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

Starting with a cardinal which is both subcompact and measurable, we produce a model in which \square_{\kappa,2} holds but \square_\kappa fails at a singular cardinal \kappa.  We will discuss several of the essential tools used, and also several ways in which this result may be extended.