Etliche stochastische Modelle haben ihre Bedeutung, Interpretation und Sinn nur, wenn sie in einen räumlichen Kontext eingebettet sind. Wir denken hier hauptsächlich an räumlich verteilte zufällige Strukturen wie Ensembles von Punktwolken, Pfade (z.B. Loops), geometrische Graphen, Verzweigungsbäume etc., die miteinander Interaktionen haben. Etliche der Modelle haben auch eine zeitliche Komponente, d.h., sie sind stochastische Prozesse solcher zufälligen Objekte. Das Ziel ist dann immer die Entwicklung von mathematischen Methoden für die makroskopische Beschreibung des Systems. Besonderes Interesse gilt Systemen, in denen Phasenübergänge versteckt sind, die mit solchen Methoden zur Oberfläche gebracht werden und deren Existenz rigoros bewiesen wird.

Besonderes Interesse gilt am WIAS Systemen, die unbeschränkt große, räumlich langreichweitige Objekte beinhalten und auf diese Weise Phasenübergänge ermöglichen, die von der Art des Hervortretens von makroskopischen Strukturen sind, sobald ein Parameter einen kritischen Schwellenwert übersteigt. Dies sind Übergänge von Kondensations- oder solche vom Gelations- oder Perkolationstyp, die alle eng zusammenhängen, aber signifikante Unterschiede aufweisen.

Einer der Hauptuntersuchungsgegenstände am WIAS sind Modelle zufälliger interagierender Loops in einer großen Box im thermodynamischen Grenzwert, wo die Gesamtlänge aller Loops von der Größenordnung des Volumens der Box ist. Der prominenteste Vertreter solcher Modelle ist das interagierende Bose-Gas, in dem der berühmte Bose-Einstein-Kondensations-Phasenübergang vermutet wird: das Auftreten von Loops sehr großer Länge, sobald die Temperatur unter eine kritische Grenze fällt. Solche Modelle sind wichtige Prototypen von Spinmodellen, also Gibbs'schen Modellen von Partikeln, deren Spinraum unbeschränkt ist und Anlass zu neuen Effekten gibt. Am WIAS werden zwei unterschiedliche Strategien verfolgt (siehe auch das mathematische Thema "Interagierende stochastische Vielteilchensysteme" und "Große Abweichungen"), und zwar die Analyse der freien Energie des Systems im thermodynamischen Grenzwert in Termen einer variationellen Beschreibung sowie mit Hilfe von unendlich langen Brown'schen Bewegungen, sowie die Anwendung von Manipulationen wie Reflektionen und die Herleitung von Korrelationsungleichungen.

Eine andere Richtung, in der das WIAS arbeitet, sind räumliche Modelle für große Partikelwolken mit Koagulationsmechanismus (siehe das Anwendungsgebiet "Koagulation"), in denen die zufällige Entstehung besonders großer (makroskopischer) Partikel für gewisse Koagulationskerne nach genügend später Zeit im Grenzwert großer Partikelsysteme untersucht wird, sogenannte Gelation. Dieser Phasenübergang kann als eine Art Explosionsübergang gesehen werden, denn alle anderen Partikel wachsen normal weiter, und ab und zu springt eines über diese Übergangsgrenze. Die Neuheit der Arbeit des WIAS besteht darin, räumliche Modelle zu betrachten. Gegenwärtig werden verinfachte Modelle betrachtet, in denen die Koagulation nicht durch eine Ortsverändreung der beiden beteiligten Partikel ausgedrückt wird, sondern durch das Einfügen einer Kante; auf diese Art entsteht einzufälliger geometrischer wachsender Graph, dessen Zusammenhangskomponenten studiert werden. Das Hauptmittel hier ist eine kombinatorische Entwicklung sowie ein Ansatz der Theorie der großen Abweichungen, siehe das gleichnamige mathematische Thema.

