60 research outputs found
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 (Οj)jâ„1 be a non-stationary Markov chain with phase space X and let gj:XâŠSL(m,R) be a sequence of functions on X with values in the unimodular group. Set gj=gj(Οj) and denote by Sn=gnâŠg1â , the product of the matrices gjâ . We provide sufficient conditions for exponential growth of the norm â„Snâ„ 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
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