We present background and the proof of universality for ergodic
Jacobi matrices with a.c. spectrum. This includes cases where the support of
the measure is a positive measure Cantor set. This describes joint work with
Artur Avila and Yoram Last.

n the talk I will study the spectral properties of a class of
SturmLiouville-type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, one can construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. I will show that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively. A key ingredient in the proof is the so- called Wonderland theorem.
The talk is based on joint work with Malcolm Brown and Karl
Michael Schmidt.

The Anderson model (which will celebrate its 50th anniversary in 2008) is among most popular topics in the random matrix and operator theory. However, so far the attention here was concentrated on single-particle models, where the random external potential is either IID or has a rapid decay of spatial correlations. Multi-particle models remained out of scope in mathematical (and, surprisingly, physical) literature. Recently, Chulaevsky and Suhov (2007) proposed a version of the multi-scale analysis (MSA) scheme tackling the multi-particle case. I'll discuss one of results in this direction: localisation in the lattice (tight binding) multi-particle models for large values of the amplitude (coupling) constant.

Ergodicity of many-particle motion is a fundamental assumption that underlies, among other things, the powerful statistical mechanics description of nature. The ergodic hypothesis can break down under some conditions, most notably in the presence of strong random potentials leading the phenomenon of Anderson localization. The theory of Anderson localization assumes no interactions among particles and it is of considerable practical interest to know whether the phenomenon can persist more generally. I review some recent ideas and results on spectral and transport properties of quantum and classical many-body systems. Within limitations of our methods we observe that localized states of classical particles are unstable against non-linearities, while interacting quantum particles can remain insulating.

We study particle systems corresponding to highly connected queuing
networks, like Internet. We examine the validity of the so called Poisson
Hypothesis (PH), which predicts that such particle system, if started
from a reasonable initial state, relaxes to its equilibrium in time
independent of the size of the network. We show that this is indeed the
case in many situations.

However, there are networks for which the relaxation process slows down.
This behavior reflects the fact that the corresponding infinite system
undergoes a phase transition. Such transition can happen only when the
load per server exceeds some critical value, while in the low load
situation the PH behavior holds. Thus, the load plays here the same role
as the inverse temperature in statistical mechanics.