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Moritz Otto

Publications and source records attributed to Moritz Otto.

18 recordsLinked to original sources
arXiv2026

Discretization-free exact recovery in geometric community detection

Geometric community detection seeks to recover latent communities in networks where connectivity depends jointly on community structure and continuous spatial geometry. Existing exact-recovery approaches typically discretize the underlying space, which can impose restrictive structural assumptions on the connectivity functions. We develop a polynomial-time, discretization-free algorithm for exact recovery in the Geometric Hidden Community Model (GHCM), operating directly on the continuous geometry. Our method succeeds even when connectivity functions coincide on a nontrivial portion of their visibility range and when two communities can be distinguished only through their connectivity to a third community. We prove exact recovery under these weaker conditions and provide experiments showing that the algorithm succeeds beyond the theoretically guaranteed regime.

cs.SI↗
arXiv2026

Central limit theorem for linear eigenvalue statistics of random geometric graphs

Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric graph, obtained by placing vertices according to a Poisson point process and connecting two vertices whenever their Euclidean distance is less than a certain threshold. Despite their broad applicability, the spectral properties of such spatial models remain far less understood than those of classical random graph models, such as Erd\H{o}s-R\'enyi graphs and Wigner matrices. The main obstacle is the presence of spatial constraints, which induce highly nontrivial dependencies among edges, placing these models outside the scope of techniques developed for purely combinatorial random graphs. In this paper, we provide the first rigorous analysis of Gaussian fluctuations for linear eigenvalue statistics of random geometric graphs. Specifically, we establish central limit theorems for $\text{Tr}[\phi(A)]$, where $A$ is the adjacency matrix and $\phi$ ranges over a broad class of suitable (possibly non-polynomial) test functions. In the polynomial setting, we moreover obtain a quantitative central limit theorem, including an explicit convergence rate to the limiting Gaussian law. We further obtain polynomial-test-function CLTs for other canonical random spatial networks, including $k$-nearest neighbor graphs and relative neighborhood graphs. Our results open new avenues for the study of spectral fluctuations in spatially embedded random structures and underscore the delicate interplay between geometry, local dependence, and spectral behavior.

math.PR↗
arXiv2026

Normal and Poisson approximation for Gibbs point processes with pair potentials

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.

math.PR↗
arXiv2024

Poisson approximation of large-lifetime cycles

In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime features on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions $d \geq 2$ under suitable model conditions.

math.PR↗
arXiv2024

Random Connection Hypergraphs

In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes, and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of various graph-theoretic and topological characteristics, including higher-order degree distributions, Betti numbers of the associated Dowker complex, and simplex counts. In particular, for the latter quantity, we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation with a simulation study and an application to the collaboration network extracted from the arXiv dataset.

math.PR↗
arXiv2024

Couplings and Poisson approximation for stabilizing functionals of determinantal point processes

We prove a Poisson process approximation result for stabilizing functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing window size that the process of points with a large nearest neighbor distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbor distance in the Ginibre process, which turns out to be different from its analogue for independent points.

math.PR↗
arXiv2023

Cumulant method for weighted random connection models

In this paper, we derive cumulant bounds for subgraph counts and power-weighted edge length in a class of spatial random networks known as weighted random connection models. This involves dealing with long-range spatial correlations induced by the profile function and the weight distribution. We start by deriving the bounds for the classical case of a Poisson vertex set, and then provide extensions to $\alpha$-determinantal processes.

math.PR↗
arXiv2023

Compound Poisson process approximation under $\beta$-mixing and stabilization

We establish Poisson and compound Poisson approximations for stabilizing statistics of $\beta$-mixing point processes and give explicit rates of convergence. Our findings are based on a general estimate of the total variation distance of a stationary $\beta$-mixing process and its Palm version. As main contributions, this article (i) extends recent results on Poisson process approximation to non-Poisson/binomial input, (ii) gives concrete bounds for compound Poisson process approximation in a Wasserstein distance and (iii) illustrates the applicability of the general result in an example on minimal angles in the stationary Poisson-Delaunay tessellation. The latter is among the first (nontrivial) situations in Stochastic Geometry, where compound Poisson approximation can be established with explicit extremal index and cluster size distribution.

