibly degenerate eigenvalues $\\mathcal{E}_1 < \\mathcal{E}_2 < \\mathcal{E}_3< \\dots$, we provide, in the limit $M\\to\\infty$, a lower bound for the gap $\\mu_2 = \\mathcal{E}_2 - \\mathcal{E}_1$ assuming that (i) the eigenvector (eigenvectors) associated to $\\mathcal{E}_1$ is ergodic (are all ergodic) and (ii) the off-diagonal terms of $\\mathcal{H}$ vanish for $M\\to\\infty$ more slowly than $M^{-2}$. Under these hypotheses, we find $\\varliminf_{M\\to\\infty} \\mu_2 \\geq \\varlimsup_{M\\to\\infty} \\min_{n} \\mathcal{H}_{n,n}$. 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