Global statistics of banded random matrices and the Poisson/Gaudin--Mehta conjecture

Abstract

The Poisson/Gaudin--Mehta conjecture, a major open problem in random matrix theory, states that in the large $N$ limit, the local eigenvalue statistics of an $N\times N$ banded symmetric Hermitian random matrix are described by Poisson statistics if the bandwidth $b \ll \sqrt{N}$, and by Gaudin--Mehta statistics if $b\gg \sqrt{N}$. The eigenvectors are also expected to undergo a localisation/delocalisation transition at the critical bandwidth $b \sim \sqrt{N}$. Given that the level density converges to the Wigner semi circle law for all $b = N^\alpha$, $0 <\alpha <1$, the Poisson/Gaudin--Mehta transition was thought to not be visible at the global scale. We prove that this is not the case by demonstrating the existence of a critical point in the fourth moment of the level density when $b = \left(\frac{3N}{2}\right)^\frac{1}{2} + o\left(N^{\frac{1}{2}}\right)$. We also find a second critical point at $b = \frac{2}{5}N + o(N)$, and present numerical evidence that this corresponds to a second, lower order, localisation/delocalisation transition in the eigenvectors. We also construct tridiagonal models of banded random matrices to investigate the possibility of capturing the Poisson/Gaudin--Mehta transition with a simpler ensemble.