Exactly Solvable Model with Two Conductor-Insulator Transitions Driven by Impurities

We present an exact analysis of two conductor-insulator transitions in the random graph model where low connectivity means high impurity concentration. The adjacency matrix of the random graph is used as a hopping Hamiltonian. We compute the height of the delta peak at zero energy in its spectrum exactly and describe analytically the structure and contribution of localized eigenvectors. The system is a conductor for average connectivities between $1.421529\dots$ and $3.154985\dots$ but an insulator in the other regimes. We explain the spectral singularity at average connectivity $e=2.718\mathrm{}281\dots$ and relate it to other enumerative problems in random graph theory.