Probabilistische Methoden für dynamische Kommunikationsnetzwerke

Publikationen

Artikel in Referierten Journalen

• M. Heida, B. Jahnel, A.D. Vu, Regularized homogenization on irregularly perforated domains, Networks and Heterogeneous Media, 20 (2025), pp. 165--212, DOI 10.3934/nhm.2025010 .
  Abstract
  We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

• J. Köppl, N. Lanchier, M. Mercer, Survival and extinction for a contact process with a density-dependent birth rate, Electronic Journal of Probability, 30 (2025), pp. 1--18, DOI 10.1214/25-EJP1335 .
  Abstract
  To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the d-dimensional integer lattice, with natural birth rate λ and natural death rate one. The process also depends on a payoff a₁₁ = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by λ times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0 ), we prove that, for all payoffs a , there is a phase transition from extinction to survival in the direction of λ. Using various block constructions, we also prove that, for all birth rates λ, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on λ . This underlines the importance of space (local interactions) and stochasticity in our model.

• P. Gracar, L. Lüchtrath, Ch. Mönch, Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension, Electronic Journal of Probability, 30 (2025), pp. 1--29, DOI 10.1214/25-EJP1399 .
  Abstract
  We consider a large class of inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights are preferred. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient, ?eff, measuring the influence of the degree-distribution on the long-range edges. We identify a sharp phase transition in ?eff for the existence of a supercritical percolation phase. In Gracar, Lüchtrath, Mörters 2021, it is shown that the soft Boolean model and the age-dependent random connection model are examples for models that have parameter regimes where there is no subcritical percolation phase. This paper completes these results and shows that in dimension one there exist parameter regimes in which the models have no supercritical phase and regimes with both super- and a subcritical phases.

• CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixed-degree nodes in weighted random connection models, Stochastic Processes and their Applications, 183 (2025), pp. 104593/1--104593/15, DOI 10.1016/j.spa.2025.104593 .
  Abstract
  We present a process-level Poisson-approximation result for the degree-$k$ vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.

• B. Jahnel, J. Köppl, B. Lodewijks, A. Tóbiás, Percolation in lattice k-neighbor graphs, Journal of Applied Probability, 62 (2025), pp. 1475--1492, DOI 10.1017/jpr.2025.15 .
  Abstract
  We define a random graph obtained via connecting each point of ℤ ^d independently to a fixed number 1 ≤ k ≤ 2d of its nearest neighbors via a directed edge. We call this graph the emphdirected k-neighbor graph. Two natural associated undirected graphs are the emphundirected and the emphbidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k=1 even the undirected k-neighbor graph never percolates, but the directed one percolates whenever k≥ d+1, k≥ 3 and d ≥5, or k ≥4 and d=4. We also show that the undirected 2-neighbor graph percolates for d=2, the undirected 3-neighbor graph percolates for d=3, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

• B. Jahnel, J. Köppl, Y. Steenbeck, A. Zass, The variational principle for a marked Gibbs point process with infinite-range multibody interactions, Electronic Journal of Probability, 30 (2025), pp. 1--32, DOI 10.1214/25-EJP1441 .
  Abstract
  We prove the Gibbs variational principle for the Asakura?Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a k-body interaction for fixed k. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.

• B. Jahnel, J. Köppl, On the long-time behaviour of reversible interacting particle systems in one and two dimensions, Probability and Mathematical Physics, 6 (2025), pp. 479--503, DOI 10.2140/pmp.2025.6.479 .
  Abstract
  By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett.

• B. Jahnel, J. Köppl, Time-periodic behaviour in one- and two-dimensional interacting particle systems, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, published online on 17.09.2025 (2025), DOI 10.1007/s00023-025-01624-5 .
  Abstract
  We provide a class of examples of interacting particle systems on $Z^d$, for $din1,2$, that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.

• L. Lüchtrath, Ch. Mönch, A very short proof of Sidorenko's inequality for counts of homomorphism between graphs, Bulletin of the Australian Mathematical Society, 113 (2025), pp. 10--14, DOI 10.1017/S000497272500019X .
  Abstract
  We provide a very elementary proof of a classical extremality result due to Sidorenko (Discrete Math. 131.1-3, 1994), which states that among all graphs G on k vertices, the k-1-edge star maximises the number of graph homomorphisms of G into any graph H.

