  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 ($0 < \beta < 2$).
