Topological data analysis provides insights to data sets from a variety of different sources by capturing its most salient properties via refined topological features. Since the mathematical field of topology is devoted to describing invariants of objects not depending on the choice of a precise metric, these features are robust against small pertubations or different embeddings of the data. One of the most classical topological invariants are the Betti numbers, which capture the number of k-dimensional holes. Topological data analysis refines this concept substantially by constructing filtrations of the space and tracing when features appear and disappear. This information is summarized in the Persistence Diagram.
Together with Christophe Biscio, Nicolas Chenavier and Anne Marie Svane, we develop goodness-of-fit tests for point processes based on the persistence diagram.

In 1934, Eugene Wigner introduced his landmark model of a gas of electrons in matter. A finite number of electrons are distributed in the Euclidean space and are subject to repulsive Coulomb forces. To confine the particles in a bounded domain, Wigner assumes the existence of a uniform background charge, approximating contributions from positive atoms. This ensures overall neutrality. The Feynman-Kac formalism makes it possible to express the quantum-mechanical statistics -- i.e., Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac -- in terms of expectation with respect to Brownian bridges subject to an interaction governed by a time-integration of the potential energy above.
Together with Paul Jung and Sabine Jansen, we derive a level-3 large deviation principle for the associated Gibbsian process on an elongated strip.

Spatial preferential attachment networks were introduced by Emmanuel Jacob and Peter Mörters to illustrate how embedding a complex network in a Euclidean space can induce a positive clustering coefficient. More precisely, the nodes $X_1,X_2,\ldots$ are distributed independently and uniformly on the unit torus and as a new node $X_i$ is born at time $T_i$, it connects to any older node $X_j$ independently with probability
$$\phi\Big(\frac{T_i\,|X_i-X_j|^2}{\gamma\,\mathsf{deg}_{T_i-}(X_j)}\Big),$$
where $\deg_{T_i-}(X_j)$ denotes the in-degree of $X_j$ at time $T_i-$. The birth times are uniformly on the interval $[0, n]$ and the parameter $\gamma>0$ controls the strength of the bias towards . Moreover, $\varphi:[0,\infty) \to [0,1]$ is a decreasing *profile function* whose integral is normalized to 1/2. For the plotting, we choose a power-law decay with exponent $\delta>1$.
Together with Christian Mönch, we analyze shortest distances in this network and work on proving a large deviation principle for the neighborhood structure.

Spatial PAM

In the paper Strongly reinforced PĆ³lya urns with graph-based competition, Remco van der Hofstad, Mark Holmes, Alexey Kuznetsov, and Wioletta Ruszel introduce a dynamical random network that provides a basic interacting-particle based model for learning in neural networks. Neurons fire randomly and the weight of one of the incident synapses is incremented. The selection of the synapse is proportional to its weight to a power $\alpha > 0$. Depending on whether $\alpha$ is larger, equal or smaller than 1, the process is in the superlinear, linear or sublinear regime. Together with Mark Holmes and Victor Kleptsyn, we study the model on countable graphs and investigate whether it is possible to arrive at a percolating but sparse network of relevant edges.

sublinear linear superlinear

The Poisson lilypond model is a model for a hard-core germ-grain system introduced by Olle Häggström and Ronald Meester in the paper Nearest neighbor and hard sphere models in continuum percolation. From each grain from a homogeneous Poisson point process a grain starts growing and the growth stops once grains get into contact. This model has been generalized by Sven Ebert and Günter Last to random birth times and by Daryl Daley, Sven Ebert and Günter Last to line-segment based grains. If the line segments are directed along the coordinate axis, I showed absence of percolation. Together with Gary Delaney and Volker Schmidt, we also considered a model with random birth times that gives rise to a fractal germ-grain model.

Watch the grains grow!

Classic model Random initial times Line segments

In classical cellular wireless networks, users communicate with a base station via direct message exchange. In order to offload some traffic from the base station or to extend the coverage of a cell, network operators have used the possibility to install certain fixed relays whose purpose is to forward messages from the base station to the user (and vice versa). In the setting of the Leibniz group on Probabilistic methods for mobile ad-hoc networks my colleagues and I are investigating new technological possibilities, where the mobile users can serve as relays themselves. In our preprint Large deviations in relay-augmented wireless networks we derive a large deviation principle for the family of frustrated transmitters in a high-density regime.

Random waypoint model