A number of stochastic models have their meaning, interpretation and sense only if they are embedded in a spatial context. We mainly think of spatially distributed random structures such as ensembles of point clouds, paths (e.g., loops), geometric graphs, branching trees, etc., which interact with one another. Many of the models also have a time component, i.e., they are stochastic processes of such random objects. The goal is then always to develop mathematical methods for the macroscopic description of the system. Of particular interest are systems in which phase transitions are hidden, which are brought to the surface with such methods and whose existence is rigorously proven.
One of the main objects of investigation at WIAS are models of randomly interacting loops in a large box at the thermodynamic limit, where the total length of all loops is of the order of magnitude of the volume of the box. The most prominent representative of such models is the interacting Bose gas, in which the famous BoseEinstein condensation phase transition is suspected: the occurrence of very long loops as soon as the temperature falls below a critical limit. Such models are important prototypes of spin models, i.e. Gibbs' models of particles whose spin space is unlimited and gives rise to new effects. Two different strategies are pursued at WIAS (see also the mathematical topic "Interacting stochastic manyparticle systems" and "Large deviations"), namely the analysis of the free energy of the system in the thermodynamic limit in terms of a variational description and with the help of infinitely long Brownian movements as well as the application of manipulations such as reflections and the derivation of correlation inequalities.
Another direction in which the WIAS works are spatial models for large particle clouds with a coagulation mechanism (see the application area "Coagulation"), in which the accidental formation of particularly large (macroscopic) particles for certain coagulation nuclei is investigated after a sufficiently late period in the limit of large particle systems; this is called gelation. This phase transition can be seen as a kind of explosion transition, because all other particles continue to grow normally, and every now and then one particle size jumps over this transition limit. The novelty of the work of the WIAS is to consider spatial models. Simplified models are currently being considered, in which the coagulation is not expressed by a change in the location of the two particles involved, but by the insertion of an edge; in this way a unique geometrical growing graph is created, the connected components of which are studied. The main means here is a combinatorial expansion, as well as an approach using the theory of large deviations, see the mathematical topic of the same name.
There are also decisive spatial influences in the asymptotic analysis of the parabolic Anderson model (see also the mathematical topic "Spectra of random operators"), the spatial randomness of which is given as Gaussian white noise. A meaningful definition of this model was a task in itself and is only possible in dimensions up to three; we are interested in temporally asymptotic behavior, especially with regard to the phenomenon of intermittency. This phenomenon is now well understood for spatially discrete models, but in the continuous case with white noise this is still a challenge that the WIAS faces in dimension two. Since the solution of this equation is not a function but a distribution, a formulation of the effect (namely that the main mass of the solution is concentrated on small islands) is a priori unclear and the proof is difficult.
Publications
Monographs

B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/9783030360900 .
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 3hour lectures or seminars which are also suitable for selfstudy. 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 10.1007/9783030527785 .
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 2stündige Lehrveranstaltung
Articles in Refereed Journals

CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multiscale infrastructureaugmented dynamic networks, Stochastic Models, published online on 06.03.2023, DOI 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. 
N. Engler, B. Jahnel, Ch. Külske, Gibbsianness of locally thinned random fields, Markov Processes and Related Fields, 28 (2022), pp. 185214, DOI 10.48550/arXiv.2201.02651 .
Abstract
We consider the locally thinned Bernoulli field on ℤ ^{d}, which is the lattice version of the TypeI 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 multiscale dynamic networks, Stochastic Processes and their Applications, 151 (2022), pp. 490518, DOI 10.1016/j.spa.2022.06.008 .
Abstract
We study the effects of mobility on two crucial characteristics in multiscale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent longtime averages are wellapproximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multiscale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birthdeath 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, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Brazilian Journal of Probability and Statistics, 3 (2022), pp. 2044, DOI 10.1214/21BJPS514 .
Abstract
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 nonexistence 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. 
B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Random Structures and Algorithms, 62 (2022), pp. 240255, 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 edgedrawing 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 twodimensional 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. 41314172, DOI 10.1007/s00023021010767 .
Abstract
We consider a onedimensional classical manybody system with interaction potential of LennardJones 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 chaseescape models on PoissonGilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/125/14, DOI 10.1214/20ECP306 .
Abstract
We present results on phase transitions of local and global survival in a twospecies model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuoustime nearestneighbor 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 welldefinedness 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 finitedegree 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. 41/141/12, DOI 10.1214/20ECP322 .
Abstract
This work develops a methodology for analyzing largedeviation 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 PoissonVoronoi cells, as well as powerweighted edge lengths in the random geometric, κnearest neighbor and relative neighborhood graph. 
A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Journal of Statistical Physics, 179 (2020), pp. 90109, DOI 10.1007/s10955020025213 .
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 PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl 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.
Contributions to Collected Editions

