Publications
Articles in Refereed Journals

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 
S.K. Iyer, S.K. Jhawar, Phase transitions and percolation at criticality in enhanced random connection models, Mathematical Physics, Analysis and Geometry, 25 (2022), pp. 140, DOI 10.1007/s1104002109409y .
Abstract
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process P?P? in R2R2 of intensity ?. In the homogeneous RCM, the vertices at x,y are connected with probability g( x ? y ), independent of everything else, where g:[0,?)?[0,1]g:[0,?)?[0,1] and ? is the Euclidean norm. In the inhomogeneous version of the model, points of P?P? are endowed with weights that are nonnegative independent random variables with distribution P(W>w)=w??1[1,?)(w)P(W>w)=w??1[1,?)(w), ? >?0. Vertices located at x,y with weights Wx,Wy are connected with probability 1?exp(??WxWy x?y ?)1?exp?(??WxWy x?y ?), ?,? >?0, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of P?P?. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of P?P?. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality. 
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. 
S.K. Jhawar, S.K. Iyer, Poisson approximation and connectivity in a scalefree random connection model, Electronic Journal of Probability, 26 (2021), pp. 123, DOI 10.1214/21EJP651 .
Abstract
We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process P s Ps of intensity s > 0 s>0 on the unit cube S = ( ? 1 2 , 1 2 ] d S=?12,12d, d ? 2 d?2 . Each vertex is endowed with an independent random weight distributed as W, where P ( W > w ) = w ? ? 1 [ 1 , ? ) ( w ) P(W>w)=w??1[1,?)(w), ? > 0 ?>0. Given the vertex set and the weights an edge exists between x , y ? P s x,y?Ps with probability ( 1 ? exp ( ? ? W x W y ( d ( x , y ) ? r ) ? ) ) , 1?exp??WxWyd(x,y)?r?, independent of everything else, where ? , ? > 0 ?,?>0, d ( ? , ? ) d(?,?) is the toroidal metric on S and r > 0 r>0 is a scaling parameter. We derive conditions on ? , ? ?,? such that under the scaling r s ( ? ) d = 1 c 0 s ( log s + ( k ? 1 ) log log s + ? + log ( ? ? k ! d ) ) , rs(?)d=1c0slogs+(k?1)loglogs+?+log??k!d, ? ? R ??R, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e ? ? e?? as s ? ? s??, where c 0 c0 is an explicitly specified constant that depends on ? , ? , d ?,?,d and ? but not on k. In particular, for k = 0 k=0 we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large s. The Poisson approximation result is derived using the Stein?s method.
Contributions to Collected Editions

Z. Benomar, Ch. Ghribi, E. Cali, A. Hinsen, B. Jahnel, Agentbased modeling and simulation for malware spreading in D2D networks, AAMAS '22: Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems, Auckland, New Zealand, May 11  13, 2022, International Foundation for Autonomous Agents and Multiagent Systems Richland, SC, 2022, pp. 9199.
Abstract
This paper presents a new multiagent model for simulating malware propagation in devicetodevice (D2D) 5G networks. This model allows to understand and analyze mobile malwarespreading dynamics in such highly dynamical networks. Additionally, we present a theoretical study to validate and benchmark our proposed approach for some basic scenarios that are less complicated to model mathematically and also to highlight the key parameters of the model. Our simulations identify critical thresholds for em no propagation and for em maximum malware propagation and make predictions on the malwarespread velocity as well as deviceinfection rates. To the best of our knowledge, this paper is the first study applying agentbased simulations for malware propagation in D2D. 
CH. Ghribi, E. Cali, Ch. Hirsch , B. Jahnel, AgentBased Simulations for Coverage Extensions in 5G Networks and Beyond, 25th Conference on Innovation in Clouds, Internet and Networks and Workshops (ICIN) (Hybrid Event), Paris, France, March 7  10, 2022, published online on 09.05.2022, DOI 10.5555/3535850.3535862 .
Preprints, Reports, Technical Reports

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. 
CH. Ghribi, E. Cali, Ch. Hirsch, B. Jahnel, Agentbased simulations for coverage extensions in 5G networks and beyond, Preprint no. 2920, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2920 .
Abstract, PDF (1032 kByte)
Devicetodevice (D2D) communications is one of the key emerging technologies for the fifth generation (5G) networks and beyond. It enables direct communication between mobile users and thereby extends coverage for devices lacking direct access to the cellular infrastructure and hence enhances network capacity. D2D networks are complex, highly dynamic and will be strongly augmented by intelligence for decision making at both the edge and core of the network, which makes them particularly difficult to predict and analyze. Conventionally, D2D systems are evaluated, investigated and analyzed using analytical and probabilistic models (e.g., from stochastic geometry). However, applying classical simulation and analytical tools to such a complex system is often hard to track and inaccurate. In this paper, we present a modeling and simulation framework from the perspective of complexsystems science and exhibit an agentbased model for the simulation of D2D coverage extensions. We also present a theoretical study to benchmark our proposed approach for a basic scenario that is less complicated to model mathematically. Our simulation results show that we are indeed able to predict coverage extensions for multihop scenarios and quantify the effects of streetsystem characteristics and pedestrian mobility on the connection time of devices to the base station (BS). To our knowledge, this is the first study that applies agentbased simulations for coverage extensions in D2D. 
Z. Benomar, Ch. Ghribi, E. Cali, A. Hinsen, B. Jahnel, Agentbased modeling and simulation for malware spreading in D2D networks, Preprint no. 2919, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2919 .
Abstract, PDF (286 kByte)
This paper presents a new multiagent model for simulating malware propagation in devicetodevice (D2D) 5G networks. This model allows to understand and analyze mobile malwarespreading dynamics in such highly dynamical networks. Additionally, we present a theoretical study to validate and benchmark our proposed approach for some basic scenarios that are less complicated to model mathematically and also to highlight the key parameters of the model. Our simulations identify critical thresholds for em no propagation and for em maximum malware propagation and make predictions on the malwarespread velocity as well as deviceinfection rates. To the best of our knowledge, this paper is the first study applying agentbased simulations for malware propagation in D2D. 
O. Collin, B. Jahnel, W. König, The free energy of a boxversion of the interacting Bose gas, Preprint no. 2914, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2914 .
Abstract, PDF (1441 kByte)
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 BoseEinstein 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 selfenergies and their entropies. The proof method comprises a twostep largedeviation 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. 
CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multiscale 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 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. 
C.F. Coletti, L.R. DE Lima, A. Hinsen, B. Jahnel, D.R. Valesin, Limiting shape for firstpassage percolation models on random geometric graphs, Preprint no. 2877, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2877 .
Abstract, PDF (2340 kByte)
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 firstpassage 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.
Talks, Poster

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, 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, Firstpassage percolation and chaseescape 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 , Stochastic geometry for telecommunications, Leibniz MMS Days 2022, April 25  27, 2022, WIAS, Potsdam, April 26, 2022.

A. Hinsen, Limiting shape for firstpassage percolation models on random geometric graphs (online talk), German Probability & Statistics Days Mannheim (Online Event), September 27  October 1, 2021, Universität Mannheim, September 27, 2021.

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, Phase transitions for the Boolean model for Cox point processes (online talk), Probability Seminar Bath (Online Event), University of Bath, Department of Mathematical Sciences, UK, October 18, 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.