Ilya Ya. Goldsheid

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 Theorem.

The Thouless formula for random non-Hermitian Jacobi matrices

Random non-Hermitian Jacobi matricesJn of increasing dimensionn are considered. We prove that the normalized eigenvalue counting measure ofJn converges weakly to a limiting measure μ asn→∞. We also extend to the non-Hermitian case the Thouless formula relating μ and the Lyapunov exponent of the second-order difference equation associated with the sequenceJn. The measure μ is shown to be log-Hölder continuous. Our proofs make use of (i) the theory of products of random matrices in the form first offered by H. Furstenberg and H. Kesten in 1960 [8], and (ii) some potential theory arguments.

Regular Spacings of Complex Eigenvalues in the One-Dimensional Non-Hermitian Anderson Model

AbstractWe 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.

Limit theorems for random walks on a strip in subdiffusive regimes

We study the asymptotic behaviour of occupation times of a transient random walk (RW) in a quenched random environment (RE) on a strip in a subdiffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is exactly the same. As a particular case, we solve a long standing problem of describing the asymptotic behaviour of a RW with bounded jumps on a one-dimensional lattice. Our technique results from the development of ideas from our previous work (Dolgopyat and Goldsheid 2012 Commun. Math. Phys. 315 241–77) on the simple RWs in RE and those used in Bolthausen and Goldsheid (2000 Commun. Math. Phys. 214 429–47; 2008 Commun. Math. Phys. 278 253–88) and Goldsheid (2008 Probab. Theory Relat. Fields 141 471–511) for the study of random walks on a strip.