A number of stochastic models have their meaning and interpretation only when they are embedded in a spatial context. We think here mainly of spatially distributed random structures such as ensembles of point clouds, paths (e.g. loops or coagulation trajectories), geometric graphs, branching trees etc. that interact with each other. These models can be static or have a temporal component, i.e. stochastic processes of such random objects. In particular, the WIAS is interested in random point processes with interactions, such as those that occur in the description of spatial telecommunications systems and also in the description of many physical systems.
The goal is then always the development of mathematical methods for the macroscopic description of the system. Systems in which phase transitions are hidden, which are brought to the surface with such methods and whose existence is rigorously proven, are of particular interest. Of particular interest in WIAS are systems that contain infinitely large, spatially longrange objects, thus enabling phase transitions that are of the nature of macroscopic structures emerging as soon as a parameter exceeds a critical threshold. These are condensation or gelation or percolationtype transitions, all closely related but with significant differences.
Contribution of the Institute
One of the main objects of study at WIAS are models of random interacting loops in a large box in the thermodynamic limit, where the total length of all loops is of the order 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 conjectured: the appearance of loops of very large length 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 the WIAS (see also the mathematical topics Large deviations and Interacting stochastic manyparticle systems), 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 motions, as well as the application of manipulations like 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 topic Coagulation), in which the accidental formation of particularly large (macroscopic) particles for certain coagulation cores after a sufficiently late period of time in the limit value of large particle systems, socalled 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 jumps over this transition limit. The novelty of the work of the WIAS consists in considering spatial models. Until recently, simplified models were 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 random geometric growing graph is created, whose connected components are studied. The main means here is a combinatorial development as well as an approach to the theory of large deviations, see Large deviations. In the upcoming work, the methods developed here will be transferred and expanded to the actual model of coagulation in space.
Random spatial graphs and the occurrence of macroscopic structures in them is also a highly active research area at WIAS. On the one hand, this deals with versions of the famous ErdősRényi graph in space and the question of how it produces a particularly large number of triangles, as well as random graphs embedded in space, which over the course of occasionally produce hubs due to their random formation, i.e. nodes with a particularly large number of edges. The WIAS focuses on graphs that are used as models for many types of social networks and that have the inherent scalefreeness effect.
There are also decisive spatial influences in the asymptotic analysis of the parabolic Anderson model, i.e. a Brownian motion in a random potential (see also the mathematical topic spectral theory random operators), whose spatial randomness is given as Gaussian white noise. A meaningful definition of this model was a task in itself and only succeeds in dimensions up to three. The WIAS is interested in temporally asymptotic behavior, especially with regard to the phenomenon of intermittency, i.e. the concentration of movement in a few small islands. This phenomenon is now well understood for spatially discrete models, but in the continuous white noise case this is still a challenge that the WIAS faces in dimension two. Since the solution of this equation here is not a function but a distribution, a formulation of the effect (namely that the bulk of the solution is concentrated on small islands) is a priori unclear and the proof is difficult, see also the mathematical topic Analysis of ordinary and partial stochastic differential equations.
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

M. Fradon, J. Kern, S. Rœlly, A. Zass, Diffusion dynamics for an infinite system of twotype spheres and the associated depletion effect, Stochastic Processes and their Applications, 171 (2024), 104319, DOI 10.1016/j.spa.2024.104319 .
Abstract
We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in ℝ^{d}, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive shortrange dynamical interaction  known in the physics literature as a depletion interaction  between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximisation of the contact number of n identical spheres in ℝ^{d}. As support material, we propose numerical simulations in the form of movies. 
A. Quitmann, L. Taggi, Macroscopic loops in the 3d doubledimer model, Electronic Communications in Probability, 28 (2023), pp. 112, DOI 10.1214/23ECP536 .
Abstract
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. 
L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, A largedeviations principle for all the components in a sparse inhomogeneous random graph, Probability Theory and Related Fields, 186 (2023), pp. 521620, DOI 10.1007/s00440022011807 .
Abstract
We study an inhomogeneous sparse random graph, G_{N}, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a largedeviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that G_{N} is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of G_{N}. In particular, we recover the criterion for the existence of the phase transition given in [BJR07]. 
O. Collin, B. Jahnel, W. König, A micromacro variational formula for the free energy of a manybody system with unbounded marks, Electronic Journal of Probability, 28 (2023), pp. 118/1118/58, DOI 10.1214/23EJP1014 .
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 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 infrastructureaugmented dynamic networks, Stochastic Models, 39 (2023), pp. 851877, 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. 131/1131/38, DOI 10.1214/23EJP1029 .
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 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, The Annals of Applied Probability, 33 (2023), pp. 45704607, DOI 10.1214/22AAP1926 .
Abstract
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. 
B. Jahnel, Ch. Külske, Gibbsianness and nonGibbsianness for Bernoulli lattice fields under removal of isolated sites, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 29 (2023), pp. 30133032, DOI 10.3150/22BEJ1572 .
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 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. 
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, 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. 
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.
Preprints, Reports, Technical Reports

