Abstract
We present a new, large-deviation approach to investigate the critical properties of the Anderson model on the Bethe lattice close to the localization transition in the thermodynamic limit. Our method allows us to study accurately the distribution of the local density of states (LDoS) down to very small probability tails as small as ${10}^{\ensuremath{-}50}$, which are completely out of reach for standard numerical techniques. We perform a thorough analysis of the functional form of the tails of the probability distributions of the LDoS, which turn out to be very well described by the functional form predicted by the supersymmetric formalism and yields a direct and transparent estimation of the correlation volume very close to the Anderson transition. Such a correlation volume is found to diverge exponentially when the localization is approached from the delocalized regime, in a singular way that is compatible with the analytic predictions of the supersymmetric treatment.
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