### March 7 - 9, 2022

# Workshop on

Analysis and Geometry

of Point Processes

### [Bielefeld University]

## Workshop on Analysis and Geometry of Point Processes

in Bielefeld, Germany

in Bielefeld, Germany

Point processes are one of the main building blocks in the description of many spatial random systems. Therefore, they are ubiquitous in the mathematical modelling of various phenomena in physics, biology, material sciences, and many more. However, the analysis of point processes often poses severe challenges. Only few tools are available to date which can be used to tackle different classes of point processes.

This workshop will gather recent developments in analytical or geometric aspects of point processes theory, such as functional inequalities, asymptotic large scale behavior, or intrinsic geometric descriptions. By bringing together scientists from these areas we aim to stimulate new interactions and foster collaborations between these different groups.

## List of Speakers

Bence Borda (Graz University of Technology)

**A smoothing inequality for the Wasserstein metric on compact manifolds**

We present a Berry-Esseen type smoothing inequality for the quadratic Wasserstein metric on compact Riemannian manifolds, which estimates the distance between two probability measures in terms of their Fourier transforms. The inequality is sharp, and has a wide range of applications in probability theory and number theory. We discuss sharp convergence rates of the empirical measure of an i.i.d. or stationary weakly dependent sample, complementing recent results of Bobkov and Ledoux on the unit cube, and Ambrosio, Stra and Trevisan on compact manifolds. We also estimate the convergence rate of random walks on compact groups to the Haar measure, and establish the functional CLT and the functional LIL for additive functionals. On compact semisimple Lie groups these hold even without a spectral gap assumption. As an application to finite point sets arising in number theory, we show that a classical construction of Lubotzky, Phillips and Sarnak on SU(2) and SO(3) achieves optimal rate in the quadratic Wasserstein metric.

Codina Cotar (London)

**Equality of the Jellium and Uniform Electron Gas next-order asymptotic terms for Coulomb and Riesz potentials**

We consider the sharp next-order asymptotics problems for: (1) the minimum energy for optimal N-point configurations; (2) the N-Marginal Optimal Transport; and (3) the Jellium problem for N-point configurations, in all three cases with Riesz costs with inverse power-law long-range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, the second appears as a semiclassical limit of DFT energy, modelling a quantum version of the same system (and is called Uniform Electron Gas in the physics literature), and the third describes charges in a uniform negative background, a rough model for electrons in a metal. Recently the second-order terms in the large-N asymptotic expansions for power s in dimension d were shown for: (1) for \max(0,d-2)\le s<d (remaining open outside this range prior to our paper, as previous methods break down); and for (2) for 0< s<d. The asymptotics expansion for (3) has long been known for s=d-2, but it has been otherwise open until now.

In the present work, we extend the sharp asymptotics for: 1) to 0< s<\max(0,d-2); and for 3) to 0< s<d. Our paper's unified proof for these sharp asymptotics for 0< s<d is based on a new and robust screening procedure, which allowed a series of improvements on the existing theory. Our methods and results are extendable to other potentials with long-range and short-range interaction.

Moreover, we show here for the first time that for inverse-power-law interactions with power 0<s<d, the second-order terms for these three problems are equal. For the Coulomb cost in d=3, our result was the first to verify the physicists' long-standing conjecture regarding the equality of the second-order terms for Jellium and Uniform Electron Gas. Moreover, if the crystallization hypothesis in d=3 holds, which is an extension of Abrikosov's conjecture originally formulated in d=2, then our result is the first to verify the physicists' conjectured 1.4442 lower bound on the famous Lieb-Oxford constant. Our work rigorously confirms some of the predictions formulated by physicists, regarding the optimal value of the Uniform Electron Gas second-order asymptotic term.

Additionally, we show that on the whole range s\in(0,d), the Uniform Electron Gas second-order constant is continuous in s.

