Publications
Articles in Refereed Journals

A. Quitmann, L. Taggi, Macroscopic loops in the Bose gas, spin O(N) and related models, Communications in Mathematical Physics, 400 (2023), pp. 20812136, DOI 10.1007/s00220023046339 .
Abstract
We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in ℤ^{d}, d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate ℤ^{d} by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hardcore constraints. 
T. Iyer, Degree distributions in recursive trees with fitnesses, Advances in Applied Probability, 55 (2023), pp. 407443, DOI 10.1017/apr.2022.40 .
Abstract
We study a general model of recursive trees where vertices are equipped with independent weights and at each timestep a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to ? newcoming vertices. Under a certain technical assumption, applying the theory of CrumpModeJagers branching processes, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we prove rigorously observations of Bianconi related to the evolving Cayley tree in [Phys.Rev.E66, 036116 (2002)]. We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call "generalised preferential attachment with fitness". We show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with maximal weight, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process. 
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.
Preprints, Reports, Technical Reports

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 
W. König, Q. Vogel, A. Zass, Offdiagonal longrange order for the free Bose gas via the FeynmanKac formula, Preprint no. 3067, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3067 .
Abstract, PDF (416 kByte)
We consider the pathintegral representation of the ideal Bose gas under various boundary conditions. We show that BoseEinstein condensation occurs at the famous critical density threshold, by proving that its $1$particlereduced density matrix exhibits offdiagonal longrange order above that threshold, but not below. Our proofs are based on the wellknown FeynmanKac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting PoissonDirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (a part from a standard eigenvalue expansion), using elementary tools like Markov's inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets and asymptotics for random walks with Paretodistributed steps. 
T. Iyer, B. Lodewijks, On the structure of genealogical trees associated with explosive CrumpModeJagers branching processes, Preprint no. 3060, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3060 .
Abstract, PDF (902 kByte)
We study the structure of genealogical trees associated with explosive CrumpModeJagers branching processes (stopped at the explosion time), proving criteria for the associated tree to contain a node of infinite degree (a emphstar) or an infinite path. Next, we provide uniqueness criteria under which with probability 1 there exists exactly one of a unique star or a unique infinite path. Under the latter uniqueness criteria we also provide an example where, with strictly positive probability less than 1, there exists a unique star in the model. We thus illustrate that this probability is not restricted to being 0 or 1. Moreover, we provide structure theorems when there is a star, where we prove that certain trees appear as subtrees in the tree infinitely often. We apply our results to a general discrete evolving tree model, named emphexplosive recursive trees with fitness. As a particular case, we study a family of emphsuperlinear preferential attachment models with fitness. For these models, we derive phase transitions in the model parameters in three different examples, leading to either exactly one star with probability 1 or one infinite path with probability 1 with every node having finite degree. Furthermore, we highlight examples where subtrees T of empharbitrary size can appear infinitely often; behaviour that is markedly distinct from superlinear preferential attachment models studied in the literature so far. 
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. 
J. Kern, B. Wiederhold, A LambdaFlemingViot type model with intrinsically varying population size, Preprint no. 3053, WIAS, Berlin, 2023.
Abstract, PDF (2956 kByte)
We propose an extension of the classical ?FlemingViot model to intrinsically varying pop ulation sizes. During events, instead of replacing a proportion of the population, a random mass dies and a, possibly different, random mass of new individuals is added. The model can also incorporate a drift term, representing infinitesimally small, but frequent events. We investigate el ementary properties of the model, analyse its relation to the ΛFlemingViot model and describe a duality relationship. Through the lookdown framework, we provide a forwardintime analysis of fixation and coming down from infinity. Furthermore, we present a new duality argument allowing one to deduce wellposedness of the measurevalued process without the necessity of proving uniqueness of the associated lookdown martingale problem. 
J. Kern, Exponential equivalence for misanthrope processes in contact with weak reservoirs and applications to totally asymmetric exclusion processes, Preprint no. 3051, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3051 .
Abstract, PDF (200 kByte)
We provide a short proof for the exponential equivalence between misanthrope processes in contact with weak reservoirs and those with impermeable boundaries. As a consequence, we can derive both the hydrodynamic limit and the large deviations of the totally asymmetric exclusion process (TASEP) in contact with weak reservoirs. This extends a recent result which proved the hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in contact with weak reservoirs. Furthermore, applications to a class of asymmetric exclusion processes with long jumps is discussed. 
A. Quitmann, A note on the monomerdimer model, Preprint no. 3046, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3046 .
Abstract, PDF (203 kByte)
We consider the monomerdimer model, whose realisations are spanning subgraphs of a given graph such that every vertex has degree zero or one. The measure depends on a parameter, the monomer activity, which rewards the total number of monomers. We consider general correlation functions including monomermonomer correlations and dimerdimer covariances. We show that these correlations decay exponentially fast with the distance if the monomer activity is strictly positive. Our result improves a previous upper bound from van den Berg and is of interest due to its relation to truncated spinspin correlations in classical spin systems. Our proof is based on the cluster expansion technique. 
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. 
M. Fradon, J. Kern, S. Rœlly, A. Zass, Diffusion dynamics for an infinite system of twotype spheres and the associated depletion effect, Preprint no. 3024, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3024 .
Abstract, PDF (2630 kByte)
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. 
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. 
W. König, Ch. Kwofie, The throughput in multichannel (slotted) ALOHA: Large deviations and analysis of bad events, Preprint no. 2991, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2991 .
Abstract, PDF (295 kByte)
We consider ALOHA and slotted ALOHA protocols as medium access rules for a multichannel message delivery system. Users decide randomly and independently with a minimal amount of knowledge about the system at random times to make a message emission attempt. We consider the two cases that the system has a fixed number of independent available channels, and that interference constraints make the delivery of too many messages at a time impossible. We derive probabilistic formulas for the most important quantities like the number of successfully delivered messages and the number of emission attempts, and we derive largedeviation principles for these quantities in the limit of many participants and many emission attempts. We analyse the rate functions and their minimizers and derive laws of large numbers for the throughput. We optimize it over the probability parameter. Furthermore, we are interested in questions like “if the number of successfully delivered messages is significantly lower than the expectation, was the reason that too many or too few sending attempts were made?”. Our main tools are basic tools from probability and the theory of (the probabilities of) large deviations.
Talks, Poster

