Section: New Results

Point Processes, Stochastic Geometry and Random Geometric Graphs

Participants : François Baccelli, Bartłomiej Błaszczyszyn, Pierre Brémaud, Kumar Gaurav, Mir Omid Haji Mirsadeghi.

stochastic geometry, point process, shot-noise, Boolean model, random tessellation, percolation, stochastic comparison

Modeling, comparison and impact of spatial irregularity of point processes on coverage, percolation, and other characteristics of random geometric models

We develop a general approach for comparison of clustering properties of point processes. It is funded on some basic observations allowing to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving the void probabilities and moment measures, thus allowing to study the impact of clustering of the underlying point process. When stronger tools are needed, $dcx$ ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is concerned. The whole approach has been worked out in a series of papers  [62] , [63] , [64] , [65] . This year we have prepared revisions of the two latter ones, from which  [65] is now accepted for the publication in Adv. Appl. Probab. We have also prepared a review article [53] for Lecture Notes in Mathematics, Springer.

AB random geometric graphs

We investigated percolation in the AB Poisson-Boolean model in $d$-dimensional Euclidean space, and asymptotic properties of AB random geometric graphs on Poisson points in ${[0,1]}^d$. The AB random geometric graph we studied is a generalization to the continuum of a bi-partite graph called the $AB$ percolation model on discrete lattices. Such an extension is motivated by applications to secure communication networks and frequency division duplex networks. The AB Poisson Boolean model is defined as a bi-partite graph on two independent Poisson point processes of intensities $\lambda$ and $\mu$ in the $d$-dimensional Euclidean space in the same manner as the usual Boolean model with a radius $r$. We showed existence of $AB$ percolation for all $d\ge 2$, and derived bounds for a critical intensity. Further, in $d=2$, we characterize a critical intensity. The set-up for $AB$ random geometric graphs is to construct a bi-partite graph on two independent Poisson point process of intensities $n$ and $cn$in the unit cube. We provided almost sure asymptotic bounds for the connectivity threshold for all $c>0$ and a suitable choice of radius cut-off functions $r_n\left(c\right)$. Further for $c<c_0$, we derived a weak law result for the largest nearest neighbor radius. This work appeared in [27] .

Random Packing Models

Random packing models (RPM) are point processes (p.p.s) where points which "contend" with each other cannot be simultaneously present. These p.p.s play an important role in many studies in physics, chemistry, material science, forestry and geology. For example, in microscopic physics, chemistry and material science, RPMs can be used to describe systems with hard-core interactions. Applications of this type range from reactions on polymer chains, chemisorption on a single-crystal surface, to absorption in colloidial systems. In these models, each point (molecule, particle,$\cdots$) in the system occupies some space, and two points with overlapping occupied space contend with each other. Another example is the study of seismic and forestry data patterns, where RPMs are used as a reference model for the data set under consideration. In wireless communications, RPMs can be used to model the users simultaneously accessing the medium in a wireless network using Carrier Sensing Medium Access (CSMA). In this context, each point (node, user, transmitter,$\cdots$) does not occupy space but instead generates interference to other points in the network. Two points contend with each other if either of them generates too much interference to the other. Motivated by this kind of application, we studied in [66] the generating functionals of several models of random packing processes: the classical Matérn hard-core model; its extensions, the $k$-Matérn models and the $\infty$-Matérn model, which is an example of random sequential packing process. The main new results are: 1) A sufficient condition for the $\infty$-Matérn model to be well-defined (unlike the other two, the $\infty$-Matérn model may not be well-defined on unbounded space); 2) the generating functional of the resulting point process which is given for each of the three models as the solution of a differential equation; 3) series representation and bounds on the generating functional of the packing models; 4) moment measures and other useful properties of the considered packing models which are derived from their generating functionals.

Extremal and Additive Matérn Point Processes

In the simplest Matérn point processes, one retains certain points of a Poisson point process in such a way that no pairs of points are at distance less than a threshold. This condition can be reinterpreted as a threshold condition on an extremal shot–noise field associated with the Poisson point process. In a joint work with P. Bermolen [Universidad de la República, Montevideo, Uruguay] [60] , we studied extensions of Matérn point processes where one retains points that satisfy a threshold condition based on an additive shot–noise field of the Poisson point process. We provide an analytical characterization of the intensity of this class of point processes and we compare the packing obtained by the extremal and additive schemes and certain combinations thereof.

Spatial Birth and Death Point Processes

In collaboration with F. Mathieu [Inria GANG] and Ilkka Norros [VTT, Finland], we continued studying a new spatial birth and death point process model where the death rate is a shot noise of the point configuration. We showed that the spatial point process describing the steady state exhibits repulsion. We studied two asymptotic regimes: the fluid regime and the hard–core regime. We derived closed form expressions for the mean (and in some cases the law) of the latency of points as well as for the spatial density of points in the steady state of each regime. A paper on the matter will be presented at Infocom 13.

A population model based on a Poisson line tessellation

In [44] , we introduce a new population model. Taking the geometry of cities into account by adding roads, we build a Cox process driven by a Poisson line tessellation. We perform several shot-noise computations according to various generalizations of our original process. This allows us to derive analytical formulas for the uplink coverage probability in each case.

Information Theory and Stochastic Geometry

In a joint work with V. Anantharam [UC Berkeley], we study the Shannon regime for the random displacement of stationary point processes. We currently investigate Multiple Access Channels.

Navigation on Point Processes and Graphs

The thesis of Mir Omid Mirsadeghi [6] studied optimal navigations in wireless networks in terms of first passage percolation on some space-time SINR graph. It established both “positive” and “negative” results on the associated percolation delay rate (delay per unit of Euclidean distance, also called time constant in the classical terminology of percolation). The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that the percolation delay rate is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the percolation delay rate is positive and finite.

A new direction of research was initiated aiming at defining a new class of measures on a point process which are invariant under the action of a navigation on this point process. This class of measures has properties similar to Palm measures of stationary point processes; but they cannot be defined in the classical framework of Palm measures.