E. Bolthausen, W. König, Ch. Mukherjee, Selfrepellent Brownian bridges in an interacting Bose gas, Preprint no. 3110, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3110 .
Abstract, PDF (478 kByte)
We consider a model of ddimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For d ≥ 5 and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the selfrepellent random walk, and build on a lace expansion method conducive to treating paths undergoing mutual repellence within each bridge. 
B. Jahnel, L. Lüchtrath, M. Ortgiese, Cluster sizes in subcritical soft Boolean models, Preprint no. 3106, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3106 .
Abstract, PDF (435 kByte)
We consider the soft Boolean model, a model that interpolates between the Boolean model and longrange percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Paretodistributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edgeweight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine pathcounting arguments identifying the precise order of decay of the probability that faraway vertices are connected. 
J. Köppl, N. Lanchier, M. Mercer, Survival and extinction for a contact process with a densitydependent birth rate, Preprint no. 3103, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3103 .
Abstract, PDF (860 kByte)
To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a densitydependent birth rate. The players are located on the ddimensional integer lattice, with natural birth rate λ and natural death rate one. The process also depends on a payoff a_{11} = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by λ times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0 ), we prove that, for all payoffs a , there is a phase transition from extinction to survival in the direction of λ. Using various block constructions, we also prove that, for all birth rates λ, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic meanfield model in which the stability of the extinction state only depends on λ . This underlines the importance of space (local interactions) and stochasticity in our model. 
E. Magnanini, G. Passuello, Statistics for the triangle density in ERGM and its meanfield approximation, Preprint no. 3102, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3102 .
Abstract, PDF (736 kByte)
We consider the edgetriangle model (or Strauss model), and focus on the asymptotic behavior of the triangle density when the size of the graph increases to infinity. In the analyticity region of the free energy, we prove a law of large numbers for the triangle density. Along the critical curve, where analyticity breaks down, we show that the triangle density concentrates with high probability in a neighborhood of its typical value. A predominant part of our work is devoted to the study of a meanfield approximation of the edgetriangle model, where explicit computations are possible. In this setting we can go further, and additionally prove a standard and nonstandard central limit theorem at the critical point, together with many concentration results obtained via large deviations and statistical mechanics techniques. Despite a rigorous comparison between these two models is still lacking, we believe that they are asymptotically equivalent in many respects, therefore we formulate conjectures on the edgetriangle model, partially supported by simulations, based on the meanfield investigation. 
B. Jahnel, U. Rozikov, Gibbs measures for hardcoreSOS models on Cayley trees, Preprint no. 3100, WIAS, Berlin, 2024.
Abstract, PDF (420 kByte)
We investigate the finitestate psolidonsolid model, for p=∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translationinvariant Gibbs measures behaves as 1→3 →5 →6 →7. This phase diagram is qualitatively similar to the one observed for threestate pSOS models with p>0 and, in the case of k=2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous. 
P.P. Ghosh, B. Jahnel, S.K. Jhawar, Large and moderate deviations in Poisson navigations, Preprint no. 3096, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3096 .
Abstract, PDF (318 kByte)
We derive large and moderatedeviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this nonMarkovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewalprocess arguments. 
M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zeroconductance model with arbitrarily strong local connectivity, Preprint no. 3095, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3095 .
Abstract, PDF (377 kByte)
We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all nontrivial choices of the connectivity parameter. The model is based on the socalled randomly stretched lattice where we additionally elongate layers containing few open edges. 
B. Jahnel, J. Köppl, Timeperiodic behaviour in one and twodimensional 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 translationinvariant stationary measure, which is not the longtime limit of all translationinvariant starting measures, due to the existence of timeperiodic orbits in the associated measurevalued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measurevalued dynamics needs to be a timestationary measure. 
L. Lüchtrath, Ch. Mönch, The directed agedependent 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, Threestate $p$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 solidonsolid 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 translationinvariant 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 nonextremality 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 nonextremal for p=1 do exhibit transitions between extremality and nonextremality. 
L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, Spatial particle processes with coagulation: Gibbsmeasure approach, gelation and Smoluchowski equation, Preprint no. 3086, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3086 .
Abstract, PDF (651 kByte)
We study a spatial Markovian particle system with pairwise coagulation, a spatial version of the MarcusLushnikov process: according to a coagulation kernel K, particle pairs merge into a single particle, and their masses are united. We introduce a statisticalmechanics approach to the study of this process. We derive an explicit formula for the empirical process of the particle configuration at a given fixed time T in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time T. The noncoagulation between any two of them induces an exponential pairinteraction, which turns the description into a manybody system with a Gibbsian pairinteraction. Based on this, we first give a largedeviation principle for the joint distribution of the particle histories (conditioning on an upper bound for particle sizes), in the limit as the number N of initial atoms diverges and the kernel scales as 1/N K. We characterise the minimiser(s) of the rate function, we give criteria for its uniqueness and prove a law of large numbers (unconditioned). Furthermore, we use the unique minimiser to construct a solution of the Smoluchowski equation and give a criterion for the occurrence of a gelation phase transition. endabstract 
CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixeddegree 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 processlevel Poissonapproximation result for the degree$k$ vertices in a highdensity weighted random connection model with preferentialattachment 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 smallweight 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 WidomRowlinson models in random environments, Preprint no. 3054, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3054 .
Abstract, PDF (606 kByte)
We consider the WidomRowlinson model in which hard disks of two possible colors are constrained to a hardcore 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 infinitevolume WidomRowlinson measure in terms of continuous (quasilocal) Papangelou inten sities. We show that this is not always possible: In the case of the symmetric WidomRowlinson model on a nonpercolating 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 continuousspace echo of a simpler nonlocality 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 longrange 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 longrange infection process on a diluted onedimensional 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 longrange correlations in nonstabilizing random environments. 
L. Andreis, T. Iyer, E. Magnanini, Gelation, hydrodynamic limits and uniqueness in cluster coagulation processes, Preprint no. 3039, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3039 .
Abstract, PDF (627 kByte)
We consider the problem of gelation in the cluster coagulation model introduced by Norris [Comm. Math. Phys., 209(2):407435 (2000)]; this model is general enough to incorporate various inhomogenieties in the evolution of clusters, for example, their shape, or their location in space. We derive general, sufficient criteria for stochastic gelation in this model, and for trajectories associated with this process to concentrate among solutions of a generalisation of the Flory equation; thus providing sufficient criteria for the equation to have gelling solutions. As particular cases, we extend results related to the classical MarcusLushnikov coagulation process and Smoluchowski coagulation equation, showing that reasonable 'homogenous' coagulation processes with exponent γ larger than 1 yield gelation. In another special case, we prove a law of large numbers for the trajectory of the empirical measure of the stochastic cluster coagulation process, by means of a uniqueness result for the solution of the aforementioned generalised Flory equation. Finally, we use coupling arguments with inhomogeneous random graphs to deduce sufficient criterion for strong gelation (the emergence of a particle of size O(N)). 
N. Forien, M. Quattropani, A. Quitmann, L. Taggi, Coexistence, enhancements and short loops in random walk loop soups, Preprint no. 3029, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3029 .
Abstract, PDF (410 kByte)
We consider a general random walk loop soup which includes, or is related to, several models of interest, such as the Spin O(N) model, the double dimer model and the Bose gas. The analysis of this model is challenging because of the presence of spatial interactions between the loops. For this model it is known from [30] that macroscopic loops occur in dimension three and higher when the inverse temperature is large enough. Our first result is that, on the d dimensional lattice, the presence of repulsive interactions is responsible for a shift of the critical inverse temperature, which is strictly greater than (1/2d), the critical value in the non interacting case. Our second result is that a positive density of microscopic loops exists for all values of the inverse temperature. This implies that, in the regime in which macroscopic loops are present, microscopic and macroscopic loops coexist. Moreover, we show that, even though the increase of the inverse temperature leads to an increase of the total loop length, the density of microscopic loops is uniformly bounded from above in the inverse temperature. Our last result is confined to the special case in which the random walk loop soup is the one associated to the Spin O(N) model with arbitrary integer values of N ≥2 and states that, on ℤ ^{2}, the probability that two vertices are connected by a loop decays at least polynomially fast with their distance. 
B. Jahnel, J. Köppl, B. Lodewijks, A. Tóbiás, Percolation in lattice kneighbor 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 kneighbor graph. Two natural associated undirected graphs are the emphundirected and the emphbidirectional kneighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed kneighbor 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 selfavoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k=1 even the undirected kneighbor 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 2neighbor graph percolates for d=2, the undirected 3neighbor 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 knearestneighbor graphs studied in continuum percolation, and our results support this interpretation. 
W. König, N. Pétrélis, R. Soares Dos Santos, W. van Zuijlen, Weakly selfavoiding walk in a Paretodistributed random potential, Preprint no. 3023, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3023 .
Abstract, PDF (604 kByte)
We investigate a model of continuoustime simple random walk paths in ℤ ^{d} undergoing two competing interactions: an attractive one towards the large values of a random potential, and a selfrepellent one in the spirit of the wellknown weakly selfavoiding random walk. We take the potential to be i.i.d. Paretodistributed with parameter α > d, and we tune the strength of the interactions in such a way that they both contribute on the same scale as t → ∞. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for α > 2d: the randomwalk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function.The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution?and is in the spirit of a standard extremevalue setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09. 
B. Jahnel, J. Köppl, On the longtime 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 nontranslationinvariant) 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 timeperiodic 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 nonattractive stochastic Ising models this answers a question due to Liggett. 
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.
Talks, Poster

