Workshop "Frontiers in Mathematical Physics" - May 26-27, 2011

A two days workshop will take place on May 26th-27th at Campus St. Martin Université de Cergy-Pontoise Amphi Colloques, Bâtiment E RdC.

Thursday May 26, 2011
09h45-10h00 Welcome

Chairman : R. Livi
10h00- 11h0 . V. Jaksic (Mc Gill University, Montreal, Canada)
Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics
11h00-11h30 Coffee Break
11h30-12h30. C.-A. Pillet (CPT Marseille & Université du Sud Toulon-Var, France)
Scattering induced current in a tight binding band
12h30-14h00. Lunch

Chairman : L. Bruneau
14h00-15h00. R. Seiringer (Mc Gill University, Montreal, Canada)
Microscopic Derivation of Ginzburg-Landau Theory
15h00-16h00. H. Kunz (Ecole Polytechnique Fédérale de Lausanne, Suisse)
Universality In Random Matrices
16h00-16h30. Co-ee Break
16h30-17h30. S. De Bievre (Université Lille 1, France)
Equilibration, generalized equipartition, and di-usion in dynamical Lorentz gases

Friday May 27, 2011
Chairman : F. Germinet
09h30-10h30. M. Aizenman (Princeton University, USA)
Random Operators on Tree Graphs I : Extended States in a Lifshitz Tail Regime
10h30-11h00. Co-ee Break
11h00-12h00. S. Warzel (Technische Universität München, Germany)
Random Operators on Tree Graphs II : Absence of Mobility Edge for Bounded Potentials at Weak Disorder
12h00-13h00. F. Klopp (Université Paris 13, France)
Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line
13h00-14h30. Lunch

Chairman : F. Koukiou
14h30-15h30. A. Klein (UC Irvine, USA) What is localization for continuous Anderson models with singular random potentials ?
15h30-16h00. Co-ee Break
16h00-17h00. A. Joye (Institut Fourier, Grenoble, France)
Random Time-Dependent Quantum Walks
17h30 Meeting with the President of the University.

26-th of May
Chair : R. Livi

V. Jaksic
Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics.
Since the days of Chernoff, the hypothesis testing has played an important role in theoretical and applied statistics. In the last decade, the mathematical structure and basic results of classical hypothesis testing have been extended to the non-commutative setting and are an important branch of quantum information theory. It was recently observed that here is a close relation (almost a a parallel) between the recent developments in quantum hypothesis testing and the developments in non-equilibrium quantum statistical mechanics. In this talk we will elaborate on this relation. ( The talk is based on the joint work with Y. Ogata, C.-A. Pillet and R. Seiringer.)

C.-A. Pillet
Scattering induced current in a tight binding band
In the single band tight-binding approximation, we consider the transport properties of an electron subject to a homogeneous static electric field. We show that repeated interactions of the electron with two-level systems in thermal equilibrium suppress the Bloch oscillations and induce a steady current, the statistical properties of which we study. (Joint work with S. De Bièvre and L. Bruneau))

Chair : L. Bruneau

R. Seirenger
Microscopic Derivation of Ginzburg-Landau Theory
We present a rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof. (Joint work with R. Frank, C. Hainzl, and J.P. Solovej)

H. Kunz
Universality In Random Matrices.
Giving a precise meaning to the concept of universality in the statistical properties of random matrices,we briefly summarise the cases where this phenomenon has been proven. We also demonstrate that a large class of random matrices belong to the Wigner class.Finally we give examples of new universality classes.

S. De Bievre
Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases
We demonstrate approach to thermal equilibrium in the fully Hamiltonian evolution of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a d-dimensional array of fixed soft scatterers that each possess an internal degree of freedom to which moving particles locally couple. We show that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature T, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics- they need not be in a state of thermal equilibrium. In our treatment, the temperature T to which the particles equilibrate is defined through a fluctuation-dissipation-like relation that emerges naturally from the microscopic Hamiltonian dynamics as a result of dynamical friction ; It obeys a generalized equipartition relation, in which the associated thermal energy kBT is equal to an appropriately defined average of the scatterers’ kinetic energy. In the equilibrated state, particle motion is diffusive. (Joint work with P. Parris, J. Stat. Phys. (2011)).

27-th May
Chair : F. Germinet

Michael Aizenman (joint work with Simone Warzel)
Random Operators on Tree Graphs ; 1. Extended States in a Lifshitz Tail Regime
We resolve an existing question concerning the location of the mobility edge for operators with a hopping term and a random potential on the Bethe lattice. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies which have been suggested with localization issues for many particle systems. For unbounded potential we find that extended states appear well beyond the energy band of the operator’s hopping term, including in a Lifshitz tail regime of very low density of states. The relevant mechanism is the formation of extended states through disorder enabled resonances, for which the exponential increase of the volume plays an essential role.

Simone Warzel (joint work with M. Aizenman)
Random Operators on Tree Graphs ; 2. Absence of Mobility Edge for Bounded Potentials at Weak Disorder
The recently established criterion for the formation of extended states on tree graphs in the presence of disorder is shown to have the surprising implication that for bounded random potentials at weak disorder there is no mobility edge in the form that was envisioned before.

Chair : F. Koukiou

F. Klopp
Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line
We present a proof of Minami type estimates for one dimensional random Schrödinger operators valid at all energies in the localization regime provided a Wegner estimate is known to hold. The Minami type estimates are then applied to various models to obtain results on their spectral statistics. The heuristics underlying our proof of Minami type estimates is that close by eigenvalues of a one-dimensional Schrödinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena. In the second case, one can undo the tunneling and thus construct quasi-modes that live far away from each other in space.

A. Klein
What is localization for continuous Anderson models with singular random potentials ?
We consider continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We will give a description of localization from which we will derive all the usual manifestations of localization, including Anderson localization (pure point spectrum with exponentially decaying eigenfunctions), dynamical localization (no spreading of wave packets under the time evolution), decay of eigenfunctions correlations (e.g., SULE, SUDEC), and decay of the Fermi projection. We will discuss how log-H\" older continuity of the integrated density of states and this description of localization follows from a single-energy multiscale analysis. (Joint work with F. Germinet.)

A. Joye
Random Time-Dependent Quantum Walks
We consider the discrete time unitary dynamics given by a quantum walk on the d-dimensional lattice performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule : a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable on the lattice when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. In particular, when averaged over the randomness, this distribution is shown to display a drift proportional to time and its centered counterpart is shown to display a diffusive behavior. A moderate deviation principle is also proven to hold.