## Concentration and Cumulants for Stabilizing Functionals on Point Processes

### Description

Many functionals of interest in stochastic geometry can be represented as sums of score functions over an underlying random point configuration, and the value of the score function at each point depends on the location of the point in space as well as on the local surrounding point configuration. The notion of stabilizing functionals formalizes this localziation idea. It has been initiated by Penrose and Yukich in the early 2000's and since then has been a very active and fruitful area of research. The theory of stabilizing functionals consist in establishing limit theorems as the intensity of the underlying point process tends to infinity.

In the proposed research program we develop further the theory of stabilizing functionals on the level of concentration inequalities and cumulant bounds. The objectives can be summarized as follows.

(a) We derive sharp cumulant bounds for so-called surface-order stabilizing functionals. We do this at first place for underlying Poisson point processes and generalize the approach in a second step to more general point processes models, for example to Gibbs point processes or to point processes with rapidly decaying correlations.

(b) We use Poisson analysis methods, especially modified log-Sobolev inequalities and covariance identities for exponential functions of Poisson point processes, to derive concentration inequalities for classical stabilizing functionals as well as for surface-order stabilizing functionals of Poisson point processes. We do this first under restrictive assumptions on the score function and compare different approaches by means of particular examples, such as the Gilbert graph, and relax the assumptions in a second step in order to include other stochastic geometry models as well.

(c) We investigate stochastic geometry models in non-Euclidean spaces as particular examples and applications of our general results. Especially, we deal with random geometric graphs and with functionals of Poisson-Voronoi tessellations in standard spaces of constant curvature +1, 0 or -1. We also develop the necessary integral-geometric tools in order to deal with the resulting geometric difficulties we are faced with in the non-Euclidean set-up.

(d) As a long-term goal we aim to prove Donsker-Varadhan-type large deviation principles for stabilizing functionals by refining and extending existing methods. In a second step these methods might be developed further to deal with surface-order stabilizing functionals as well. Again, our general results have consequences for specific models, especially for stochastic geometry models in non-Euclidean spaces.

### Preprints/Publications

Moritz Otto, Christoph Thäle: Large nearest neighbour balls in hyperbolic stochastic geometry (09/2022)

Christoph Thäle, Holger Sambale, Nils Heerten: Probabilistic Limit Theorems for the Coefficients of a Class of Root-Unitary Polynomials (06/2022)

Anna Gusakova, Johannes Heiny, Christoph Thäle: The volume of random simplices from elliptical distributions in high dimension (06/2022)

Zakhar Kabluchko, Daniel Rosen, Christoph Thäle: Fluctuations of λ-geodesic Poisson hyperplanes in hyperbolic space (05/2022)

Florian Besau, Daniel Rosen, Christoph Thäle: Random inscribed polytopes in projective geometries (05/2022) published

Carina Betken, Tom Kaufmann, Kathrin Meier, Christoph Thäle: Second-order properties for planar Mondrian tessellations (04/2022)

Anna Gusakova, Matthias Reitzner, Christoph Thäle: Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon (04/2022)

Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schütt, Christoph Thäle, Elisabeth Werner: Spherical convex hull of random points on a wedge (03/2022)

Carina Betken, Christoph Thäle: Approaching the coupon collector's problem with group drawings via Stein's method (02/2022)

Steven Hoehner, Ben Li, Michael Roysdon, Christoph Thäle: Asymptotic expected $T$-functionals of random polytopes with applications to $L_p$ surface areas (02/2022)

Carina Betken, Matthias Schulte, Christoph Thäle: Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes (11/2021) published

Thomas Godland , Zakhar Kabluchko, Christoph Thäle: Beta-star polytopes and hyperbolic stochastic geometry. (09/2021) published

Anna Gusakova, Zakhar Kabluchko, Christoph Thäle: The $\beta$-Delaunay tessellation IV: Mixing properties and central limit theorems. (08/2021)

Peter Eichelsbacher, Benedikt Rednoß, Christoph Thäle, Guangqu Zheng: A simplified second-order Gaussian Poincaré inequality in discrete setting with applications (08/2021)

Thomas Godland , Zakhar Kabluchko , Christoph Thäle: Random cones in high dimensions II: Weyl cones (06/2021) published

Anna Gusakova, Zakhar Kabluchko, Christoph Thäle: The β-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions (04/2021) published

Anna Gusakova, Zakhar Kabluchko, Christoph Thäle: The β-Delaunay tessellation II: The Gaussian limit tessellation (01/2021) published

Thomas Godland , Zakhar Kabluchko , Christoph Thäle: Random cones in high dimensions I: Donoho-Tanner and Cover-Efron cones (12/2020) published

Anna Gusakova, Christoph Thäle: On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems (11/2020) published

Anna Gusakova, Holger Sambale, Christoph Thäle: Concentration on Poisson spaces via modified Φ-Sobolev inequalities (09/2020) published

Felix Herold, Daniel Hug, Christoph Thäle: Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space? (11/2019) published