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

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### Type

Thesis### Thesis type

Masters by Research### Author/s

Swan, Andrew### 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 $b \ll \sqrt{N}$, ...

See moreThe 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.

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See moreThe 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.

See less

### Date

2016-10-11### Licence

The author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission.### Faculty/School

Faculty of Science, School of Mathematics and Statistics### Awarding institution

The University of Sydney## Share