Volume 5 (2005), Number 3. Abstracts

M. Aizenman and S. Warzel. Persistence under Weak Disorder of AC Spectra of Quasi-Periodic Schrödinger Operators on Trees Graphs [PDF]

We consider radial tree extensions of one-dimensional quasi-periodic Schrödinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch–Floquet states for the one dimensional operator corresponding to the radial problem.

Keywords. Random operators, absolutely continuous spectrum, quasi-periodic cocycles, Bloch states

2000 Mathematics Subject Classification. 47B80, 37E10

C. Boldrighini, R. Minlos, F. Nardi, and A. Pellegrinotti. Asymptotic Decay of Correlations for a Random Walk in Interaction with a Markov Field [PDF]

We consider a random walk on Z in a random environment independent in space and with a Markov evolution in time. We study the decay in time of correlations of the increments of the annealed random walk. We prove that for small stochasticity they fall off as ≈t^−1/2e^−α₁t, for α>0. The analysis shows that, as the parameters of the model vary, a transition to a fall-off of the type ≈e^{− \bar αt}, for \bar α ∈(0,α₁), may occur.

Keywords. Random walk, correlations, Markov chain, increments, field “from the point of view of the particle”

2000 Mathematics Subject Classification. 60J15, 82B10, 82B41

J. Bourgain. Anderson–Bernoulli Models [PDF]

We prove the exponential localization of the eigenfunctions of the Anderson model in R^d in the regime of large coupling constant for the random potentials which values are independent and Bernoulli distributed.

Keywords. Anderson localization, random Bernoulli potential

2000 Mathematics Subject Classification. 82B44 (60H25, 81Q10, 82B10)

D. Dolgopyat. Averaging and Invariant Measures [PDF]

An important approach to establishing stochastic behavior of dynamical systems is based on the study of systems expanding a foliation and of measures having smooth densities along the leaves of this foliation. We review recent results on this subject and present some extensions and open questions.

Keywords. Averaging, Sinai–Ruelle–Bowen measures, partial hyperbolicity, decay of correlations

2000 Mathematics Subject Classification. Primary 34C29; Secondary 37D30

M. Goldstein and W. Schlag. On Schrödinger Operators with Dynamically Defined Potentials [PDF]

The purpose of this article is to review some of the recent work on the operator

(H_x)_n = −ψ_n−1 − ψ_n+1 + λ V(Tⁿx) ψ_n

on l²(Z), where T: X→X is an ergodic transformation on (X,ν) and V is a real-valued function. λ is a real parameter called coupling constant. Typically, X=T^d=(R/Z)^d with Lebesgue measure, and V will be a trigonometric polynomial or analytic. We shall focus on our earlier papers, as well as other work which was obtained jointly with Jean Bourgain. Our goal is to explain some of the methods and results from these references. Some of the material in this paper has not appeared elsewhere in print.

Keywords. Eigenfunction, localization, Lyapunov exponent

2000 Mathematics Subject Classification. 47B80

D. Gomes, R. Iturriaga, K. Khanin, and P. Padilla. Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation [PDF]

We prove convergence of stationary distributions for the randomly forced Burgers and Hamilton–Jacobi equations in the limit when viscosity tends to zero. It turns out that for all values of the viscosity ν there exists a unique (up to an additive constant) global stationary solution to the randomly forced Hamilton–Jacobi equation. The main result follows from the convergence of these solutions in a limit when ν tends to zero without changing its sign. The two limiting solutions (for different signs of the viscosity term) correspond to unique backward and forward viscosity solutions. Our approach, which is an extension of the previous work, is based on the stochastic version of Lax formula for solutions to the initial and final value problems for the viscous Hamilton–Jacobi equation.

Keywords. Random Burgers equation, random Hamilton–Jacobi equation, convergence to viscosity solutions

2000 Mathematics Subject Classification. 35L65, 37H10, 37D99

S. Novikov. Topology of Generic Hamiltonian Foliations on Riemann Surfaces [PDF]

The topology of generic Hamiltonian dynamical systems given by the real parts of generic holomorphic 1-forms on Riemann surfaces is studied. Our approach is based on the notion of transversal canonical basis of cycles. This approach allows us to present a convenient combinatorial model of the whole topology of the flow, especially effective for g=2. A maximal abelian covering over the Riemann surface is needed here. The complete combinatorial model of the flow is constructed. It consists of the plane diagram and g straight-line flows in 2-tori “with obstacles.” The fundamental semigroup of positive closed paths transversal to the foliation is studied. This work contains an improved exposition of the results presented in the author's recent preprint and new results concerning the calculation of all transversal canonical bases of cycles in the 2-torus with obstacle in terms of continued fractions.

Keywords. Hamiltonian system, Riemann surface, transversal semigroup

2000 Mathematics Subject Classification. 37D40

Ya. Pesin and S. Senti. Thermodynamical Formalism Associated with Inducing Schemes for One-Dimensional Maps [PDF]

For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions φ on I which admit a unique equilibrium measure μ_φ. Our results apply to unimodal maps corresponding to a positive Lebesgue measure set of parameters in a one-parameter transverse family.

Keywords. Equilibrium measures, Gibbs measures, inducing schemes, thermodynamic formalism, unimodal maps

2000 Mathematics Subject Classification. 37D25, 37D35, 37E05, 37E10

A. Rybko and S. Shlosman. Poisson Hypothesis for Information Networks. I [PDF]

In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queuing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of heavy-tail service times the PH can be violated.

Keywords. Mean-field models, server, waiting time, phase transition, limit theorem, self-averaging property, attractor

2000 Mathematics Subject Classification. Primary: 82C20; Secondary: 60J25

A. Soshnikov. Statistics of Extreme Spacings in Determinantal Random Point Processes [PDF]

We study determinantal translation-invariant random point processes on the real line. Under some technical assumptions on the correlation kernel, we prove that the smallest nearest spacings in a large interval have Poisson statistics as the length of the interval goes to infinity.

Keywords. Determinantal random point processes, cluster functions, Poisson statistics

2000 Mathematics Subject Classification. 60G55, 60G70

A. Vershik. Towards the Definition of Metric Hyperbolicity [PDF]

We introduce measure-theoretic definitions of hyperbolic structure for measure-preserving automorphisms. A wide class of K-automorphisms possesses a hyperbolic structure; we prove that all K-automorphisms have a slightly weaker structure of semi-hyperbolicity. Instead of the notions of stable and unstable foliations and other notions from smooth theory, we use the tools of the theory of polymorphisms. The central role is played by polymorphisms associated with a special invariant equivalence relation, more exactly, with a homoclinic equivalence relation. We call an automorphism with given hyperbolic structure a hyperbolic automorphism and prove that it is canonically quasi-similar to a so-called prime nonmixing polymorphism. We present a short but necessary vocabulary of polymorphisms and Markov operators.

Keywords. Polymorphisms, Markov operator, hyperbolic structure, quasisimilarity

2000 Mathematics Subject Classification. Primary: 37A05, 47A45, 60J27; Secondary: 37A25, 37H10, 47A40

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