The most basic capacity and error exponent questions of information theory can be expressed in terms of random geometric objects living in Euclidean spaces with dimensions tending to infinity. This approach was introduced by Venkat Anantharam and François Baccelli to evaluate random coding error exponents in channels with additive stationary and ergodic noise. More generally, the analysis of stochastic geometry in the Shannon regime leads to new high dimension stochastic geometry questions that are currently investigated.
A compatible point-shift f maps, in a translation invariant way, each point of a stationary point process N to some point of N. It is fully determined by its associated point-map, g, which gives the image of the origin by f. The initial question studied by Mir-Omid Haji-Mirsadeghi and François Baccelli is whether there exist probability measures which are left invariant by the translation of -g. The point map probabilities of N are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to the initial question. Point-map probabilities are shown to be a strict generalization of Palm probabilities: when the considered point-shift f is bijective, the point-map probability of N boils down to the Palm probability of N. When it is not bijective, there exist cases where the point-map probability of N is absolutely continuous with respect to its Palm probability, but there also exist cases where it is singular with respect to the latter.
Each such point-shift defines a random graph on the points of the point process. The connected components of this graph can be split into a collection of foils, which are the analogue of the stable manifold of the point-shift dynamics.
The same authors give a general classification of point-shifts in terms of the cardinality of the foils of these connected components. There are three types: F/F, I/F and I/I as shown in the paper Point-Shift Foliation of a Point Process. Using the framework of Günter Last, James Murphy has extended the cardinality classification to the case of point processes on unimodular groups. Murphy has studied point-shifts of point processes on topological groups at length.
Anup Biswas and François Baccelli studied the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, they showed that the shot-noise field can be scaled suitably to have a non degenerate alpha-stable limit, as the intensity of the underlying point process goes to infinity. More precisely, finite dimensional distributions converge and the finite dimensional distributions of the limiting random field have i.i.d. stable random components. This limit is hence called the alpha- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fréchet white noise field.
Spatial point processes involving birth and death dynamics are ubiquitous in networks. Such dynamics are particularly important in peer-to-peer networks and in wireless networks. In the paper “Mutual Service Processes in R^d, Existence and Ergodicity”, Fabien Mathieu (Bell Laboratories), Ilkka Norros (VTT) and François Baccelli proposed a way to analyze the long term behavior of such dynamics on Euclidean spaces using coupling techniques. This line of though is continued by Mayank Manjrekar on other classes of spatial birth and death processes.
Ngoc Mai Tran and François Baccelli derived a tropical version of the result of Kac on the zeros of polynomials with random coefficients (Zeros of Random Tropical Polynomials, Random Polygons and Stick-Breaking).
The common roots of tropical of a class of random polynomials in two variables is analyzed in a recent work by the same authors in Iterated Gilbert Mosaics and Poisson Tropical Plane Curves using a stochastic geometry approach based on iterated random tessellations.