This term, all seminars take place Thursdays at 2pm, room MS.04 (Zeeman Building), unless indicated otherwise.



Schedule for

Wolfgang König, 28.02.2008

Seminars

12.01.2012 Antti Knowles (Harvard University)
Quantum diffusion and eigenvector delocalization for random band matrices
The general formulation of the universality conjecture for disordered systems states that there are two distinctive regimes depending on the energy and the disorder strength. In the strong disorder regime, the eigenvectors are localized and the local spectral statistics are Poisson. In the weak disorder regime, the eigenvectors are delocalized and the local statistics coincide with those of a Gaussian matrix ensemble. Random band matrices represent natural intermediate models to study the eigenvalue statistics and quantum propagation in disordered systems, as they interpolate between mean-field-type Wigner matrices and random Schrödinger operators. In particular, band matrices provide a means of probing the localization-delocalization transition.

I shall report on recent joint work with L. Erdos. We consider a large class of random band matrices H with band width W, and prove that the quantum time evolution generated by H is diffusive up to time scales of order W^{d/3}, where d is the number of spatial dimensions. We also derive an explicit formula for the diffusion constant. As a corollary, we prove that the localization length of an arbitrarily large majority of eigenvectors is larger than a factor W^{d/6} times the band width W.
Contact: Daniel/Volker
19.01.2012 Dimitrios Tsagkarogiannis (Bonn)
Phase transitions and the cluster expansion in the canonical ensemble
In this talk we discuss an extension of a previous work by Lebowitz-Mazel-Presutti ('99) proving a liquid-vapour phase transition for a system of particles in R^d interacting via a Kac (local mean field type) potential. We enrich the model by adding a repulsive potential of small but finite order which on physical grounds may be responsible for the occurence of crystalline structure while (due to its smallness) it should not affect the liquid-vapour transition and we focus on the second issue. A main step in the treatment of the microscopic Hamiltonian is to construct a coarse-grained Hamiltonian with multi-canonical constraints as required by the Lebowitz- Penrose theory. For this, the introduction of the small range interaction creates an extra complication which is directly related to the question of validity of the cluster expansion in the canonical ensemble that we shall also discuss. This is work in progress in collaboration with E. Presutti and E. Pulvirenti.
Contact: Stefan A
26.01.2012


02.02.2012


09.02.2012 Ellen Saada (CNRS, Universite Paris Descartes)
Couplings and Attractiveness for Interacting Particle Systems
Attractiveness is a fundamental tool to study interacting particle systems. It corresponds to the existence of a coupling of two processes with the same infinitesimal generator, that stay ordered as soon as it is the case for their initial states. This property supplemented with some irreducibility conditions enables to determine the extremal invariant and translation invariant measures for the model. This is the first step to derive the hydrodynamic behavior of the system.

We consider two classes of models for which the ``basic coupling'' construction is not possible under necessary and sufficient conditions for attractiveness:
On the one hand, generalized misanthrope models which are conservative particle systems on $\Z^d$ such that, in each transition, $k$ particles may jump from a site $x$ to another site $y$, with $k\geq 1$;
on the other hand, exclusion processes with speed change, for which the jump rate from site $x$ to site $y$ also depends on the number of particles on other sites.

In both cases, under attractiveness conditions, we construct an increasing coupling which also enables to control discrepancies when the coupled initial states are not ordered. We finally apply these results to various unidimensional examples.

This is joint work with Thierry Gobron (CNRS, Cergy-Pontoise).
Contact: Stefan G
16.02.2012


23.02.2012


01.03.2012


08.03.2012 Vitali Wachtel (LMU Munich)
Random walks in cones
We study the asymptotic behaviour of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing of a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of all asymptotic relations we use the strong approximation of random walks by the Brownian motion.
Contact: Stefan A
15.03.2012