Non-equilibrium Dynamics and Random Matrices (nedrm)

Geometry of metrics and measure concentration in abstract ergodic theory

Tim Austin
New York University
April 30, 2014
Many of the major results of modern ergodic theory can be understood in terms of a sequence of finite metric measure spaces constructed from the marginal distributions of a shift-invariant process. Most simply, the growth rate of their covering numbers gives the entropy of the process, and then one finds that more refined geometric invariants determine other properties of the process.

Landau damping: Gevrey regularity and paraproducts

Clément Mouhot
University of Cambridge
April 30, 2014
We present the key ideas of a new proof of Landau damping for the Vlasov-Poisson equation obtained in a joint work with Bedrossian and Masmoudi. This nonlinear transport equation is a fundamental model for describing self-interacting plasmas or galaxies, and Landau damping is a nonlinear stability mechanism based on phase mixing. The new method does not use a Newton scheme and seems to capture a critical Gevrey regularity for the damping. Moreover its greater flexibility should open the way to further work, and we will finish by sketching some open questions.

Nonlinear Brownian motion and nonlinear Feynman-Kac formula of path-functions

Shige Peng
Shandon University
April 23, 2014
We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated nonlinear i.i.d random variables of order \(1/n\) tend to a maximal distribution according a new law of large number, whereas, with a new central limit theorem, the accumulation of order \(1/\sqrt{n}\) tends to a nonlinear normal distribution.

Free entropy

Philippe Biane
Université Paris-Est Marne-la-Vallée
April 22, 2014
Free entropy is a quantity introduced 20 years ago by D. Voiculescu in order to investigate noncommutative probability spaces (e.g. von Neumann algebras). It is based on approximation by finite size matrices. I will describe the definition and main properties of this quantity as well as applications to von Neumann algebras. I will also explain a new approach based on work with Y. Dabrowski, using random matrices, which leads to the solution of some problems concerning this quantity.

Limiting Eigenvalue Distribution of Random Matrices Involving Tensor Product

Leonid Pastur
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
April 16, 2014
We consider two classes of \(n \times n\) sample covariance matrices arising in quantum informatics. The first class consists of matrices whose data matrix has \(m\) independent columns each of which is the tensor product of \(k\) independent \(d\)-dimensional vectors, thus \(n=d^k\). The matrices of the second class belong to \(\mathcal{M}_n(\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}), \ n=d_1 d_2\) and are obtained from the standard sample covariance matrices by the partial transposition in \(\mathbb{C}^{d_2}\).