  We study numerically Anderson localization on lattices that are tree-like
except for the presence of one loop of varying length $L$. The resulting
expressions allow us to compute corrections to the Bethe lattice solution on i)
Random-Regular-Graph (RRG) of finite size $N$ and ii) euclidean lattices in
finite dimension. In the first case we show that the $1/N$ corrections to to
the average values of observables such as the typical density of states and the
inverse participation ratio have prefactors that diverge exponentially
approaching the critical point, which explains the puzzling observation that
the numerical simulations on finite RRGs deviate spectacularly from the
expected asymptotic behavior. In the second case our results, combined with the
$M$-layer expansion, predict that corrections destroy the exotic critical
behavior of the Bethe lattice solution in any finite dimension, strengthening
the suggestion that the upper critical dimension of Anderson localization is
infinity. This approach opens the way to the computation of non-mean-field
critical exponents by resumming the series of diverging diagrams through the
same recipes of the field-theoretical perturbative expansion.
