Section: Research Program

Stochastic Geometry

Stochastic geometry  [40] is a rich branch of applied probability which allows one to quantify random phenomena on the plane or in higher dimension. It is intrinsically related to the theory of point processes. Initially its development was stimulated by applications to biology, astronomy and material sciences. Nowadays it is also widely used in image analysis. It provides a way of estimating and computing “spatial averages”. A typical example, with obvious communication implications, is the so called Boolean model, which is defined as the union of discs with random radii (communication ranges) centered at the points of a Poisson point process (user locations) of the Euclidean plane (e.g., a city). A first typical question is that of the prediction of the fraction of the plane which is covered by this union (statistics of coverage). A second one is whether this union has an infinite component or not (connectivity). Further classical models include shot noise processes and random tessellations. Our research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory.

F. Baccelli, B. Blaszczyszyn in collaboration with M. Karray (Orange Labs) are preparing a new book focusing on the mathematical tools at the basis of stochastic geometry. The book will cover the main mathematical foundations of the field, namely the theory of point processes and random measures as well as the theory of random closed sets. The basis will be the graduate classes and the research courses taught by the authors at a variety of places worldwide.

The collaboration of F. Baccelli with V. Anantharam (UC Berkeley) continues in new directions on high dimensional stochastic geometry, primarily in relation with Information Theory, cf. Section 7.23.

The collaboration of B. Blaszczyszyn with D. Yogeshwaran (Indian Statistical Institute) and Y. Yukich (Lehigh University) led to the development of the limit theory for geometric statistics on general input processes, cf. Section 7.22.