  Random operators may acquire extended states formed from a multitude of
mutually resonating local quasi-modes. This mechanics is explored here in the
context of the random Schr\"odinger operator on the complete graph. The
operators exhibits local quasi modes mixed through a single channel. While most
of its spectrum consists of localized eigenfunctions, under appropriate
conditions it includes also bands of states which are delocalized in the
$\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the
statistics of \v{S}eba spectra. The analysis proceeds through some general
observations on the scaling limits of random functions in the Herglotz-Pick
class. The results are in agreement with a heuristic condition for the
emergence of resonant delocalization, which is stated in terms of the tunneling
amplitude among quasi-modes.
