Ph. Biane, F. Bornemann, J. Najim, P. Šeba, O. Zeitouni

Free probability theory allows to make prediction concerning spectra of large matrices obtained from independent matrice, for example sums or products of such matrices. We show that it gives also some insight into eigenvectors. I will assume no knowledge of free probability and only basic knowledge of random matrix theory.

Folkmar Bornemann (Müaut;nchen): Numerical Evaluation of Distributions in Random Matrix Theory.

Many probability distributions of Random Matrix Theory can be expressed as operator determinants and their derivatives. However, their numerical evaluation (necessary, e.g., for comparing empirical distributions with their theoretical counter-parts) has generally been thought to rely on alternative analytic expressions, most famously in terms of the Painlevé transcendents like the Tracy-Widom distribution. We will discuss a simple and extremely effective numerical method for operator determinants and its application to many distributions in Random Matrix Theory. Special attention will be paid to questions of numerical accuracy. Using this numerical method we were able to provide compelling numerical evidence that the Airy1-process is not the large matrix limit of the largest eigenvalue in GOE matrix diffusion.

Jamal Najim (Paris): Asymptotic independence of the extreme eigenvalues in the GUE.

Consider an n by n matrix from the Gaussian Unitary Ensemble (GUE). We prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE. We will mention applications of related results to digital communications.

Petr Šeba (Hradec Králové): The universality distribution of distances in the human and animal traffic.

We show that the mechanism of distance perception which is evolutionarily very old and shared with mammals and birds leads finally to an universal behavior of spacings in the human/animal traffic. We demonstrate this fact on data referring to car parking and stopping, highway and pedestrian traffic, bird perching on electric poles and the distances within a sheep herd. A simple theory of this phenomenon will be presented.

Ofer Zeitouni (Rehovot and Minneapolis): The Wyner model of communication and sparse random matrices.

The Wyner model of cellular communication has cells arranged in a line, with receiving antennas connected to a single cell and its d neighbors. Under natural assumptions, the capacity involves the density of states of large Jacobi matrices, and hence is related via Thouless' formula in the strip to certain Lyapunov exponents. From this, some assymptotics of the capacity can be deduced. (Joint work with N. Levy and S. Shamai)