RESEARCH INTERESTS

• Stein's Method

• Random Graphs

• Malliavin Calculus

• Poisson Measures

• Central Limit Theorems

These are four types of graphs that share two characteristics: they have been built on a Poisson process, and their construction is in some sense 'local', either because they look for their nearest spatial neighbour under some constraints, or because they only connect to points within a certain deterministic radius. Using the Malliavin-Stein method (in particular Last, Peccati, Schulte 2016, AAP), one can asses the behaviour of these graphs on a global scale by studying what happens locally when you add a point. In this way, we derive Central Limit Theorems. During my PhD, I studied what happened when you look at the sum of the edge-lengths to some negative power. In this case, the function only satisfies reduced moment conditions, and one needs a lower-moment version of the Malliavin-Stein method, see (T. 2025 AAP).

These graphs are equally interesting in non-Euclidean geometries. This is a nearest neighbour embracing graph (see Sambale, Thäle, T. 2024 arXiv), where a point keeps connecting to its closest neighbours until it is included in the convex hull of its neighbours. On the left, the graph is constructed in Euclidean space, on the right in the hyperbolic plane (represented in the Poincaré disk model). In both cases, we can show that the 'local construction' prevails in some sense, and derive quantitative central limit theorems of speed Var(F)^{-1/2}. The variances are of the order 'volume of the observation window' in which we observe the graph. If this window is a ball of radius r, then in the Euclidean case the speed of convergence is r^{-d/2}, while in the hyperbolic case it is e^{-(d-1)/2 r}.