Concentration and Cumulants for Stabilizing Functionals on Point Processes

• Applicant: Professor Dr. Christoph Thäle
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 443779210

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2265: Random geometric systems