Random discrete Schr\"odinger operators from Random Matrix Theory

We investigate random, discrete Schr\"odiner operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature $\beta$. They belong to the class of "critical" random Schr\"odiner operators with random potentials which diminish as $|x|^{-{1/2}}$. We show that as a function of $\beta$ their eigenstates undergo a transition from extended ($\beta \ge 2 $) to power-law localized ($0 < \beta < 2$).