Resonances and Partial Delocalization on the Complete Graph

Abstract

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 Schrodinger operator on the complete graph. The operator 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 -though not in -sense, where the eigenvalues have the statistics of 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.