math.PR↗
arXiv2023

Normal approximation for Gibbs processes via disagreement couplings

This work improves the existing central limit theorems (CLTs) for geometric functionals of Gibbs processes in three aspects. First, we derive a CLT for weakly stabilizing functionals, thereby improving on the previously used assumption of exponential stabilization. Second, we show that this CLT holds for interaction ranges up to the percolation threshold of the dominating Poisson process. This avoids imprecise branching bounds from graphical construction. Third, by constructing simultaneous couplings of several Palm processes for Gibbs functionals, we provide a quantitative CLT in terms of Kolmogorov bounds for normal approximation. An important conceptual ingredient in these advances is the extension of disagreement coupling adapted to unbounded windows and to the comparison at multiple spatial locations.

math.PR↗
arXiv2023

Large deviations for hyperbolic $k$-nearest neighbor balls

We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.

math.PR↗
arXiv2022

Large nearest neighbour balls in hyperbolic stochastic geometry

Consider a stationary Poisson process in a $d$-dimensional hyperbolic space. For $R>0$ define the point process $\xi_R^{(k)}$ of exceedance heights over a suitable threshold of the hyperbolic volumes of $k$th nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius $R$ centred at a fixed point. The point process $\xi_R^{(k)}$ is compared to an inhomogeneous Poisson process on the real line with intensity function $e^{-u}$ and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum $k$th nearest neighbour ball with a limiting Gumbel distribution is derived.

math.PR↗
arXiv2021

Extremal behavior of large cells in the Poisson hyperplane mosaic

We study the asymptotic behavior of a size-marked point process of centers of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension $d \ge 2$. The sizes of the cells are measured by their inradius or their $k$th intrinsic volume ($k \ge 2$), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalize a result in Chenavier and Hemsley (2016) in various directions. Our proof is based on a general Poisson process approximation result that extends a theorem in Bobrowski, Schulte and Yogeshwaran (2021).

math.PR↗
arXiv2021

Limit laws for large kth-nearest neighbor balls

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability $k$th-nearest neighbor balls of $X_1,\ldots,X_n$. Our result generalizes Theorem 2. of [10], which refers to the special case $k=1$. Our proof is completely different since it employs the Chen-Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.

math.PR↗
arXiv2021

Disagreement coupling of Gibbs processes with an application to Poisson approximation

We discuss a thinning and an embedding procedure to construct finite Gibbs processes with a given Papangelou intensity. Extending the approach in Hofer-Temmel (2019) and Hofer-Temmel and Houdebert (2019) we will use this to couple two finite Gibbs processes with different boundary conditions. As one application we will establish Poisson approximation of point processes derived from certain infinite volume Gibbs processes via dependent thinning. As another application we shall discuss empty space probabilities of certain Gibbs processes.

math.PR↗
arXiv2020

Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry

We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall's problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Voronoi and -Delaunay mosaic. As a consequence, we establish Gumbel limits for the asymptotic distribution of concrete size functionals and specify the rate of convergence. This extends extreme value results from Calka and Chenavier (2014) and Chenavier (2014).

math.PR↗
arXiv2017

Tail dependence of recursive max-linear models with regularly varying noise variables

Recursive max-linear structural equation models with regularly varying noise variables are considered. Their causal structure is represented by a directed acyclic graph (DAG). The problem of identifying a recursive max-linear model and its associated DAG from its matrix of pairwise tail dependence coefficients is discussed. For example, it is shown that if a causal ordering of the associated DAG is additionally known, then the minimum DAG representing the recursive structural equations can be recovered from the tail dependence matrix. For the relevant subclass of recursive max-linear models, identifiability of the associated minimum DAG from the tail dependence matrix and the initial nodes is shown. Algorithms find the associated minimum DAG for the different situations. Furthermore, given a tail dependence matrix, an algorithm outputs all compatible recursive max-linear models and their associated minimum DAGs.

stat.ME↗