We discuss recently established criteria for the formation of extended states on tree graphs in the presence of disorder. These criteria have the surprising implication that for bounded random potentials, as in the Anderson model, in the weak disorder regime there is no transition to a spectral regime of Anderson localization in the form usually…

I shall describe a theory of adiabatic response for controlled open systems governed by Lindblad evolutions. The theory gives quantum response a geometric interpretation induced from the geometry of Hilbert space projections. For a two level system the metric turns out to be the Fubini-Study metric and the symplectic form the adiabatic curvature…

The fluctuations of a physical quantity can be described by its moments. In many cases, these diverge as an asymptotic parameter becomes large (or small), through the influence of geometric singularities. These large moments are described by power laws whose exponents can be determined from a knowledge of the singularities. Examples are twinkling…

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this talk I will present a formalism which also allows to study nonlinear systems. I will in particular define a weak topology on many-body states, which appropriately describes the physical behavior of the…

We consider the planar Ising model on bounded domains from a conformal invariance point of view. We are interested in the scaling limit of the model at critical temperature. Physics theories, notably Conformal Field Theory, predict the existence of two conformal local fields describing the model in the continuum: the spin and the energy density…

Trace formulas for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulas depend on a parameter (w) which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only…

Trace formulas for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulas depend on a parameter (w) which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only…

We will discuss the spectral properties of random operators on regular tree graphs. The models have have been among the earliest studied for Anderson localization, and they continue to attract attention because of analogies with localization issues for many particles. The talk will focus on the location of the mobility edge. Somewhat…