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

Constructive approach to limit theorems for recurrent diffusive random walks on a strip.

We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed

Exponential Growth of Products of Non-Stationary Markov-Dependent Matrices

Let (ξj)j≥1 be a non-stationary Markov chain with phase space X and let gj:X↦SL(m,R) be a sequence of functions on X with values in the unimodular group. Set gj=gj(ξj) and denote by Sn=gn…g1⁠, the product of the matrices gj⁠. We provide sufficient conditions for exponential growth of the norm ∥Sn∥ when the Markov chain is not supposed to be stationary. This generalizes the classical theorem of Furstenberg on the exponential growth of products of independent identically distributed matrices as well as its extension by Virtser to products of stationary Markov-dependent matrices

LOCAL LIMIT THEOREMS FOR RANDOM WALKS IN A RANDOM ENVIRONMENT ON A STRIP

The paper consists of two parts. In the first part we review recent work on limit theorems for random walks in random environment (RWRE) on a strip with jumps to the nearest layers. In the second part, we prove the quenched Local Limit Theorem (LLT) for the position of the walk in the transient diffusive regime. This fills an important gap in the literature. We then obtain two corollaries of the quenched LLT. The first one is the annealed version of the LLT on a strip. The second one is the proof of the fact that the distribution of the environment viewed from the particle (EVFP) has a limit for a. e. environment. In the case of the random walk with jumps to nearest neighbours in dimension one, the latter result is a theorem of Lally \cite{L}. Since the strip model incorporates the walks with bounded jumps on a one-dimensional lattice, the second corollary also solves the long standing problem of extending Lalley's result to this case

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