Universite Paris Sud

Joint Work with Philippe G. LeFloch. We consider vacuum
spacetimes with two spatial Killing vectors and with initial data
prescribed on $T^3$. The main results that we will present concern the
future asymptotic behaviour of the so-called polarized solutions. Under
a smallness assumption, we derive a full set of asymptotics for these
solutions. Within this symetry class, the Einstein equations reduce to a
system of wave equations coupled to a system of ordinary differential
equations. The main difficulty, not present in previous study of similar
systems, is that, even in the limit of large times, the two systems do
not directly decouple. We overcome this problem by the introduction of a
new system of ordinary differential equations, whose unknown are
renormalized variables with renormalization depending on the solution of
the non-linear wave equations.

Abstract: Given a sequence of random variables X_n that converge toward a Gaussian distribution, by looking at the next terms in the asymptotic E[exp(zX_n)] = exp(z^2 / 2) (1+ ...), one can often state a principle of moderate deviations. This happens in particular for sums of dependent random variables, and in this setting, it becomes useful to develop techniques that allow to compute the precise asymptotics of exponential generating series. Thus, we shall present a method of cumulants, which gives new results for the deviations of certain observables in statistical mechanics:

- the number of triangles in a random Erdos-Renyi graph;

- and the magnetization of the one-dimensional Ising model.

The talk will recall the results of three preprints, first two authored by my former student Mickael Launay, and the final coauthored by Mickael and myself. All three works are available on arXiv. At this point it is not clear that they will ever get published (or submitted for review) but hopefully this does not make their contents less interesting. This class of interacting urn processes was introduced in Launay's thesis, in an attempt to model more realistically the memory sharing that occurs in food trail pheromone marking or in similar collective learning phenomena. An interesting critical behavior occurs already in the case of exponential reinforcement. No prior knowledge of strong urns will be assumed, and I will try to explain the reason behind the phase transition.