Publications

Articles in Refereed Journals

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Connectivity in mobile device-to-device networks in urban environments, IEEE Transactions on Information Theory, 69 (2023), pp. 7132--7148, DOI 10.1109/TIT.2023.3298278 .
    Abstract
    In this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model. Every device independently picks a target location on the street system using a general waypoint kernel, and travels to the target along the shortest path on the streets with an individual velocity. Then, any pair of devices becomes connected whenever they are on the same street in sufficiently close proximity, for a sufficiently long time. After presenting some general properties of the multi-parameter system, we focus on an analysis of the clustering behavior of the random connectivity graph. In our main results we isolate regimes for the almost-sure absence of percolation if, for example, the device intensity is too small, or the connectivity time is too large. On the other hand, we exhibit parameter regimes of sufficiently large intensities of devices, under favorable choices of the other parameters, such that percolation is possible with positive probability. Most interestingly, we also show an in-and-out of percolation as the velocity increases. The rigorous analysis of the system mainly rests on comparison arguments with simplified models via spatial coarse graining and thinning approaches. Here we also make contact to geostatistical percolation models with infinite-range dependencies.

  • C.F. Coletti, L.R. DE Lima, A. Hinsen, B. Jahnel, D.R. Valesin, Limiting shape for first-passage percolation models on random geometric graphs, Journal of Applied Probability, published online on 24.04.2023, DOI 10.1017/jpr.2023.5 .
    Abstract
    Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times.

  • O. Collin, B. Jahnel, W. König, A micro-macro variational formula for the free energy of a many-body system with unbounded marks, Electronic Journal of Probability, 28 (2023), pp. 118/1--118/58, DOI 10.1214/23-EJP1014 .
    Abstract
    The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose--Einstein condensation phase transition is sought for. We introduce a simplified version of the model with an organisation of the particles in deterministic boxes instead of Brownian cycles as the marks of a reference Poisson point process (for simplicity, in Z d, instead of R d). We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks. In the characteristic formula, each of the microscopic particles and the statistics of the macroscopic part of the configuration are seen explicitly; the latter receives the interpretation of the condensate. The formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Chase-escape in dynamic device-to-device networks, Journal of Applied Probability, published online in August 2023, DOI 10.1017/jpr.2023.47 .
    Abstract
    The present paper features results on global survival and extinction of an infection in a multi-layer network of mobile agents. Expanding on a model first presented in CHJW22, we consider an urban environment, represented by line-segments in the plane, in which agents move according to a random waypoint model based on a Poisson point process. Whenever two agents are at sufficiently close proximity for a sufficiently long time the infection can be transmitted and then propagates into the system according to the same rule starting from a typical device. Inspired by wireless network architectures, the network is additionally equipped with a second class of agents that is able to transmit a patch to neighboring infected agents that in turn can further distribute the patch, leading to a chase-escape dynamics. We give conditions for parameter configurations that guarantee existence and absence of global survival as well as an in-and-out of the survival regime, depending on the speed of the devices. We also provide complementary results for the setting in which the chase-escape dynamics is defined as an independent process on the connectivity graph. The proofs mainly rest on percolation arguments via discretization and multiscale analysis.

  • CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multi-scale infrastructure-augmented dynamic networks, Stochastic Models, 39 (2023), pp. 851--877, DOI 10.1080/15326349.2023.2184832 .
    Abstract
    We consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes illustrating that the limiting distribution depends sensitively on the scaling of the time horizon.

  • B. Jahnel, S.K. Jhawar, A.D. Vu, Continuum percolation in a nonstabilizing environment, Electronic Journal of Probability, 28 (2023), pp. 131/1--131/38, DOI 10.1214/23-EJP1029 .
    Abstract
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].

  • B. Jahnel, J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, The Annals of Applied Probability, 33 (2023), pp. 4570--4607, DOI 10.1214/22-AAP1926 .
    Abstract
    We consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.