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. 19.
Abstract
We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary CoxGilbert 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 edgelength measure of a realization of a PoissonVoronoi 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 nonMarkovian 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 nonMarkovian 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. 16, DOI 10.23919/WiOPT47501.2019.9144122 .
Abstract
The typical cell is a key concept for stochasticgeometry 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 Manhattantype 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 doubledimer 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 selfavoiding 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 doubledimer model, namely the doubledimer 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 selfavoiding 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 selfavoiding 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 PoissonBoolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features longrange dependencies in the environment, leading to absence of a sharp phase transition for the associated CoxBoolean 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 translationinvariant interacting particle systems on the ddimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a timestationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translationinvariant measure implies, that the measure is Gibbs w.r.t. the same specification as the timestationary 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 translationinvariant measures is a Gibbs measure w.r.t. the same specification as the timestationary measure. This extends previously known results to fairly general irreversible interacting particle systems. 
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 nonGibbsianness 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 noninvasive for large p, as it changes only a small fraction p(1p)^{2d} of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a nonGibbsian measure, i.e., it does not possess a continuous version of its finitevolume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.
Talks, Poster

B. Jahnel, Continuum percolation in random environments, Topics in High Dimensional Probability, January 2  13, 2023, International Centre for Theoretical Sciences, Bangalore, India, January 3, 2023.

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, March 31, 2022.

A. Quitmann, Macroscopic loops in the Bose gas and related models, Random Geometric Systems, First Annual Conference of SPP2265, April 11  14, 2022, HarnackHaus, Tagungsstätte der MaxPlanckGesellschaft, April 14, 2022.

T. Iyer, Spatial coagulation and gelation, Random Geometric Systems, First Annual Conference of SPP2265, April 11  14, 2022, HarnackHaus, Tagungsstätte der MaxPlanckGesellschaft, April 13, 2022.

S.K. Jhawar, Poisson approximation and connectivity in a scalefree network, Spring School on Random geometric graphs, March 28  April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 31, 2022.

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, Existence of Infinite Cluster on the Manhattan Grid, Processes on Random Geometric Graphs, September 12  16, 2022, Universität zu Köln, Mathematisches Institut.

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.

A. Zass, Interacting diffusions as marked Gibbs point processes, Random Point Processes in Statistical Physics, June 29  July 1, 2022, HarnackHaus, Tagungsstätte der MaxPlanckGesellschaft, June 30, 2022.

A. Zass, Marked Gibbs point processes (crash course), Random Geometric Systems, First Annual Conference of SPP2265, April 11  14, 2022, HarnackHaus, Tagungsstätte der MaxPlanckGesellschaft, April 12, 2022.

B. Jahnel, Firstpassage percolation and chaseescape 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, Malware propagation in mobile devicetodevice networks (online talk), Joint H2020 AI@EDGE and INSPIRE5G Project Workshop (Online Event), Workshop of the IEEE International Conference on Cloud Networking, Nov. 710, 2022, Paris, France, November 8  9, 2022, November 8, 2022.

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

W. König, Manybody Systems and the Interacting Bose Gas (minicourse), Random Point Processes in Statistical Physics, June 29  July 1, 2022, Harnack House, Berlin.

W. König, Spatial coagulation and gelation, Random Geometric Systems, First Annual Conference of SPP2265, April 11  14, 2022, HarnackHaus, Tagungsstätte der MaxPlanckGesellschaft, April 13, 2022.

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

B. Jahnel, Firstpassage percolation and chaseescape 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), BernoulliIMS One World Symposium 2020 (Online Event), August 24  28, 2020, August 27, 2020.