**Ergodic learning of point processes**

Almost surely, any infinite realization of an ergodic point process, say in Euclidean space, makes it possible to completely characterize the distribution of the point process and thus (in principle) to sample from this distribution new realizations. In practice, we only see a partial realization in a finite window. If this window (the number of points init) is large enough, can we come up with an approximation of the unknown original distribution and use it to sample new realizations? Inspired by recent advances in gradient descent methods for maximum entropy models, ina joint work A. Brochard, S. Mallat, and S. Zhang, B. Blaszczyszyn proposed a method to generate similar point patterns by jointly moving particles of an initial Poisson process realization towards a target counting measure. The overall quality

of this model is evaluated on point processes with various geometric structures through spectral and topological data analysis, compared in particular to Tscheschel, Stoyan~(2006).

- A. Brochard, B., Błaszczyszyn, S. Mallat, and S. Zhang, Particle gradient descent model for point process generation Statistics and Computing volume 32, Article number: 49 (2022).

Many specific problems ranging from theoretical probability to applications in statistical physics, combinatorial optimization and communications can be formulated as an optimal tuning of local parameters in large systems of interacting particles. Using the framework of stationary point processes in the Euclidean space, Bartłomiej Błaszczyszyn and Christian Hirsch (Bernoulli Institute, University of Groningen, The Netherlands) posed this problem as an optimal stationary marking of a given stationary point process. The quality of a given marking is evaluated in terms of scores calculated in a covariant manner for all points in function of the proposed marked configuration. In the absence of total order of the configurations of scores, one identifies intensity-optimality and local optimality as two natural ways for defining optimal stationary marking. One derives tightness and integrability conditions under which intensity-optimal markings exist and further stabilization conditions making them equivalent to locally optimal ones. Several motivating examples are discussed together with various possible approaches leading to uniqueness results.

- Bartłomiej Błaszczyszyn, Christian Hirsch “Optimal stationary markings”, Stochastic Processes and their Applications Volume 138, August 2021, Pages 153-185.

**Limit theory for asymptotically de-correlated dynamic spatial random models**

In an ongoing work with D. Yogeshwaran and J. E. Yukich, B .Blaszczyszyn considers statistics of spatial random models evolving over a fixed time domain and which are asymptotically de-correlated over spatial domains. They establish the limit theory (LLNm CLT) of these statistics as the spatial domain increases up to R^d. The models involve three sources of randomness, namely the random collection of particle locations (sites), the initial states (covariates), and the particle system evolution. They adopt the language of particle systems, but the set-up is general and yieldsthe limit theory for statistics of

other stochastic geometric structures such as interacting diffusions, geostatistical models, empirical random fields and Gibbsian models.

**Nash equilibrium structure of Cox process Hotelling games**

Venkat Anantharam (UC Berkeley) and François Baccelli studied an N-player game where a pure action of each player is to select a non-negative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of its open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and comprised of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

- V. Anantharam and F. Baccelli, ‘Nash equilibrium structure of Cox process Hotelling games’, Advances in Applied Probability, Applied Probability Trust, volume 54, pages 570-598, (June 2022). DOI : 10.1017/apr.2021.45. URL : https://hal.science/hal-03107798

Consider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. This note by F. Baccelli and S. Kalamkar discusses the properties of two stationary point processes associated with the latter and depending on a parameter theta. The first one is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and are such that the angle with which they see the two nuclei defining the facet is theta. The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here is about the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. The paper gives answers to these two questions and briefly discusses their practical motivations. It also discusses natural extensions to dimension three.

- F. Baccelli and S. Kalamkar, “On Point Processes Defined by Angular Conditions on Delaunay Neighbors in the Poisson-Voronoi Tessellation”, Journal of Applied Probability, 58(4), 2021. https://arxiv.org/abs/2010.16116

**The stochastic geometry of unconstrained one-bit data compression**

A stationary stochastic geometric model was proposed by F. Baccelli and E. O’Reilly (Caltech) for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of ℝn or a stationary Poisson point process in ℝn. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension n to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding.

- F. Baccelli and E. O’Reilly “ The stochastic geometry of unconstrained one-bit data compression”,

Electronic Journal of Probability, 24(138): 1-27, 2019. https://arxiv.org/abs/1810.06095