T. Iyer, Properties of recursive trees with independent fitnesses (online talk), Seminar Complex Systems, Queen Mary University of London, School of Mathematical Sciences, London, UK, February 14, 2023.

J. Kern, Mini Course: A unified approach to nongradient systems, Seminar Interacting Random Systems (Hybrid Event), May 11  15, 2023, WIAS Berlin.

J. Kern, Young measures and the hydrodynamic limit of asymmetric exclusion processes, Seminar Stochastics, Technical University of Lisbon, Lisbon, Portugal, July 10, 2023.

C. ZarcoRomero, Wilson's current idea using Markov Chains (online talk), 56th National Congress of the Mexican Mathematical Society, Autonomous University of San Luis Potosí, San Luis Potosí, Mexico, October 9, 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.

E. Magnanini, Gelation and hydrodynamic limits in a spatial MarcusLushnikov process, In Search of Model Structures for Nonequilibrium Systems, Münster, April 24  28, 2023.

E. Magnanini, Gelation and hydrodynamic limits in a spatial MarcusLushnikov process, Workshop ``In search of model structures for nonequilibrium systems'', April 24  28, 2023, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

E. Magnanini, Gelation in a spatial MarcusLushnikov process, Workshop MathMicS 2023: Mathematics and Microscopic Theory for Random Soft Matter Systems, Düsseldorf, February 13  15, 2023.

E. Magnanini, Gelation in a spatial MarcusLushnikov process, 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.

E. Magnanini, Spatial coagulation and gelation, SPP2265ReviewerKolloquium, Köln, August 29, 2023.

W. König, Spatial particle processes with coagulation: Large deviations and gelation, 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 15, 2023.

W. König, Survey of 1st phase of the SPP2265, SPP2265ReviewerKolloquium, August 29, 2023, Deutsches Zentrum für Luft und Raumfahrt, Köln, August 29, 2023.

W. König, The interacting Bose gas in the semiclassical limit, Workshop ``Recent Advances in BoseEinstein Condensation'', August 30  September 1, 2023, Technische Universität München, Department of Mathematics, August 31, 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.

W. van Zuijlen, Anderson Hamiltonians with singular potentials, 16th German Probability and Statistics Days (GPSD) 2023, March 7  10, 2023, Universität DuisburgEssen, March 9, 2023.

W. van Zuijlen, Anderson models, from Schrödinger operators to singular SPDEs, Oberseminar Analysis und Theoretische Physik, Leibniz Universität Hannover, Institut für Angewandte Mathematik, December 12, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, 16th German Probability and Statistics Days (GPSD) 2023, March 7  10, 2023, Universität DuisburgEssen, March 9, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, iPOD Seminar, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 4, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, Probability Seminar, University of Warwick, Mathematics Institute, UK, June 14, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, Seminar Dipartimento di Matematica e Applicazioni, Università degli Studi di MilanoBicocca, Milano, Italy, September 7, 2023.

W. van Zuijlen, Weakly selfavoiding walk in a random potential (part I and II), Workshop ``Polymers and selfavoiding walks'', May 31  June 2, 2023, Henri Lebesgue, Centre de Mathematiques, Nantes, France.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations