A surprising prediction is that the eigenfunctions of random band matrices have similar spatial spread as cycles in various models of random permutations. In this talk we discuss various ideas in the proofs of localization in these models.

Since the times of Ludwig Boltzmann, most physicists take it for granted that generic non-integrable closed dynamical systems, whether classical or quantum, thermalize at long times for generic initial conditions. Whether this belief is true or false depends on the type of systems one is willing to consider and the meaning of “generic” and “thermalizes”. In this talk I will discuss two types of many-body dynamical systems where thermalization (i.e. weak convergence to the state of maximal entropy) can be established for a large class of initial states. The first one is a system of an infinite number of spins with evolution generated by a repeated application of a Clifford Cellular Automaton. The second one is the classical counterpart of the first one and can be thought of as a mixing automorphism of an infinite-dimensional torus. 

Abstract: In this lecture, we will present several dynamical and fractal-dimensional ways of characterizing the spectral measures of Schrödinger operators, such as Rajchman behavior and Hausdorff/packing dimensions, and discuss the extent to which these properties are stable under rank-one perturbations.
We begin with the concrete setting of half-line Schrödinger operators, where a theorem of Gordon shows that generic rank-one perturbations eliminate pure point spectrum, ruling out the most extreme dynamical and dimensional behavior. I will then describe constructions demonstrating that properties only slightly weaker than pure point spectrum can, in fact, be entirely stable: for certain sparse half-line models, both packing-dimension-zero and non-Rajchman behavior persist for every rank-one perturbation.
In the second part, we examine how spectral dimensions behave when passing from the whole line to the half-line. I will present an operator whose spectral measure on the line has Hausdorff dimension one, whereas every half-line restriction - under any boundary condition - has dimension zero, even though the two settings differ only by a finite-rank perturbation.

We will consider a nonstationary random walk on a compact topological group. Under a classical strict aperiodicity assumption, we will establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate. I'll provide an overview of results known for stationary random walks and will describe the main idea of the proof for the nonstationary case.