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

Evidence of the Poisson/Gaudin-Mehta phase transition for banded matrices on global scales