Stochastische Systeme mit Wechselwirkung

Publikationen

Artikel in Referierten Journalen

• T. Iyer, On a sufficient condition for explosion in CMJ branching processes and applications to recursive trees, Electronic Communications in Probability, 29 (2024), pp. 46/1--46/12, DOI 10.1214/24-ECP616 .
  Abstract
  We provide sufficient criteria for explosion in Crump-Mode-Jagers branching process, via the process producing an infinite path in finite time. As an application, we deduce a curious phase-transition in the infinite tree associated with a class of recursive tree models with fitness, showing that in one regime every node in the tree has infinite degree, whilst in another, the tree is locally finite, with a unique infinite path. The latter class encompasses many models studied in the literature, including the weighted random recursive tree, the preferential attachment tree with additive fitness, and the Bianconi-Barab´asi model, or preferential attachment tree with multiplicative fitness.

• J. Kern, B. Wiederhold, A Lambda-Fleming--Viot type model with intrinsically varying population size, Electronic Journal of Probability, 29 (2024), pp. 125/1--125/28, DOI 10.1214/24-EJP1185 .
  Abstract
  We propose an extension of the classical ?-Fleming-Viot 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 Λ-Fleming-Viot model and describe a duality relationship. Through the lookdown framework, we provide a forward-in-time analysis of fixation and coming down from infinity. Furthermore, we present a new duality argument allowing one to deduce well-posedness of the measure-valued process without the necessity of proving uniqueness of the associated lookdown martingale problem.

• M. Fradon, J. Kern, S. Rœlly, A. Zass, Diffusion dynamics for an infinite system of two-type 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 short-range 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.

• R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuation-theory perspective, Journal of Statistical Physics, 191 (2024), pp. 1--60, DOI 10.1007/s10955-024-03233-8 .
  Abstract
  Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

• A. Quitmann, L. Taggi, Macroscopic loops in the 3d double-dimer model, Electronic Communications in Probability, 28 (2023), pp. 1--12, DOI 10.1214/23-ECP536 .
  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 self-avoiding loops. Our first result is that in ℤ ^d, d>2, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [Betz, Taggi] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when d >2, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.

• A. Quitmann, L. Taggi, Macroscopic loops in the Bose gas, spin O(N) and related models, Communications in Mathematical Physics, 400 (2023), pp. 2081--2136, DOI 10.1007/s00220-023-04633-9 .
  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 hard-core constraints.

• T. Iyer, Degree distributions in recursive trees with fitnesses, Advances in Applied Probability, 55 (2023), pp. 407--443, 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 time-step a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to ? new-coming vertices. Under a certain technical assumption, applying the theory of Crump-Mode-Jagers 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 large-deviations principle for all the components in a sparse inhomogeneous random graph, Probability Theory and Related Fields, 186 (2023), pp. 521--620, DOI 10.1007/s00440-022-01180-7 .
  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 large-deviations 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 micro-macro variational formula for the free energy of a many-body system with unbounded marks, Electronic Journal of Probability, 28 (2023), pp. 118/1--118/58, DOI 10.1214/23-EJP1014 .
  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 Bose--Einstein 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 self-energies and their entropies. The proof method comprises a two-step large-deviation 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

• T. Iyer, Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees, Preprint no. 3138, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3138 .
  Abstract, PDF (512 kByte)
  A sequence of trees (T_n) _{n ∈ N} contains a emphpersistent hub, or displays emphdegree centrality, if there is a fixed node of maximal degree for all sufficiently large n ∈ N. We derive sufficient criteria for the emergence of a persistent hub in genealogical trees associated with Crump-Mode-Jagers branching processes with independent waiting times between births of individuals, and sufficient criteria for the non-emergence of a persistent hub. We also derive criteria for uniqueness of these persistent hubs. As an application, we improve results in the l iterature concerning the emergence of unique persistent hubs in generalised preferential attachment trees, in particular, allowing for cases where there may not be a emphMalthusian parameter associated with the process. The approach we use is mostly self-contained, and does not rely on prior results about Crump-Mode-Jagers branching processes

• T. Iyer, Fixation of leadership in non-Markovian growth processes, Preprint no. 3137, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3137 .
  Abstract, PDF (462 kByte)
  Consider a model where N equal agents possess `values', belonging to N<sub>0</sub>, that are subject to incremental growth over time. More precisely, the values of the agents are represented by N independent, increasing N<sub>0</sub> valued processes with random, independent waiting times between jumps. We show that the event that a single agent possesses the maximum value for all sufficiently large values of time (called `leadership') occurs with probability zero or one, and provide necessary and sufficient conditions for this to occur. Under mild conditions we also provide criteria for a single agent to become the unique agent of maximum value for all sufficiently large times, and also conditions for the emergence of a unique agent having value that tends to infinity before `explosion' occurs (i.e. conditions for `strict leadership' or `monopoly' to occur almost surely). The novelty of this model lies in allowing non-exponentially distributed waiting times between jumps in value. In the particular case when waiting times are mixtures of exponential distributions, we improve a well-established result on the `balls in bins' model with feedback, removing the requirement that the feedback function be bounded from below and also allowing random feedback functions. As part of the proofs we derive necessary and sufficient conditions for the distribution of a convergent series of independent random variables to have an atom on the real line, a result which we believe may be of interest in its own right.