  • B. Jahnel, J. Köppl, Trajectorial dissipation of Phi-entropies for interacting particle systems, Journal of Statistical Physics, 190 (2023), pp. 119/1--119/22, DOI 10.1007/s10955-023-03136-0 .
    Abstract
    A classical approach for the analysis of the longtime behavior of Markov processes is to consider suitable Lyapunov functionals like the variance or more generally Φ -entropies. Via purely analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite general assumptions on the process. We complement these classical results via a more probabilistic approach and show that dissipation is already present on the level of individual trajectories for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry and compact local spin spaces. This extends previous results from the setting of finite-state Markov chains or diffusions in Rn to an infinite-dimensional setting with weak assumptions on the dynamics.

  • B. Jahnel, Ch. Külske, Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 29 (2023), pp. 3013--3032, DOI 10.3150/22-BEJ1572 .
    Abstract
    We consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [0,1]. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large p, as it changes only a small fraction p(1-p)2d of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.

Contributions to Collected Editions

  • P. Gracar, L. Lüchtrath, Ch. Mönch, The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects, 18th International Workshop on Algorithms and Models for the Web-Graph, Toronto, Canada, May 23 - 26, 2023, M. Dewar, P. Prałat, P. Szufel, F. Théberge, M. Wrzosek, eds., 13894 of Lecture Notes in Computer Science, Springer, Cham, 2023, pp. 19--35, DOI 10.1007/978-3-031-32296-9_2 .
    Abstract
    We study the weight-dependent random connection model, a class of sparse graphs featuring many real-world properties such as heavy-tailed degree distributions and clustering. We introduce a coefficient, (deltaf), measuring the effect of the degree-distribution on the occurrence of long edges. We identify a sharp phase transition in (deltaf) for the existence of a giant component in dimension (d=1).

  • B. Jahnel, A.J. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, in: Proceedings of the 12th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, T. Jordán, G.Y. Katona, C. Király, G. Wiener, eds., Budapest University of Technology and Economics, Department of Computer Science and Information Theory, 2023, pp. 537--542.

Preprints, Reports, Technical Reports

  • B. Jahnel, J. Köppl, Time-periodic behaviour in one- and two-dimensional interacting particle systems, Preprint no. 3092, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3092 .
    Abstract, PDF (311 kByte)
    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, The directed age-dependent random connection model with arc reciprocity, Preprint no. 3090, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3090 .
    Abstract, PDF (304 kByte)
    We introduce a directed spatial random graph model aimed at modelling certain aspects of social media networks. We provide two variants of the model: an infinite version and an increasing sequence of finite graphs that locally converge to the infinite model. Both variants have in common that each vertex is placed into Euclidean space and carries a birth time. Given locations and birth times of two vertices, an arc is formed from younger to older vertex with a probability depending on both birth times and the spatial distance of the vertices. If such an arc is formed, a reverse arc is formed with probability depending on the ratio of the endpoints' birth times. Aside from the local limit result connecting the models, we investigate degree distributions, two different clustering metrics and directed percolation.

  • B. Jahnel, U. Rozikov, Three-state $p$-SOS models on binary Cayley trees, Preprint no. 3089, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3089 .
    Abstract, PDF (640 kByte)
    We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value 0,1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p>0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p=1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p=1 do exhibit transitions between extremality and non-extremality.

  • CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixed-degree nodes in weighted random connection models, Preprint no. 3057, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3057 .
    Abstract, PDF (474 kByte)
    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, Ch. Külske, A. Zass, Locality properties for discrete and continuum Widom--Rowlinson models in random environments, Preprint no. 3054, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3054 .
    Abstract, PDF (606 kByte)
    We consider the Widom--Rowlinson model in which hard disks of two possible colors are constrained to a hard-core repulsion between particles of different colors, in quenched random environments. These random environments model spatially dependent preferences for the attach- ment of disks. We investigate the possibility to represent the joint process of environment and infinite-volume Widom--Rowlinson measure in terms of continuous (quasilocal) Papangelou inten- sities. We show that this is not always possible: In the case of the symmetric Widom-Rowlinson model on a non-percolating environment, we can explicitly construct a discontinuity coming from the environment. This is a new phenomenon for systems of continuous particles, but it can be understood as a continuous-space echo of a simpler non-locality phenomenon known to appear for the diluted Ising model (Griffiths singularity random field [ EMSS00]) on the lattice, as we explain in the course of the proof.