• L. Lüchtrath, All spatial graphs with weak long-range effects have chemical distance comparable to Euclidean distance, Journal of Theoretical Probability, 39 (2025), pp. 1--18, DOI 10959-025-01467-0 .
  Abstract
  This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

Preprints, Reports, Technical Reports

• B. Jahnel, L. Lüchtrath, A.D. Vu, Oriented bond-site percolation in random environment and contact processes with periodic recovery, Preprint no. 3203, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3203 .
  Abstract, PDF (4664 kByte)
  We investigate oriented bond-site percolation on the planar lattice in which entire columns are stretched. Generalising recent results by Hilário et al., we establish non-trivial percolation under a (1+ ε)-th moment condition on the stretches and use this to prove survival of contact processes with periodic recoveries as well as in random environments.

• D. Kamecke, B. Jahnel, Phase transitions for the Widom--Rowlinson model in random environments, Preprint no. 3200, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3200 .
  Abstract, PDF (434 kByte)
  We establish non-uniqueness regimes for the infinite-volume two-colored Widom--Rowlinson model based on inhomogeneous Poisson point processes with locally finite intensity measures featuring percolation. As an application, we provide almost-sure phase-transition results for the Widom--Rowlinson model based on translation-invariant and ergodic Cox point processes with stabilizing and non-stabilizing directing measures.

• L. Lüchtrath, B. Jahnel, Ch. Mönch, Phase transitions for contact processes on sparse random graphs via metastability and local limits, Preprint no. 3199, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3199 .
  Abstract, PDF (322 kByte)
  We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with recent results by Nam, Nguyen and Sly (Trans. Am. Math. Soc. 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

• B. Jahnel, L. Lüchtrath, Ch. Mönch, Phase transitions for contact processes on one-dimensional networks, Preprint no. 3170, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3170 .
  Abstract, PDF (292 kByte)
  We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with integer indexed vertices that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (Ménard et al., Ann. Sci. Éc. Norm. Supér., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: First, for Gilbert graphs derived from stationary point processes on the real line marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Second, for independent Bernoulli long-range percolation on the integers, where the edge probabilities are given via a polynomial in the edges length', we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever the power of said polynomial is large than two. Although our approach utilises the restrictive topology of the line, we believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in higher dimensions.

• B. Jahnel, L. Lüchtrath, A.D. Vu, First contact percolation, Preprint no. 3164, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3164 .
  Abstract, PDF (506 kByte)
  We study a version of first passage percolation on Z^d where the random passage times on the edges are replaced by contact times represented by random closed sets on R. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models -- all calibrated to have one expected contact per day -- that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments.

Vorträge, Poster

• J. Köppl, Condensation with Optimal Rate and Excitation Spectrum for Bose Gases in the Gross-Pitaevskii Limit, Arbeitsgemeinschaft: Analysis of Many-body Quantum Systems, October 5 - 10, 2025, Mathematisches Forschungsinstitut Oberwolfach.

• J. Köppl, Spontaneous time-symmetry breaking in interacting particle systems, Oberseminar im Institut für Angewandte Analysis, Universität Ulm, Institut für Angewandte Analysis, Ulm, November 24, 2025.

• A.D. Vu, An oriented percolation model in random environment, Workshop on Stochastic Processes on Random Geometries, February 17 - 21, 2025, Technische Universiät Braunschweig, Institut für Mathematische Stochastik, February 20, 2025.

• A.D. Vu, Survival of an infection under dilutions in time an space, 17th German Probability and Statistics Days (GPSD), March 11 - 14, 2025, Technische Universität Dresden, March 12, 2025.

• A.D. Vu, Oriented percolation in random spatial environment, 44th Conference on Stochastic Processes and their Applications, July 14 - 18, 2025, Wroclaw University of Science and Technology and University of Wrocław, Faculty of Mathematics and Computer Science, Wrocław, July 17, 2025.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, International Scientific Conference on Mathematical Analysis and Dynamical Systems, online, May 20 - 21, 2025, University of Exact and Social Sciences, Uzbekistan, Tashkent, Uzbekistan, May 20, 2025.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Stat-Mech in Créteil 2025, June 10 - 11, 2025, Université Paris-Est Créteil, Créteil, France, June 10, 2025.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, SPP2265-Meeting on Stochastic Geometry and Point Processes, May 22, 2025, Universität Münster, Institut für Mathematische Stochastik, May 22, 2025.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Colloquium Mathematics, University of Groningen, Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Groningen, Netherlands, April 23, 2025.

• B. Jahnel, First Contact Percolation, Rhein-Main-Kolloquium Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, April 25, 2025.

• B. Jahnel, First contact percolation, Workshop ``Recent Advances in Evolving and Spatial Random Graphs'', June 2 - 4, 2025, Universität Augsburg, Institut für Mathematik, Bayrischzell, June 2, 2025.

• B. Jahnel, Probabilistic Methods for Dynamic Communication Networks, Seminar der Fakultät für Elektrotechnik und Informationstechnik, Technische Universität Dresden, Fakultät für Elektrotechnik und Informationstechnik, July 2, 2025.