• H. Shafigh, A spatial model for dormancy in random environment, Preprint no. 3136, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3136 .
  Abstract, PDF (627 kByte)
  In this paper, we introduce a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of the random environment. However, the branching mechanism is governed by a random environment which dictates the branching rates. We consider three specific choices for random environments composed of particles: (1) a Bernoulli field of immobile particles, (2) one moving particle, and (3) a Poisson field of moving particles. In each case, the particles of the random environment can either be interpreted as emphcatalysts, accelerating the branching mechanism, or as emphtraps, aiming to kill the individuals. The different between active and dormant individuals is defined in such a way that dormant individuals are protected from being trapped, but do not participate in migration or branching. We quantify the influence of dormancy on the growth resp.,survival of the population by identifying the large-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman-Kac formula. Especially, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk.

• T. Bai, W. König, Q. Vogel, Proof of off-diagonal long-range order in a mean-field trapped Bose gas via the Feynman--Kac formula, Preprint no. 3119, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3119 .
  Abstract, PDF (392 kByte)
  We consider the non-interacting Bose gas of many bosons in dimension larger than two in a trap in a mean-field setting with a vanishing factor in front of the kinetic energy. The semi-classical setting is a particular case and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman--Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose?Einstein condensation) for the vanishing factor above a certain threshold and non-occurrence of ODLRO for below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman--Kac formula, the order parameter that Feynman suggested in [Fe53]. For small values of the factor, we prove that all loops have the minimal length one, and for large ones we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson--Dirichlet distribution.

• M. Gösgens, L. Lüchtrath, E. Magnanini, M. Noy, É. DE Panafieu, The Erdős--Rényi random graph conditioned on every component being a clique, Preprint no. 3111, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3111 .
  Abstract, PDF (2166 kByte)
  We consider an Erdős-Rényi random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability p as the number of vertices n diverges. We prove that there is a phase transition at p=1/2 in these observables. We additionally study the near-critical regime as well as the sparse regime

• E. Bolthausen, W. König, Ch. Mukherjee, Self-repellent 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 d-dimensional 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 self-repellent random walk, and build on a lace expansion method conducive to treating paths undergoing mutual repellence within each bridge.

• E. Magnanini, G. Passuello, Statistics for the triangle density in ERGM and its mean-field approximation, Preprint no. 3102, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3102 .
  Abstract, PDF (736 kByte)
  We consider the edge-triangle 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 mean-field approximation of the edge-triangle model, where explicit computations are possible. In this setting we can go further, and additionally prove a standard and non-standard 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 edge-triangle model, partially supported by simulations, based on the mean-field investigation.

• L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, Spatial particle processes with coagulation: Gibbs-measure 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 Marcus--Lushnikov process: according to a coagulation kernel K, particle pairs merge into a single particle, and their masses are united. We introduce a statistical-mechanics 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 non-coagulation between any two of them induces an exponential pair-interaction, which turns the description into a many-body system with a Gibbsian pair-interaction. Based on this, we first give a large-deviation 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, Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula, Preprint no. 3067, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3067 .
  Abstract, PDF (416 kByte)
  We consider the path-integral representation of the ideal Bose gas under various boundary conditions. We show that Bose--Einstein condensation occurs at the famous critical density threshold, by proving that its $1$-particle-reduced density matrix exhibits off-diagonal long-range order above that threshold, but not below. Our proofs are based on the well-known Feynman--Kac 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 Poisson--Dirichlet 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 Pareto-distributed steps.

• T. Iyer, B. Lodewijks, On the structure of genealogical trees associated with explosive Crump--Mode--Jagers 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 Crump--Mode-Jagers 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 sub-trees 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 emphsuper-linear 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 sub-trees T of empharbitrary size can appear infinitely often; behaviour that is markedly distinct from super-linear preferential attachment models studied in the literature so far.

