  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 $10^{-50}$ which are completely out of reach for
standard numerical techniques. We perform a thorough analysis of the functional
form and of the tails of the probability distribution of the LDoS which yields
for the first time a direct, transparent, and precise estimation of the
correlation volume close to the Anderson transition. Such correlation volume is
found to diverge exponentially when the localization is approached from the
delocalized regime, in a singular way that is in agreement with the analytic
predictions of the supersymmetric treatment.
