Schedule

The intensity of a spatial point process is one of the first quantity of interest to estimate in presence of real-data. When no covariate is observed, non-parametric kernel estimation is routinely used but comes with some drawbacks: it adapts poorly to nonconvex domain, and the estimation is not consistent in an increasing domain asymptotic regime. When the intensity depends on observed covariates, most estimation methods are parametric. Non-parametric kernel estimation has been extended to this situation, but it appears to be efficient only for a few numbers of covariates, and provides no indication on the importance of each covariate, which is crucial for interpretation. 

In this talk, we show how to adapt random forest regression to circumvent these drawbacks and estimate non-parametrically the intensity of a spatial point process with or without covariates, while measuring the importance of each variable in the latter case. Our approach allows to handle non-convex domain together with a large number of covariates. From a theoretical side, we prove that in the case of purely random forests, our method is consistent in both infill and increasing domain asymptotic regime, and may achieve a minimax rate of convergence.

The anisotropy of 2D Gaussian random fields through their Lipschitz-Killing curvature densities

I will present a joint work with Agnès Desolneux (CNRS, Centre Borelli, ENS Paris-Saclay) on the geometry of excursion sets of smooth stationary Gaussian fields.

In this work we focus on the effect of anisotropy on their Lipschitz-Killing curvature densities (close from area, perimeter and Euler characteristics of excursion sets) and propose a new geometrical index of anisotropy, closely linked with isoperimetric inequalities.

Hyperuniformity characterizes random spatial structures that exhibit slower growth in variance at large scales compared to Poisson structures. Initially conceptualized in statistical physics by Torquato and Stillinger (2003), hyperuniform systems have garnered significant interest due to their unique position between perfect crystals, liquids, and glasses. These distinctive properties make them valuable for designing innovative materials and Monte Carlo numerical integration, and they have gained attention in various applied contexts, offering insights into phenomena ranging from DNA and the immune system to photoreceptors, urban systems, and cosmology.

Detecting and quantifying hyperuniformity is crucial across these diverse domains. Despite this, statistical tests for hyperuniformity have only recently gained attention. In joint work with F. Lavancier and G. Mastrilli, reported in arXiv:2407.16797, we address the challenge of estimating the "strength" of hyperuniformity, which is related to the exponent of the spectrum at zero wavelength, in a class of stationary point processes in Euclidean space. The core mathematical concept is that the variance of linear statistics, which are based on smooth and rapidly decreasing functions, grows in a way that explicitly involves this exponent. By leveraging the multivariate central limit theorem for many of these statistics, which involve orthogonal functions at different scales (wavelets), we obtain an asymptotically consistent estimator of the "strength" of hyperuniformity, based on a single realization of the point process, with explicit confidence intervals.

We investigate this approach through numerical simulations using point process models and a real dataset.

Random geometric complexes are simplicial complexes (high-dimensional graphs) whose vertices are generated by a random point process in a metric space. In this talk we will focus on the homology (cycles/holes in various dimensions) of these complexes. Our main results show that the lifetime distribution of homological cycles obeys a universal law, that depends on neither the support nor the original distribution of the point process. We will focus on the notion of “weak universality”, addressing Poisson or binomial processes. We will present the main universality statement and the key steps for proving it. In fact, we will show that this notion of universality applies in a much broader context to scale-invariant geometric functionals (for example, the degree distribution in the k-NN graph). In addition, we will briefly discuss “strong universality”, which applies for a much wider class of point-cloud distributions, and is currently an open conjecture.

In this talk, we consider the convex hull of the intersection of the $d$-dimensional unit ball with a homogeneous Poisson point process of intensity $\lambda$ in ${\mathbb R}^d$. We are interested in the behavior when $d$ is large of the support function of such polytope in an arbitrary direction. We identify three different regimes (sub-critical, critical, and supercritical) in terms of the intensity $\lambda(d)$ and derive in each regime the precise distributional convergence after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once and deduce partial counterparts for the radius-vector function of the polytope. This complements recent works due to Bonnet, Kabluchko and Turchi and to Bonnet and O'Reilly on the volume and typical facet height respectively and strengthens our understanding of the geometry of high-dimensional random polytopes.