• B. Jahnel, Ch. Külske, A. Zass, Locality properties for discrete and continuum Widom--Rowlinson models in random environments, Preprint no. 3054, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3054 .
  Abstract, PDF (606 kByte)
  We consider the Widom--Rowlinson model in which hard disks of two possible colors are constrained to a hard-core 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 infinite-volume Widom--Rowlinson measure in terms of continuous (quasilocal) Papangelou inten- sities. We show that this is not always possible: In the case of the symmetric Widom-Rowlinson model on a non-percolating 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 continuous-space echo of a simpler non-locality 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, 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 monomer-dimer model, Preprint no. 3046, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3046 .
  Abstract, PDF (203 kByte)
  We consider the monomer-dimer model, whose realisations are spanning sub-graphs 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 monomer-monomer correlations and dimer-dimer 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 spin-spin 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):407-435 (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 Marcus-Lushnikov 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.

• W. König, N. Pétrélis, R. Soares Dos Santos, W. van Zuijlen, Weakly self-avoiding walk in a Pareto-distributed random potential, Preprint no. 3023, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3023 .
  Abstract, PDF (604 kByte)
  We investigate a model of continuous-time simple random walk paths in ℤ ^d undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d. Pareto-distributed 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 random-walk 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 extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09.

• W. König, Ch. Kwofie, The throughput in multi-channel (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 multi-channel 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 large-deviation 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.

Vorträge, Poster

• T. Iyer, Persistent hubs in generalised preferential attachment trees, Seminar series in Probability and Combinatorics, Uppsala University, Department of Mathematics, Uppsala, Sweden, May 23, 2024.

• T. Iyer, Persistent hubs in generalised preferential attachment trees, Seminar Series in Mathematical Statistics, Stockholm University, Department of Mathematics, Stockholm, Sweden, May 22, 2024.

• T. Iyer, Persistent hubs in preferential attachment trees, Oberseminar AG Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, June 20, 2024.

• J. Kern, Modelling populations with fluctuating size, Mathematical Models in Ecology and Evolution (MMEE), July 15 - 18, 2024, Universität Wien, Fakultät für Mathematik, Wien, Austria, July 15, 2024.

• A. Zass, A model for colloids: Diffusion dynamics for two-type hard spheres and the associated depletion effect, 4th Italian Meeting on Probability and Mathematical Statistics, June 10 - 14, 2024, University of Rome, Tor Vergata, Sapienza University of Rome, The University of Roma Tre, LUISS, Rome, Italy, June 10, 2024.

• A. Zass, A model for colloids: Diffusion dynamics for two-type hard spheres and the associated depletion effect, Kolloquium des Instituts für Mathematische Stochastik, Technische Universität Braunschweig, Institut für Mathematische Stochastik, January 17, 2024.

• A. Zass, Discrete and continuous Widom--Rowlinson models in random environment, A lifelong journey in stochastic analysis: from branching processes to statistical mechanics, May 27 - 28, 2024, Institut Henri Poincaré, Paris, May 28, 2024.

• E. Magnanini, Gelation and hydrodynamic limits in cluster coagulation processes, 4th Italian Meeting on Probability and Mathematical Statistics, June 10 - 14, 2024, University of Rome, Tor Vergata, Sapienza University of Rome, The University of Roma Tre, LUISS, Rome, Italy, June 14, 2024.

• W. König, Off-diagonal long-range order for the free Bose gas via the Feynman-Kac formula, Forschungsseminar Analysis, FernUniversität in Hagen , Fakultät für Mathematik und Informatik, online, null, April 24, 2024.

• 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 non-gradient 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. Zarco-Romero, 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 two-type speres and the associated depletion effect, Workshop MathMicS 2023: Mathematics and microscopic theory for random Soft Matter systems, February 13 - 15, 2023, Heinrich-Heine-Universität 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 large-deviations 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 Marcus--Lushnikov process, In Search of Model Structures for Non-equilibrium Systems, Münster, April 24 - 28, 2023.

• E. Magnanini, Gelation and hydrodynamic limits in a spatial Marcus--Lushnikov process, Workshop ``In search of model structures for non-equilibrium systems'', April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

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

• E. Magnanini, Gelation in a spatial Marcus--Lushnikov process, Workshop MathMicS 2023: Mathematics and microscopic theory for random Soft Matter systems, February 13 - 15, 2023, Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik II - Soft Matter, February 14, 2023.

• E. Magnanini, Spatial coagulation and gelation, SPP2265-Reviewer-Kolloquium, 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, Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik II -- Soft Matter, February 15, 2023.

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

• W. König, The interacting Bose gas in the semi-classical limit, Workshop ``Recent Advances in Bose--Einstein 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 Duisburg-Essen, 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 Duisburg-Essen, 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 Milano-Bicocca, Milano, Italy, September 7, 2023.

• W. van Zuijlen, Weakly self-avoiding walk in a random potential (part I and II), Workshop ``Polymers and self-avoiding walks'', May 31 - June 2, 2023, Henri Lebesgue, Centre de Mathematiques, Nantes, France.