It is a classical result, by Dyson, that the behavior of the eigenvalues of a random unitary matrix following uniform measure tend, when the dimension goes to infinity, after a suitable scaling, to a random set of points, called adeterminantal sine-kernel process. By defining the model in all dimensions on a single probability space, we are able to show that the convergence stated above can occur almost surely. Moreover, in an article with K. Maples and A. Nikeghbali, we interpret the limiting point process as the spectrum of a random operator.

We consider the contact process on a random graph chosen with a fixed degree, power law distribution, according to a model proposed by Newman, Strogatz and Watts (2001). We follow the work of Chatterjee and Durrett (2009) who showed that for arbitrarily small infection parameter λ

In this talk we aim at emphasizing the role of information in financial markets (public information versus insider information). In particular, if the information about a particular event (as for instance the default event of a company) is incorporated into a pricing model, then by a change of the underlying filtration, one can compute risk premiums attached to particular events. We also show that modeling of the information leads eventually to modeling of dependencies.

We start with showing how rearrangement inequalities may be used in probabilistic contexts such as e.g. for obtaining bounds on survival probabilities in trapping models. This naturally motivates the need for a new rearrangement inequality which can be interpreted as involving symmetric rearrangements around infinity. After outlining the proof of this inequality we proceed to give some further applications to the volume of Lévy sausages as well as to capacities for Lévy processes.
(Joint work with P. Sousi and R. Sun)