  We prove that the Poisson/Gaudin--Mehta phase transition conjectured to occur
when the bandwidth of an $N \times N$ symmetric banded matrix grows like $\sqrt
N$ is observable as a critical point in the fourth moment of the level density
for a wide class of symmetric banded matrices. A second critical point when the
bandwidth grows like ${2 \over 5} N$ leads to a new conjectured phase
transition in the eigenvalue localization, whose existence we demonstrate in
numerical experiments.