  • B. Jahnel, A.D. Vu, A long-range contact process in a random environment, Preprint no. 3047, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3047 .
    Abstract, PDF (3735 kByte)
    We study survival and extinction of a long-range infection process on a diluted one-dimensional lattice in discrete time. The infection can spread to distant vertices according to a Pareto distribution, however spreading is also prohibited at random times. We prove a phase transition in the recovery parameter via block arguments. This contributes to a line of research on directed percolation with long-range correlations in nonstabilizing random environments.

  • B. Jahnel, J. Köppl, B. Lodewijks, A. Tóbiás, Percolation in lattice k-neighbor graphs, Preprint no. 3028, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3028 .
    Abstract, PDF (437 kByte)
    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.

  • P. Gracar, L. Lüchtrath, Ch. Mönch, The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects, Preprint no. 3011, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3011 .
    Abstract, PDF (5064 kByte)
    We study the weight-dependent random connection model, a class of sparse graphs featuring many real-world properties such as heavy-tailed degree distributions and clustering. We introduce a coefficient, (deltaf), measuring the effect of the degree-distribution on the occurrence of long edges. We identify a sharp phase transition in (deltaf) for the existence of a giant component in dimension (d=1).

  • B. Jahnel, J. Köppl, On the long-time behaviour of reversible interacting particle systems in one and two dimensions, Preprint no. 3004, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3004 .
    Abstract, PDF (287 kByte)
    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, L. Lüchtrath, Existence of subcritical percolation phases for generalised weight-dependent random connection models, Preprint no. 2993, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2993 .
    Abstract, PDF (299 kByte)
    We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space and carries an independent weight. In contrast to many established models, the presence of an edge is not only allowed to depend on the distance and weights of its end vertices but can also depend on the surrounding vertex set. Our result can be applied in particular to models combining heavy-tailed degree distributions and long-range effects, which are typically well connected. Moreover, we establish bounds on the tail-distribution of the number of points and the diameter of the subcritical component of a typical point. The proofs rest on a multi-scale argument.

Talks, Poster

  • J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems and applications to attractor properties, Analysis and Probability Seminar Passau, Universität Passau, Fakultät für Informatik und Mathematik, January 17, 2023.

  • J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, 16th German Probability and Statistics Days (GPSD) 2023, March 7 - 10, 2023, Universität Duisburg-Essen, March 9, 2023.

  • J. Köppl, The long-time behaviour of interacting particle systems: A Lyapunov functional approach, In Search of Model Structures for Non-equilibrium Systems, Münster, April 24 - 28, 2023.

  • J. Köppl, The long-time behaviour of interacting particle systems: A Lyapunov functional approach, In search of model structures for non-equilibrium systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

  • J. Köppl, The long-time behaviour of interacting particle systems, Stochastic Processes and Related Fields, Kyoto, September 4 - 8, 2023.

  • J. Köppl, The longe-time behavior of interacting particle systems: A Lyapunov functional approach, Mathematics of Random Systems: Summer School 2023, September 11 - 15, 2023, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, November 13, 2023.

  • A.D. Vu, Discrete contact process in random environment, Mathematics of Random Systems: Summer School 2023, September 11 - 15, 2023, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, September 15, 2023.

  • A.D. Vu, Percolation on the Manhattan grid, Stochastic Processes and Related Fields, Kyoto, Japan, September 4 - 8, 2023.

  • A.D. Vu, Percolation on the Manhattan grid, 16th German Probability and Statistics Days (GPSD) 2023, March 7 - 10, 2023, Universität Duisburg-Essen, March 9, 2023.

