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

Abstract

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.