Random matrix approach to a special kind of quantum random hopping

We use the random matrix method to study one kind of quantum random hopping. The Hamiltonian is a non-Hermitian matrix with some negative subdiagonal elements. Using potential theory, we calculate the eigenvalue density of an N x N matrix when N goes to infinity. We also obtain a least upper bound of the module of eigenvalues. In view of phase string theory in high-temperature superconductors, this model connects with localization-delocalization transition.