This is joint work with Benjamin Dadoun (Université du Mans).

A graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on the distance between them. We derive bounds on the probability that the graph is fully connected by analysing key modes of disconnection. In particular, analytic expressions are given for the mean and variance of the number of isolated nodes, and a sharp threshold established for their occurrence. Bounds are also derived for uncrossed gaps, and it is shown analytically that uncrossed gaps have negligible probability in the scaling at which isolated nodes appear. This is in stark contrast to the hard RGG in which uncrossed gaps are the most important factor when considering network connectivity. (Joint work with Michael Wilsher and Carl Dettmann.)

In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order Lχ, while the transversal fluctuations, known as wandering, grow as Lξ . We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3/5 and χ = 1/5, or ξ = 7/10 and χ = 2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find ξ = 0.60 ± 0.01 and χ = 0.20 ± 0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β-skeletons and the Delaunay triangulation and are characterized by the values ξ = 0.70 ± 0.01 and χ = 0.40 ± 0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation χ = 2ξ − 1 is satisfied for all these models. These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

Starting from the points of a homogeneous Poisson point process in the plane, we let independent and identically distributed random continuous paths grow. Each path stops growing at time t if it intersects the trace of any other path up to that time. We let time go to infinity and then consider the union of all paths to investigate the connectivity properties of this set. Under mild assumptions, we show the absence of percolation provided that a certain loop condition is satisfied. Joint work with David Coupier and David Dereudre.

In this talk, I will describe a model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In this stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, the model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, I will discuss the large-window asymptotics of graph-theoretic and topological characteristics such as higher-order degree distribution, Betti numbers of the associated Dowker complex as well as simplex counts. In particular, for the latter quantity there are regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. I will conclude by a simulation study and an application to the collaboration network extracted from the arXiv dataset. 

This talk is based on joint work with M. Brun, P. Juhász, and M. Otto

We study a broad class of geometric random graph models, with scale-free degrees, under the condition that the number of edges exceeds the average. These models include many known models, such as Scale-free percolation, Boolean models with heavy-tailed radii and age-dependent connection models. We identify the upper bound for the large deviation probability of having more edges than expected and establish limit theorems for the empirical distribution of the degrees and edges lengths, under this rare event. Our work shows that the conditional graphs exhibit a condensation phenomenon. A fixed and finite number of vertices is responsible for the linear number of additional edges, while the other vertices retain a degree close to the expected value. We make this observation more explicit by showing that the distribution of edge lengths splits into a bulk and traveling wave part, each of which has an asymptotic positive fraction of the total mass.

Joint work with: Remco van der Hofstad, Céline Kerriou, Neeladri Maitra and Peter Mörters

The union of the particles of a stationary Poisson process of compact (convex) sets in Euclidean space is called Boolean model and is a classical topic of stochastic geometry. We consider Boolean models in hyperbolic space, where one takes the union of the particles of a stationary Poisson process in the space of compact (convex) subsets of the hyperbolic space. Geometric functionals such as the volume of the intersection of the Boolean model with a compact convex observation window are studied. In particular, the asymptotic behavior for balls with increasing radii as observation windows is investigated. Exact and asymptotic formulas for expectation, variances and covariances are shown and univariate and multivariate central limit theorems are derived. Compared to the Euclidean framework, some new phenomena can be observed.

Joint work with Günter Last and Matthias Schulte.

We consider a random connection model (RCM) driven by a stationary marked Poisson process whose intensity measure is scaled by a parameter $t\ge 0$. Important special cases are the Boolean model with general compact grains and the so-called weighted RCM. We prove that the infinite clusters are deletion stable, that is the removal of a Poisson point cannot split a cluster in two or more infinite clusters. We then show that this stability together with a natural irreducibility assumption implies uniqueness of the infinite cluster. This extends and unifies several results in the literature. An important ingredient of our proofs are differentiability and convexity properties of the cluster density which are of interest in their own right. Some of the main ideas come from a seminal paper by Aizenman, Kesten and Newman (1987), treating discrete percolation models. If time allows we will also discuss sharp phase transition.

