Regular Spacings of Complex Eigenvalues in the One-dimensional non-Hermitian Anderson Model

We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.Comment: 21 pages, 1 figur

Random walks in a random environment on a strip: a renormalization group approach

We present a real space renormalization group scheme for the problem of random walks in a random environment on a strip, which includes one-dimensional random walk in random environment with bounded non-nearest-neighbor jumps. We show that the model renormalizes to an effective one-dimensional random walk problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in the recurrent case, while in the sub-linear transient phase, the displacement grows as a power of the time.Comment: 9 page

Spectral Curves and Localization in Random Non-Hermitian Tridiagonal Matrices

Eigenvalues and eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano-Nelson deformation. The deformed spectrum is annular-shaped, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are stuctures that correspond to the two-arc and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the Argument principle.Comment: 5 pages, 9 figures, typographical error corrected in reference

Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte

Density of states in the non-hermitian Lloyd model

We reconsider the recently proposed connection between density of states in the so-called ``non-hermitian quantum mechanics'' and the localization length for a particle moving in random potential. We argue that it is indeed possible to find the localization length from the density of states of a non-hermitian random ``Hamiltonian''. However, finding the density of states of a non-hermitian random ``Hamiltonian'' remains an open problem, contrary to previous findings in the literature.Comment: 6 pages, RevTex, two-column