Interacting Random Systems

Publications

Articles in Refereed Journals

• 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.

Preprints, Reports, Technical Reports

• B. Jahnel, L. Lüchtrath, Ch. Mönch, Phase transitions for contact processes on one-dimensional networks, Preprint no. 3170, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3170 .
  Abstract, PDF (292 kByte)
  We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with integer indexed vertices that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (Ménard et al., Ann. Sci. Éc. Norm. Supér., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: First, for Gilbert graphs derived from stationary point processes on the real line marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Second, for independent Bernoulli long-range percolation on the integers, where the edge probabilities are given via a polynomial in the edges length', we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever the power of said polynomial is large than two. Although our approach utilises the restrictive topology of the line, we believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in higher dimensions.

• H. Shafigh, Dormancy in random environment: Symmetric exclusion, Preprint no. 3166, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3166 .
  Abstract, PDF (368 kByte)
  In this paper, we study 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, namely the simple symmetric exclusion process. We will interpret the presence of the exclusion particles either as emphcatalysts, accelerating the branching mechanism, or as emphtraps, aiming to kill the individuals. The difference 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. In particular, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk

• 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

Talks, Poster

• E. Magnanini, A Markovian particle system with coagulation, Oberseminar Stochastik, Universität zu Kö ln, Department Mathematik/Informatik, January 22, 2025.

• H. Langhammer, Spatial Coagulation and Gelation, Random Geometric Systems, Third Annual Conference of SPP2265, October 28 - 30, 2024, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, October 29, 2024.

• H. Langhammer, Large deviations for a spatial particle progress with coagulation, Seminar Dipartimento di Matematica, Universit`a di Roma ``La Sapienza'', Dipartimento di Matematica, Rome, Italy, September 25, 2024.

• T. Iyer, Fixiation of leadership in growth processes, Probability Seminar, University of Melbourne, School of Mathematics and Statistics, Melbourne, Australia, October 1, 2024.

• 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 generalised preferential attachment trees, Monash Probability and Statistics Seminar, Monash University, School of Mathematics, Melbourne, Australia, November 3, 2024.

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

• T. Iyer, Spatial Coagulation and Gelation, Random Geometric Systems, Third Annual Conference of SPP2265, October 28 - 30, 2024, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, October 29, 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.

• C. Zarco Romero, Large deviations for inhomogeneous random graphs, Probability-Autumn-School on Point Processes and their dynamics, November 19 - 22, 2024, Universität Münster, Institut für Mathematische Stochastik.

• C. Zarco Romero, Large deviations for inhomogeneous random graphs, Berlin-Oxford Summer School in Mathematics of Random Systems 2024, September 9 - 13, 2024, Oxford University, Mathematical Institute, Oxford, UK, September 10, 2024.

• C. Zarco Romero, Statistical mechanics for spatial dense random graphs, Particle systems in random environments, August 26 - 30, 2024.

• C. Zarco Romero, The underlying structure of dense graph limits, 52th Probability Summer School Saint-Flour, July 1 - 13, 2024, Université Clermont Auvergne, Saint-Flour, France.

• 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, A model for colloids: Diffusion dynamics for two-type hard spheres and the associated depletion effect, Colloquium in Probability, Technische Universität München, Department of Mathematics, November 11, 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.

• A. Zass, Discrete and continuous Widom--Rowlinson models in random environment, 52th Probability Summer School Saint-Flour, July 1 - 13, 2024, Université Clermont Auvergne, Saint-Flour, France, July 10, 2024.

• A. Zass, Marked Gibbs point processes (crash course), Random Geometric Systems, Third Annual Conference of SPP2265, October 28 - 30, 2024, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, October 29, 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.

• E. Magnanini, Spatial Coagulation and Gelation, Random Geometric Systems, Third Annual Conference of SPP2265, October 28 - 30, 2024, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, October 29, 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.

• A. Zass, Discrete and continous Widow--Rowlinson models in random environment, Interacting particles in the continuum, September 9 - 13, 2024, EURANDOM, Eindhoven, Netherlands, September 10, 2024.

External Preprints

• L. Andreis, T. Iyer, E. Magnanini, Convergence of cluster coagulation dynamics, Preprint no. arXiv:2406.12401, Cornell University, 2024, DOI 10.48550/arXiv.2406.12401 .
  Abstract
  We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [Comm. Math. Phys., 209(2):407-435 (2000)]. In this process, pairs of particles x,y in a measure space E, merge to form a single new particle z according to a transition kernel K(x,y,dz), in such a manner that a quantity, one may regard as the total mass of the system, is conserved. This model is general enough to incorporate various inhomogeneities in the evolution of clusters, for example, their shape, or their location in space. We derive sufficient criteria for trajectories associated with this process to concentrate among solutions of a generalisation of the Flory equation, and, in some special cases, by means of a uniqueness result for solutions of this equation, prove a weak law of large numbers. This multi-type Flory equation is associated with conserved quantities associated with the process, which may encode different information to conservation of mass (for example, conservation of centre of mass in spatial models). We also apply criteria for gelation in this process to derive sufficient criteria for this equation to exhibit gelling solutions. When this occurs, this multi-type Flory equation encodes, via the associated conserved property, the interaction between the gel and the finite size sol particles.

• P. Gonçalves, J. Kern, L. Xu, A novel approach to hydrodynamics for long-range generalized exclusion, Preprint no. arXiv:2410.17899 ?, Cornell University, 2024, DOI 10.48550/arXiv.2410.17899 .
  Abstract
  We consider a class of generalized long-range exclusion processes evolving either on Z or on a finite lattice with an open boundary. The jump rates are given in terms of a general kernel depending on both the departure and destination sites, and it is such that the particle displacement has an infinite expectation, but some tail bounds are satisfied. We study the superballisitic scaling limit of the particle density and prove that its space-time evolution is concentrated on the set of weak solutions to a nonlocal transport equation. Since the stationary states of the dynamics are unknown, we develop a new approach to such a limit relying only on the algebraic structure of the Markovian generator.