The PoPoPoP team is using point processes to create more efficient networks

Point processes bringing together mathematicians from Lille

If you were asked to draw dots at random in a circle, you will instinctively space them out fairly evenly. However, if this were done randomly, some points would form clusters by ending up very close to each other, or conversely, would leave large empty areas with no apparent structure. No dependence or regularity here! This is known as the Poisson point process in random geometry. 

“It is this notion of point processes that is the scientific basis behind the Inria PoPoPoP (Point Processes from Probability and Physics) project team”, says Mylène Maïda, professor of mathematics at the Paul Painlevé laboratory at the University of Lille. The team is made up of mathematicians from the Paul Painlevé Laboratory and the Institut Mines-Télécom (IMT) Nord Europe, and is the result of the merger of two working groups, one focusing on random geometry, initiated 20 years ago by Emeritus Professor Youri Davydovand the other on random matrices, created around ten years ago.  “Since our research areas overlapped to a large extent, we decided it would be scientifically relevant to come together in the same team,” continues the researcher. The advantage  is that this broadens our fields of research, providing better coverage of subjects relating to point processes.”

Modeling dependency

The PoPoPoP team’s ambition is to design and analyse new probabilistic models. Traditionally, the mathematical objects that are studied are independent of each other. This independence is now well understood by researchers and illustrated by the famous bell-shaped Gaussian curve, for example.

Although this dependency is difficult to handle mathematically, it has the advantage of producing more realistic models. This approach therefore offers a compromise between a deterministic system, perfectly organised like a crystal, and a more disordered system. By studying this dependency, the PoPoPoP researchers can adjust the degree of randomness to create better organised, more “rigid” and more efficient mathematical models. Above all, these models are proving more relevant for various applications where randomness is essential: 5G networks, numerical integration and metamaterials.

More efficient telecommunications networks 

Telecommunications networks are one of PoPoPoP’s key areas of exploration. The use of random geometry and point processes has long contributed to optimising telecoms coverage across a given area. But until now, models have been static because antennas are fixed. 

5G is a game-changer, as David Coupier, Head of PoPoPoP and professor at IMT, explains: “It introduces a new Device-to-Device (D2D) functionality. The principle is that two devices, such as smartphones or IoT sensors, can communicate directly without going through a relay antenna, provided they are close enough to each other.” For this user-to-user network to work, the signal must be able to propagate from one device to another.

This propagation of information is described by percolation theory. Just as boiling water passes through the percolator filled with ground coffee in an Italian coffee maker, the signal passes through a medium in real time, carried by a multitude of moving points. 

And that’s where PoPoPoP comes in, translating these dynamics into mathematical models. This is a complex challenge, but the promise of efficient telecommunications networks driven by the users themselves is well worth the effort!

More precise numerical integration algorithms

Another field of application explored by the team is the improvement of numerical integration algorithms, used to give approximate values of large integrals. In practical terms, numerical integration is essential in a variety of applications, such as engineering and environmental sciences.  By improving these algorithms, we can increase the accuracy of our estimates and use computing resources more sparingly. “The Monte Carlo method is the classic method for these algorithms,” says Mylène Maïda. “It is based on the random drawing of independent points.” Although this approach works, it is subject to significant fluctuations and is risky in terms of accuracy. For example, when two points fall in almost the same place, the second point does not provide any additional information. Conversely, entire areas may be empty, omitting potentially important values.

One PoPoPoP project involves using dependent point processes, such as the Ginibre point process, to select points that are better distributed in space. Through their mutual interaction, these processes significantly improve the accuracy and reliability of numerical integral approximations. Within the team, a thesis is currently being written in collaboration with Rémi Bardenet, research director at the CNRS and the CRIStAL laboratory, on the quantification of these approximations. The medium-term objective is to offer Python libraries integrating these dependent processes for certain numerical integration tasks.

Metamaterials with controlled properties

The team’s latest area of research, metamaterials, is still in its infancy. “These are artificial materials with microscopic patterns that offer physical properties that no natural material possesses,” reveals Mylène Maïda. “These properties radically change the way light, sound and electromagnetic waves propagate. The most striking example is the “invisibility cloak!” By enveloping an object in a metamaterial capable of deflecting optical waves, light bypasses it and comes out the other side as if it hadn’t been blocked. The object disappears from view. While the invisibility cloak remains a novelty in the laboratory today, the concept could, for example, be used to guide electromagnetic waves in medical imaging devices to detect tumours and neurodegenerative diseases at an earlier stage.

Specifically, the PoPoPoP researchers are working to distribute tiny elements called resonators in a way that enables the material to absorb waves. This organisation must be precise: if it is too irregular, the material will not absorb waves sufficiently; if it is too regular, it will only absorb a limited frequency band. Between the two, the specific processes studied by the team offer an interesting alternative, with resonators that repel each other slightly without becoming completely ordered, giving the metamaterials the desired properties.

“Our goal is to integrate dependency into randomness, where independence is not sufficient for modelling,” says David Coupier. “Dependency provides a better representation of reality, improves mathematical models and paves the way for technological innovations.” In other words, the random component of all these systems is not to be replaced, but organised and controlled.