  • A.D. Vu, Percolation on the Manhattan grid, 18. Doktorand:innentreffen der Stochastik 2023, August 21 - 23, 2023, Universität Heidelberg, Fakultät für Mathematik und Informatik, August 23, 2023.

  • B. Jahnel, Continuum percolation in random environment, Oberseminar zur Stochastik, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, June 22, 2023.

  • B. Jahnel, Continuum percolation in random environments, SPP 2265 Summer School on Probability and Geometry on Configuration Spaces, July 17 - 21, 2023, WIAS Berlin (in Harnack House).

  • B. Jahnel, Continuum percolation in random environments, Topics in High Dimensional Probability, January 2 - 13, 2023, International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India, January 3, 2023.

  • B. Jahnel, Dynamical Gibbs variational principles and attractor properties, Mathematisches Kolloquium, Universität zu Köln, Abteilung Mathematik, June 14, 2023.

  • B. Jahnel, Percolation, Oberseminar, Technische Universität Braunschweig, Institut für Mathematische Stochastik, November 8, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Seminar der Fakultät Informatik, Hochschule Reutlingen, October 6, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, November 2, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, January 30, 2023.

  • B. Jahnel, Subcritical percolation phases for generalized weight-dependent random connection models, BOS-Workshop on Stochastic Geometry, February 22 - 24, 2023, Universität Osnabrück, Institut für Mathematik, February 24, 2023.

  • B. Jahnel, Subcritical percolation phases for generalized weight-dependent random connection models, 21st INFORMS Applied Probability Society Conference, June 28 - 30, 2023, Centre Prouvé, Nancy, France, June 29, 2023.

  • B. Jahnel, Survey of 1st phase of the SPP2265, SPP2265-Reviewer-Kolloquium, August 29, 2023, Deutsches Zentrum für Luft- und Raumfahrt, Köln, August 29, 2023.

  • B. Jahnel, The statistical mechanics of the interlacement point process, Second Annual Conference of the SPP 2265, March 27 - 30, 2023, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Köln, March 29, 2023.

  • B. Jahnel, Subcritical percolation phases for generalized weight-dependent random connection models, DMV Annual Meeting 2023, Minisymposium MS 12 ``Random Graphs and Statistical Network Analysis'', September 25 - 28, 2023, Technische Universität Ilmenau, September 25, 2023.

  • B. Jahnel, Connectivity and chase-escape in mobile device-to-device networks in urban environments, 29th Nordic Congress of Mathematicians with EMS, July 3 - 7, 2023, Aalborg University, Department of Mathematical Sciences, Denmark, July 5, 2023.

  • L. Lüchtrath, Euclidean diameter of the soft Boolean model, 29th Nordic Congress of Mathematicians with EMS, July 3 - 7, 2023, Aalborg University, Department of Mathematical Sciences, Denmark, July 6, 2023.

  • L. Lüchtrath, Evolving networks, their limits and global properties, Oberseminar Stochastik, Universität Augsburg, Institut für Mathematik, January 25, 2023.

  • L. Lüchtrath, Finite percolation thresholds in one dimensional inhomogeneous random graphs, 16th German Probability and Statistics Days (GPSD) 2023, March 7 - 10, 2023, Universität Duisburg-Essen, March 7, 2023.

  • L. Lüchtrath, Percolation in weight-dependent random connection models, Workshop on Random Discrete Structures, Münster, March 20 - 24, 2023.

  • L. Lüchtrath, The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects, 18th Workshop on Algorithms and Models for Web Graphs, May 23 - 26, 2023, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, May 25, 2023.

  • L. Lüchtrath, The random cluster graph, Probability and Financical Mathematics Seminar, University of Leeds, School of Mathematics, Leeds, UK, November 23, 2023.

  • L. Lüchtrath, The random cluster graph, Probability Seminar, University of Sheffield, School of Mathematics and Statistics, Sheffield, UK, November 15, 2023.