  The three-dimensional Anderson model represents a paradigmatic model to
understand the Anderson localization transition. In this work we first review
some key results obtained for this model in the past 50 years, and then study
its properties from the perspective of modern numerical approaches. Our main
focus is on the quantitative comparison between the level sensitivity
statistics and the level statistics. While the former studies the sensitivity
of Hamiltonian eigenlevels upon inserting a magnetic flux, the latter studies
the properties of unperturbed eigenlevels. We define two versions of
dimensionless conductance, the first corresponding to the width of the level
curvature distribution relative to the mean level spacing, and the second
corresponding to the ratio of the Heisenberg and the Thouless time obtained
from the spectral form factor. We show that both conductances look remarkably
similar around the localization transition, in particular, they predict a
nearly identical critical point consistent with other measures of the
transition. We then study some further properties of those quantities: for
level curvatures, we discuss particular similarities and differences between
the width of the level curvature distribution and the characteristic energy
studied by Edwards and Thouless in their pioneering work [J. Phys. C. 5, 807
(1972)]. In the context of the spectral form factor, we show that at the
critical point it enters a broad time-independent regime, in which its value is
consistent with the level compressibility obtained from the level variance.
Finally, we test the scaling solution of the average level spacing ratio in the
crossover regime using the cost function minimization approach introduced in
[Phys. Rev. B. 102, 064207 (2020)]. We find that the extracted transition point
and the scaling coefficient agree with those from the literature to high
numerical accuracy.
