We study how the spectral properties of ergodic Schr\”odinger operators are
reflected in the asymptotic properties of its periodic approximation as the
period tends to infinity. The first property we address is the asymptotics of
the bandwidths on the logarithmic scale, which quantifies the sensitivity of
the finite volume restriction to the boundary conditions. We show that the
bandwidths can always be bounded from below in terms of the Lyapunov exponent.
Under an additional assumption satisfied by i.i.d potentials, we also prove a
matching upper bound. Finally, we provide an additional assumption which is
also satisfied in the i.i.d case, under which the corresponding eigenvectors
are exponentially localised with a localisation centre independent of the
Floquet number.