Research
Eine Simulation eines Systems Brown'scher Loops als Marken an den Punkten eines Punktprozesses

Entscheidende räumliche Einflüsse gibt es auch bei der asymptotischen Analyse des parabolischen Anderson-Modells (siehe auch das mathematische Thema "Spektra zufälliger Operatoren"), dess räumlicher Zufall als ein Gauß'sches weißes Rauschen gegeben ist. Eine sinnvolle Definition dieses Modells ist eine Aufgabe für sich gewesen und gelingt nur in Dimensionen bis zu drei; wir sind am zeitlich asymptotischen Verhalten interessiert, insbesondere im Hinblick auf das Phänomen der Intermittenz. Für räumlich diskrete Modelle ist dieses Phänomen mittlerweile gut verstanden, aber im kontinuierlichen Fall mit weißem Rauschen ist dies noch eine Herausforderung, der das WIAS sich in Dimension zwei stellt. Da die Lösung dieser Gleichung hier keine Funktion, sondern eine Distribution ist, ist eine Formulierung des Effektes (nämlich dass sich die Hauptmasse der Lösung auf kleinen Inseln konzentriert) a priori unklar und der Beweis schwierig.


Publikationen

  Monografien

  • B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/978-3-030-36090-0 .
    Abstract
    This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They may feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance.

  • W. König, Große Abweichungen, Techniken und Anwendungen, M. Brokate, A. Heinze , K.-H. Hoffmann , M. Kang , G. Götz , M. Kerz , S. Otmar, eds., Mathematik Kompakt, Birkhäuser Basel, 2020, VIII, 167 pages, (Monograph Published), DOI 10.1007/978-3-030-52778-5 .
    Abstract
    Die Lehrbuchreihe Mathematik Kompakt ist eine Reaktion auf die Umstellung der Diplomstudiengänge in Mathematik zu Bachelor- und Masterabschlüssen. Inhaltlich werden unter Berücksichtigung der neuen Studienstrukturen die aktuellen Entwicklungen des Faches aufgegriffen und kompakt dargestellt. Die modular aufgebaute Reihe richtet sich an Dozenten und ihre Studierenden in Bachelor- und Masterstudiengängen und alle, die einen kompakten Einstieg in aktuelle Themenfelder der Mathematik suchen. Zahlreiche Beispiele und Übungsaufgaben stehen zur Verfügung, um die Anwendung der Inhalte zu veranschaulichen. Kompakt: relevantes Wissen auf 150 Seiten Lernen leicht gemacht: Beispiele und Übungsaufgaben veranschaulichen die Anwendung der Inhalte Praktisch für Dozenten: jeder Band dient als Vorlage für eine 2-stündige Lehrveranstaltung

  Artikel in Referierten Journalen

  • N. Engler, B. Jahnel, Ch. Külske, Gibbsianness of locally thinned random fields, Markov Processes and Related Fields, 28 (2022), pp. 185--214, DOI 10.48550/arXiv.2201.02651 .
    Abstract
    We consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions

  • B. Jahnel, Ch. Hirsch, E. Cali, Percolation and connection times in multi-scale dynamic networks, Stochastic Processes and their Applications, 151 (2022), pp. 490--518, DOI 10.1016/j.spa.2022.06.008 .
    Abstract
    We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.

  • B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Random Structures and Algorithms, published online on 30.03.2022 (2022), DOI 10.1002/rsa.21084 .
    Abstract
    We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Distribution of cracks in a chain of atoms at low temperature, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 22 (2021), pp. 4131--4172, DOI 10.1007/s00023-021-01076-7 .
    Abstract
    We consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Poisson--Gilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/1--25/14, DOI 10.1214/20-ECP306 .
    Abstract
    We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

  • CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Electronic Communications in Probability, 25 (2020), pp. 41/1--41/12, DOI 10.1214/20-ECP322 .
    Abstract
    This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

  • A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Journal of Statistical Physics, 179 (2020), pp. 90--109, DOI 10.1007/s10955-020-02521-3 .
    Abstract
    In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the Poisson--Voronoi, Poisson--Delaunay and Poisson line tessellation, we also treat the Johnson--Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

  Beiträge zu Sammelwerken

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, in: IEEE 18th International Symposium on on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks, (WiOpt), Volos, Greece, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 1--9.
    Abstract
    We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

  • A. Hinsen, Ch. Hirsch, B. Jahnel, E. Cali, Typical Voronoi cells for Cox point processes on Manhatten grids, in: 2019 International Symposium on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks (WiOPT), Avignon, France, 2019, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 1--6, DOI 10.23919/WiOPT47501.2019.9144122 .
    Abstract
    The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.

