Random discrete Schr̈odinger operators from random matrix theory

We investigate random, discrete Schr̈odinger operators which arise naturally in the theory of random matrices, and depend parametrically on Dysonś Coulomb gas inverse temperatureβ. They are similar to the class of cŕitical' random Schr̈odinger operators with random potentials which diminish as |x|-12 . We show that as a function ofβthey undergo a transition from a regime of (powerlaw) localized eigenstates with a pure point spectrum forβ < 2 to a regime of extended states with a singular continuous spectrum for≤2.