The talk is mostly based on recent joint work with Mikhail Chebunin: https://arxiv.org/abs/2403.17762 

Geometric functionals of smooth Gaussian fields have statistical applications in different areas of science (e.g. medical imaging, cosmology etc). I will give an overview of one such application and the general theory behind it. This theory relies crucially on a 'locality' property of the functional which fails to hold for many natural topological functionals. In the final part of the talk, I will describe some recent progress in extending classical results for geometric functionals to this non-local setting.

We study the asymptotic behaviour of a size-marked point process of centres of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension $d \ge 2$. The sizes of the cells are measured by their inradius or their $k$th intrinsic volume ($k \ge 2$), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalise a result in Chenavier and Hemsley (2016) in various directions.

The random geometric graph (RGG) is obtained by placing $n$ vertices uniformly at random in a  bounded region of  $R^d$ and connecting any two vertices distant at most $r$ apart.  We discuss large-$n$ asymptotics with $r= r_n$ a specified sequence.

Given a positive integer $k$, let $S_{n,k}$ be the number of components of  order $k$ in this graph, and let $S_n := \sum_k S_{n,k}$, the total number of components.  Let $L_n = \max\{k: S_{n,k} >0\}$, the order of the largest component.

In the `thermodynamic limit' where $ n r_n^d \to c \in (0,\infty)$, a law of large numbers and central limit theorem were already known for $S_{n,k}$, and for $S_n$, and for $L_n$.  We discuss newer  results of this type in the `dense  limit' where $nr_n^d \to \infty$ slowly.

In a related result, we determine the large-$\lambda$ asymptotics for the probability that the origin lies in a cluster of order $k$ in a Poisson Boolean model with intensity $\lambda$.

This is joint work with Xiaochuan Yang.

The random connection model with weights is a spatial random graph whose vertices are given by the points of a marked stationary Poisson process. For each pair of vertices it is decided independently if they are connected by an edge and the corresponding probability is a function of their distance and their marks, the so-called connection function. Throughout this talk, we consider a particular choice of the mark distribution and the connection function such that the resulting random graph exhibits a scale-free behaviour. We study large degrees of vertices within an observation window and - in case that the intensity of the underlying Poisson process is sufficiently small - the sizes of large components belonging to the observation window. For increasing observation windows we show that the corresponding point processes converge to Poisson processes after suitable rescalings. In particular, we obtain the limiting distributions of the largest degree and the size of the largest component. The proofs rely on comparing the point processes of degrees and component sizes with the point process of the marks.

This talk is based on joint works with Chinmoy Bhattacharjee and Matthias Lienau.

In this talk, we present new bounds on the $d_2$ and $d_3$ distances between multivariate Poisson functionals and a limiting Gaussian distribution. The bounds involve $(2+\epsilon)$-moments of the so-called add-one costs, which are the differences to the functional induced when adding one point to the underlying Poisson measure. Applications of these bounds include quantitative CLTs for multivariate functionals of the random geometric graph.

We examine optimal matchings or transport between two stationary point processes, and in particular, from a point process to a lattice or Lebesgue measure, respectively. The main focus of the talk will be on the implication of hyperuniformity (reduced variance fluctuations in point processes) to optimal transport: in dimension 2 , we show that the typical matching cost has a finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are 𝐿2-perturbed lattices. Our method does not formally require assumptions on the correlation measure or the variance behavior, and it retrieves known sharp bounds for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems. The proof relies on the estimation of the optimal transport cost between point processes restricted to large windows for a well-chosen cost through their Fourier-Stieltjes transforms, related to their structure factor. This is a joint project with Raphael Lachieze-Rey.

Denote by $K_t$ the convex hull of $t$ i.i.d. random variables uniformly distributed in a smooth $d$-dimensional convex set $K$. When $d = 2$, it is shown that the growth process given by the space-time defect radius vector function for $K_t$ displays 1 : 2 : 3 scaling characterizing KPZ growth processes and converges to a two-parameter limit process given by a Hopf-Lax variational formula. The one point marginal distribution of the defect radius vector function asymptotically converges to an explicit limit distribution with Tracy-Widom tails, whereas in all dimensions the maximal defect radius vector function converges to a Gumbel distribution. 

This talk is based on joint work with Pierre Calka.