Évènements

L’orateur du seminaire donnera un exposé introductif sur les processus ponctuels de Gibbs avec interaction sommable (même à portée finie) basé sur le minicours https://arxiv.org/abs/1701.08105 

The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated to the so-called Poisson point process. In a first part, we present several aspects of finite volume GPP defined on a bounded window in Rd. In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space Rd. Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we will discuss in the talk. The DLR equations, the GNZ equations will be presented as well.

In this talk, we discuss the question of Gibbs point processes in R^d with pairwise interactions that are not integrable at infinity. A standard example is the Riesz potential of the form g(x)=1/|x|^s where s<d. This setting has a long history, notably because the case s=d-2 corresponds to the classical Coulomb potential, which arises from electrostatic theory. We will first address the existence of the process in the infinite volume regime when a neutralizing background is introduced (this model is known as Jellium in theoretical physics). Subsequently, we will discuss the rigidity of such point processes, specifically hyper-uniformity and number rigidity. We will provide a state-of-the-art review and present numerous conjectures and open problems.