David Dereudre (Lille)

**Number-Rigidity and $\beta$-Circular Riesz gas**

For an inverse temperature $\beta>0$, we define the $\beta$-circular Riesz gas on $R^d$ as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential $g(x) = \Vert x \Vert^{-s}$. We focus on the non integrable case $d-1<s<d$. Our main result ensures, for any dimension $d\ge 1$ and inverse temperature $\beta>0$, the existence of a $\beta$-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set $\Delta$ is a function of the point configuration outside $\Delta$. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations.

David García-Zelada (Paris)

**Electrostatic particles in an uncharged region**

We will be interested in a system at equilibrium of n electrostatic particles living in the plane and confined by a fixed background of opposite charge. As n goes to infinity this background attracts our particles so that the total charge is zero. It turns out that there are particles surviving this cancellation and living in the uncharged part of the plane. This can be rigorously proved for a specific temperature that makes the system integrable, and we will see that its description is related to the space of holomorphic functions in that region. This is joint work with Rapha\"el Butez, Alon Nishry and Aron Wennman arXiv:2104.03959.

Antoine Gloria (Paris)

** A functional-analytic version of hyperuniformity and its application to sedimentation**

Consider a suspension of rigid particles in a fluid in a large tank. When particles are heavier than the fluid, they sediment.

Defining an effective sedimentation speed has been an open problem in physics until the contribution of Batchelor in the 1970’s, who used stochastic cancellations (assuming positions of particles are weakly correlated) in form of a Debye screening effect to define this effective sedimentation speed (in the limit of large tanks) in dimensions 3 and larger.

Further formal calculations by Caflisch and Luke then suggested that the variance of the sedimentation should however blow up with the size of a tank, which is not observed in mechanical experiments. Later on Koch and Shaqfeh argued that large-scale order of the point process could yield stronger cancellations and help define a finite limiting variance.

In this talk I will present hyperuniform point processes as defined by Stillinger and Torquato, introduce a functional-analytic version thereof, and show how it helps to rigorously revisit the Batchelor, Caflisch-Luke and Koch-Shaqfeh arguments.

Anna Gusakova (Münster)

**Random simplicial tessellations**

A tessellation in $\mathbb{R}^d$ is a locally finite collection of convex polytopes, which cover the space and have disjoin interior. In this talk we consider three models of random simplicial tessellations ($\beta$-, $\beta'$- and Gaussian-Delaunay tessellations), whose construction is based on Poisson point process $\eta$ in $\mathbb{R}^{d+1}$. Each point of the process $\eta$ is assumed to have two coordinates of different nature, namely spacial coordinate $v\in\mathbb{R}^d$ and hight coordinate $h\in\mathbb{R}$. With respect to the coordinate $v$ the intensity measure of $\eta$ is just a Lebesgue measure and with respect to the coordinate $h$ the intensity measure has a density of the form $h^{\beta}$, $h^{-\beta}$ or $e^{h/2}$ for $\beta$-, $\beta'$- or Gaussian-model respectively. The corresponding simplicial tessellation arises as a dual to the Laguerre tessellation constructed for the process $\eta$.

Various properties of the above tessellations, starting from the description of the typical cells and finishing by the central limit theorems for the number of $k$-faces in a growing window will be presented. We will also consider the connection between these three models and the relation to classical Poisson-Delaunay tessellation.

Günter Last (Karlsruhe Institute of Technology)

**Invariant allocations of stationary diffuse random measures**

We consider two jointly stationary and ergodic random measures $\xi$ and $\eta$ on $\mathbb R^d$ with equal finite intensities and assume $\xi$ to be diffuse. We construct (invariant) allocations transporting $\xi$ to arbitrary $\eta$. If either $\xi$ or $\eta$ has a non-trivial discrete factor (e.g. if $\eta$ has a non-trivial discrete part), then our allocation is a function of the pair $(\xi,\eta)$ and does not require additional randomness. When that condition does not hold, we show by a counterexample that a factor allocation transporting $\xi$ to $\eta$ need not exist.

