Real eigenvalues in the non-Hermitian Anderson model

The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.Comment: 21 pp., 2 fig; to appear in Ann. Appl. Proba

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

Sets of non-Lyapunov behaviour for scalar and matrix Schr\"odinger cocycles

We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schr\"odinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_n$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_n^{-1}$. On the positive side, for a large class of cocycles including the i.i.d.\ ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $\rho(t)$ such that $\rho(t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure

Lingering Random Walks in Random Environment on a Strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the (log t)2 asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the (log t)2 asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theore