B. Jahnel, Timeperiodic behavior in one and twodimensional interacting particle systems (online talk), International Scientific Conference on Gibbs Measures and the Theory of Dynamical Systems (online event), May 20  21, 2024, Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan, Romanovskiy Institut of Mathematics and University of Exact and Social Sciences, Tashkent, Uzbekistan, May 20, 2024.

L. Lüchtrath, Cluster sizes in soft Boolean models, Probability and Analysis 2024, April 22  26, 2024, Wroclaw University of Science and Technology, Będlewo, Poland, April 22, 2024.

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 DuisburgEssen, 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.

A. Zass, Diffusion dynamics for an system of twotype speres and the associated depletion effect, Workshop MathMicS 2023: Mathematics and microscopic theory for random Soft Matter systems, February 13  15, 2023, HeinrichHeineUniversität Düsseldorf, Institut für Theoretische Physik II  Soft Matter, February 14, 2023.

A. Zass, The statistical mechanics of the interlacement point process, Second Annual Conference of the SPP2265, March 27  30, 2023, Deutsches Zentrum für Luft und Raumfahrt (DLR), Köln, March 30, 2023.

H. Langhammer, A largedeviations principle for all the components in a sparse inhomogeneous random graph, Workshop ``Random Graphs: Combinatorics, Complex Networks and Disordered Systems", March 27  31, 2023, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, March 28, 2023.

B. Jahnel, Dynamical Gibbs variational principles and attractor properties, Mathematisches Kolloquium, Universität zu Köln, Abteilung Mathematik, June 14, 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 weightdependent random connection models, 21st INFORMS Applied Probability Society Conference, June 28  30, 2023, Centre Prouvé, Nancy, France, June 29, 2023.

B. Jahnel, Subcritical percolation phases for generalized weightdependent 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, 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.

W. König, 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 30, 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 DuisburgEssen, March 7, 2023.

L. Lüchtrath, Percolation in weightdependent random connection models, Workshop on Random Discrete Structures, Münster, March 20  24, 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  Platforms and Mathematical Optimization for Secure and Resilient Future Networks (Online Event), 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.