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. The average connectivity is related to the concentration of impurities. The adjacency matrix of a large random graph is used as a hopping Hamiltonian. Its spectrum has a delta peak at zero energy. Our analysis is based on an explicit expression for the height of this peak, and a detailed description of the localized eigenvectors and of their contribution to the peak. Starting from the low connectivity (high impurity density) regime, one encounters an insulator-conductor transition for average connectivity 1.421529... and a conductor-insulator transition for average connectivity 3.154985.... We explain the spectral singularity at average connectivity e=2.718281... and relate it to another enumerative problem in random graph theory, the minimal vertex cover problem.