Harvard-MIT Random Matrix Theory Afternoon

Massachusetts Institute of Technology, February 13, 2013



Charles Bordenave [abstract]
Tea break
Patrik Ferrari [abstract]
Benjamin Schlein [abstract]

Practical Information
The talks will take place at MIT in room 4-163, on February 13, 2013. No registration necessary. However, if you would like to attend the dinner after the talks, please RSVP by February 10, 2013.

Directions to MIT. Event poster.

For further information, please contact the organizers.

Titles and Abstracts

Charles Bordenave (Toulouse)
Large deviations for Wigner matrices without gaussian tails

Abstract: We consider a Wigner matrix: a random Hermitian matrix X of size n whose entries above the diagonal are independent and identically distributed with unit variance. Since the seminal work of Wigner in the 50's, it is known that the empirical distribution of the eigenvalues of X / sqrt n converges to the semi-circular law.

In 1997, Ben Arous and Guionnet have established a large deviation principle (LDP) around the semi-circular law when the entries are Gaussian. The associated rate function is the Voiculescu's non-commutative entropy. Their proof was based on the explicit formula for the joint law of the eigenvalues, and beyond this result, establishing LDP's for Wigner matrices remains largely open.

When the entries are of Weibull type but not subgaussian (for example exponential) we will see that it is however possible to prove such LDP using ideas coming from random graphs. This is a joint work with Pietro Caputo (Univ. Roma Tre).

Patrik Ferrari (Bonn)
Interlacing patterns in particle systems and random matrices

Abstract: We will discuss some probability measures on interlacing configurations and show that they arise naturally both from the evolution of some particle systems (resp. systems of Brownian motions) and in random matrix theory.

Benjamin Schlein (Bonn)
Mean field and Gross-Pitaevskii limits of quantum dynamics

Abstract: The derivation of effective evolution equations is a central question in non-equilibrium statistical mechanics. It turns out that, in the mean field limit, the many body quantum evolution can be approximated by the nonlinear Hartree equation. We will describe how optimal estimates on the rate of the convergence and a central limit theorem for the fluctuations around the Hartree dynamics can be obtained making use of so called coherent states. The Gross-Pitaevskii regime is a very singular mean field limit, relevant for the description of the dynamics of Bose-Einstein condensates. In this case, we will show how the convergence towards the limiting dynamics can be established by correcting the coherent states with appropriate Bogoliubov transformations, describing the correlation structure developed by the many body evolution.