  Preprints, Reports, Technical Reports

  • A. Quitmann, L. Taggi, Macroscopic loops in the $3d$ double-dimer model, Preprint no. 2944, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2944 .
    Abstract, PDF (265 kByte)
    The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in ℤ d, d>2, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [Betz, Taggi] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when d >2, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.

  • B. Jahnel, S.K. Jhawar, A.D. Vu, Continuum percolation in a nonstabilizing environment, Preprint no. 2943, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2943 .
    Abstract, PDF (2463 kByte)
    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, Preprint no. 2935, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2935 .
    Abstract, PDF (355 kByte)
    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.

  • CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multi-scale dynamic networks, Preprint no. 2895, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2895 .
    Abstract, PDF (1634 kByte)
    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.

  • M. Heida, B. Jahnel, A.D. Vu, Stochastic homogenization on irregularly perforated domains, Preprint no. 2880, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2880 .
    Abstract, PDF (668 kByte)
    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.

  • B. Jahnel, Ch. Külske, Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites, Preprint no. 2878, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2878 .
    Abstract, PDF (426 kByte)
    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.

  • B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Preprint no. 2704, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2704 .
    Abstract, PDF (389 kByte)
    We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, Preprint no. 2674, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2674 .
    Abstract, PDF (3133 kByte)
    We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

  Vorträge, Poster

  • A. Quitmann, Macroscopic loops in a random walk loop soup, Spring School on Random geometric graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik.

  • A.D. Vu, An Application for Percolation Theory in Analysis, Spring School on Random geometric graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 31, 2022.

  • A.D. Vu, Percolation theory and the effective conductivity, 21st Workshop on Stochastic Geometry, Stereology and Image Analysis, June 5 - 10, 2022, Nesuchyne, Czech Republic, June 6, 2022.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Spring School on Random geometric graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 30, 2022.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Spring School: Random geometric graphs, Technische Universität Darmstadt, Fachbereich Mathematik, March 30, 2022.

  • B. Jahnel, Phase transitions and large deviations for the Boolean model of continuum percolation for Cox point processes, Probability Seminar University Padua, Università di Padova, Dipartimento di Matematica, Italy, March 25, 2022.

  • B. Jahnel, Connectivity improvements in mobile device-to-device networks (online talk), Telecom Orange Paris, France, July 6, 2021.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Stochastic Geometry Days, November 15 - 19, 2021, Dunkerque, France, November 17, 2021.

  • B. Jahnel, Gibbsian representation for point processes via hyperedge potentials (online talk), Thematic Einstein Semester on Geometric and Topological Structure of Materials, Summer Semester 2021, Technische Universität Berlin, May 20, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes, Workshop on Randomness Unleashed Geometry, Topology, and Data, September 22 - 24, 2021, University of Groningen, Faculty of Science and Engineering, Groningen, Netherlands, September 23, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), DYOGENE Seminar (Online Event), INRIA Paris, France, January 11, 2021.

  • B. Jahnel, Stochastic geometry for epidemiology (online talk), Monday Biostatistics Roundtable, Institute of Biometry and Clinical Epidemiology (Online Event), Campus Charité, June 14, 2021.

  • W. König, A box version of the interacting Bose gas, Workshop on Randomness Unleashed Geometry, Topology, and Data, September 22 - 24, 2021, University of Groningen, Faculty of Science and Engineering, Groningen, Netherlands, September 23, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), Bernoulli-IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, August 27, 2020.