The talk is based on joint work with Hermann Thorisson.

Thomas Leblé (Paris)

**Two “physical” characterizations of the Sine-beta process**

The Sine-beta point process describes the limiting microscopic behavior of eigenvalues for certain random matrices. One can also see it as an infinite-volume Gibbs measure for a certain statistical physics system in dimension 1 (Dyson's “log-gas”). With this point of view, we give two "physical" characterizations: through DLR equations, and as the unique minimiser of a free energy functional at the level of point processes. Joint works with Dereudre-Hardy-Maïda and Erbar-Huesmann.

Giovanni Peccati (Luxembourg)

**Functional scaling limits of nodal intersections.**

We consider the random point process given by the nodal intersections of planar Gaussian random waves, and study its scaling limit over growing domains - with specific emphasis on the exact variance asymptotics and functional convergence of its chaotic components. One intriguing result is that the second chaotic component converges to a total disorder field (that is, to a random field containing uncountably many independent standard Gaussian random variables) indexed by closed curves. Such a field already appears in works by Lebowitz (1983) and Buckley and Sodin (2017).

Joint work with M. Notarnicola and A. Vidotto.

Matthias Schulte (Hamburg)

** Multivariate normal approximation of stabilising functionals of Poisson processes**

In this talk functionals of Poisson processes are studied which are sums of scores of the underlying points. It is assumed that the scores stabilise in the sense that the score of a point is determined by the points in a random neighbourhood given by a so-called radius of stabilisation. We consider a vector of sums of stabilising scores. For the situation that the radii of stabilisation decay exponentially fast and some moment assumptions are satisfied, quantitative bounds for the multivariate normal approximation are established. The results deal with several distances including a multivariate generalisation of the Kolmogorov distance and lead to optimal rates of convergence. Several examples concerning spatial random graphs will be discussed. The proofs of the main results rely on the Malliavin-Stein method for multivariate normal approximation and a careful analysis of the covariance structure.

This talk is based on joint work with J.E. Yukich (Lehigh University).

Christoph Thäle (Bochum)

**Typical cell of the Poisson-Voronoi tessellation in constant curvature geometries**

The Poisson-Voronoi tessellation is one of the most classical objects studied in stochastic geometry, and was mainly studied in the $d$-dimensional Euclidean space. The focus of this talk will be on the typical cell of the Poisson-Voronoi tessellation. Moreover generally, we consider in parallel the spherical and the hyperbolic model as well. By means of a connection to so-called beta polytopes we will be able to describe the distribution of these cells and to determine its expected f-vector.

Dario Trevisan (Pisa)

**On Minimal Spanning Trees for Random Euclidean Bipartite Graphs**

The minimum spanning tree (MST) problem is a combinatorial optimization problem with many applications, well beyond its historical introduction for network design. The study of its random instances on Euclidean models, e.g., on complete graphs obtained by sampling i.i.d. uniform points on a d-dimensional cube, is classical, with many limit results as the number of the points grows. In this talk, I will present two new results for its bipartite counterpart, i.e., with an additional colouring (red/blue) of the points and allowing connections only between different colours. First, we prove that the maximum vertex degree of the MST grows logarithmically, in contrast with the non-bipartite case, where a uniform bound holds, depending on d only -- a fact crucially used in many classical results. Despite this difference, we then argue that the cost of the MST, suitably normalized, converge a.s. to a limiting constant that can be represented as a series of integrals, thus extending a result of Avram and Bertsimas to the bipartite case and confirming a conjecture by Riva, Malatesta and Caracciolo. (Joint work with M. Correddu, Università di Pisa)

## Schedule

**Organizers:**

Matthias Erbar (Bielefeld University)

Martin Huesmann (Münster University)

Jonas Jalowy (Münster University)

The Workshop is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the SPP 2265 *Random Geometric Systems*.