• B. Jahnel, Subcritical annulus crossing in spatial random graphs, Seminar, Institut National de Recherche en Informatique et en Automatique (INRIA), Paris, France, June 11, 2025.

• B. Jahnel, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Probability Colloquium Augsburg--Munich, Universität Augsburg, Institut für Mathematik, January 24, 2025.

• L. Lüchtrath, What is...?? Geometric inhomogeneous random graphs, Random Geometric Systems, Fourth Annual Conference of SPP2265, June 23 - 26, 2025, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, June 23, 2025.

• L. Lüchtrath, Cluster sizes in subcritical soft Boolean models, 17th German Probability and Statistics Days (GPSD), March 11 - 14, 2025, Technische Universität Dresden, March 11, 2025.

• L. Lüchtrath, Existence of an extinction phase for the contact process on one-dimensional networks with heavy-tailed degree distribution, Workshop on Stochastic Processes on Random Geometries, February 17 - 21, 2025, Technische Universiät Braunschweig, Institut für Mathematische Stochastik, February 20, 2025.

• L. Lüchtrath, The contact process on one-dimensional scale-free networks, Workshop ``Recent Advances in Evolving and Spatial Random Graphs'', June 2 - 4, 2025, Universität Augsburg, Institut für Mathematik, Bayrischzell, June 4, 2025.

• L. Lüchtrath, The contact process on one-dimensional scale-free networks, Bath Probability Lectures 2025, September 16 - 19, 2025, Department of Mathematical Sciences, Bath, France, September 19, 2025.

• L. Lüchtrath, The contact process on one-dimensional scale-free networks, Probability Seminar, University of Leeds, School of Mathematics, Leeds, UK, September 25, 2025.

Preprints im Fremdverlag

• J. Bäumler, B. Jahnel, J. Köppl, B. Lodewijks, L. Reeves, A. Tóbiás, Local criteria for global connectivity comparisons: beyond stochastic domination, Preprint no. arXiv:2510.03934, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2510.03934 .
  Abstract
  We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

• G. Ghosh, B. Jahnel, Coexistence for Competing Branching Random Walks with Identical Asymptotic Shape on Zd, Preprint no. arXiv:2512.09153, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2512.09153 .
  Abstract
  We consider two independent branching random walks that start next to each other on the d-dimensional hypercubic lattice and that carry two different colors. Vertices of the lattice are colored according to the color of the walker cloud that first visits the vertex, leading to the question of possible coexistence in the sense that both colors appear on infinitely many vertices. Under mild conditions, we prove the coexistence for two independently distributed branching random walks obeying the same first- and second-order behavior for their extremal particles. To complement this result, we also exhibit examples for the almost-sure absence of coexistence, for d=1, in cases where the asymptotic shapes of the walker clouds are calibrated to coincide, thereby answering a question by Deijfen and Vilkas (ECP 28(15):1-11, 2023). As a main tool we employ second-order and large-deviation approximations for the position of the extremal particles in one-dimensional branching random walks.

• P.P. Ghosh, B. Jahnel, S. Steenbeck, Throughput in inhomogeneous planar drainage networks, Preprint no. arXiv:2506.18132, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2506.18132 .
  Abstract
  We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of Z, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length.

• CH. Hirsch, B. Jahnel, P. Juhász, Functional limit theorems for edge counts in dynamic random connection hypergraphs, Preprint no. arXiv:2507.16270, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2507.16270 .
  Abstract
  We introduce a dynamic random hypergraph model constructed from a bipartite graph. In this model, both vertex sets of the bipartite graph are generated by marked Poisson point processes. Vertices of both vertex sets are equipped with marks representing their weight that influence their connection radii. Additionally, we assign the vertices of the first vertex set a birth-death process with exponential lifetimes and the vertices of the second vertex set a time instant representing the occurrence of the corresponding vertices. Connections between vertices are established based on the marks and the birth-death processes, leading to a weighted dynamic hypergraph model featuring power-law degree distributions. We analyze the edge-count process in two distinct regimes. In the case of finite fourth moments, we establish a functional central limit theorem for the normalized edge count, showing convergence to a Gaussian AR(2)-type process as the observation window increases. In the challenging case of the heavy-tailed regime with infinite variance, we prove convergence to a novel stable process that is not L´evy and not even Markov.

• Y. Steenbeck, A. Zass, J. Köppl, B. Jahnel, Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties, Preprint no. arXiv:2508.21196, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2508.21196 .
  Abstract
  We consider continuous-time birth-and-death dynamics in Rd that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure