Interacting Random Systems

Publications

Articles in Refereed Journals

• J. Kern, Weak reservoirs are superexponentially irrelevant for misanthrope processes, Electronic Communications in Probability, 30 (2025), pp. 24/1--24/10, DOI 10.1214/25-ECP671 .

• 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, Electronic Journal of Probability, 30 (2025), pp. 1--26, DOI 10.1214/25-EJP1370 .
  Abstract
  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.

• W. König, Q. Vogel, A. Zass, Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula, The Annals of Applied Probability, 35 (2025), pp. 4030--4066, DOI 10.1214/25-AAP2213 .
  Abstract
  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.

Preprints, Reports, Technical Reports

• H. Shafigh, L. Tyrpak, Responsive dormancy of a spatial population among a moving trap, Preprint no. 3224, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3224 .
  Abstract, PDF (270 kByte)
  In this paper, we study a spatial model for dormancy in a random environment via a two-type branching random walk in continuous-time, where individuals switch between dormant and active states depending on the current state of a fluctuating environment (responsive switching). The branching mechanism is governed by the same random environment, which is here taken to be a simple symmetric random walk. We will interpret the presence of this random walk as a emphtrap which attempts to kill the individuals whenever it meets them. The responsive switching between the active and dormant state is defined so that active individuals become dormant only when a trap is present at their location and remain active otherwise. Conversely, dormant individuals can only wake up once the environment becomes trap-free again. We quantify the influence of dormancy on population survival by analyzing the long-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. Specifically, we investigate the quantitative role of dormancy by extending the Parabolic Anderson model to a two-type random walk framework.

• J. Bäumler, T. Iyer, Permutations in competing growth processes and balls-in-bins, Preprint no. 3220, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3220 .
  Abstract, PDF (393 kByte)
  Consider a model of N independent, increasing ℕ₀-valued processes, with random, independent waiting times between jumps. It is known that there is either an emergent `leader', in which a single process possesses the maximal value for all sufficiently large times, or every pair of processes alternates leadership infinitely often. We show that in the latter regime, almost surely, one sees every possible permutation of rankings of processes infinitely often. In the case that the waiting times are exponentially distributed, this proves a conjecture from Spencer (appearing in a paper from Oliveira) on the `balls-in-bins' process with feedback [8, Conjecture1].

• E. Magnanini, G. Passuello, A standard CLT for triangles in a class of ERGMs, Preprint no. 3211, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3211 .
  Abstract, PDF (256 kByte)
  We prove a standard Central Limit Theorem for the (normalized) number of triangles in a class of Exponential Random Graphs derived from a slight modification of the edge-triangle model. Our main theorem covers the whole analyticity region of the free energy, and is based on a polynomial representation of the partition function.

• L. Andreis, T. Iyer, E. Magnanini, Convergence of cluster coagulation dynamics, Preprint no. 3182, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3182 .
  Abstract, PDF (435 kByte)
  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 emphmass 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 emphFlory 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 emphconserved 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 emphgelation in this process to derive sufficient criteria for this equation to exhibit emphgelling solutions. When this occurs, this multi-type Flory equation encodes, via the associated conserved property, the interaction between the emphgel and the finite size emphsol particles.

• M. Fradon, A. Zass, Infinite-dimensional diffusions and depletion interaction for a model of colloids, Preprint no. 3181, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3181 .
  Abstract, PDF (398 kByte)
  We consider infinite-dimensional random diffusion dynamics for the Asakura--Oosawa model of interacting hard spheres of two different sizes. We construct a solution to the corresponding SDE with collision local times, analyse its reversible measures, and observe the emergence of an attractive short-range depletion interaction between the large spheres. We study the Gibbs measures associated to this new interaction, exploring connections to percolation and optimal packing.

• 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

Talks, Poster

• T. Iyer, Recent progress concerning generalized preferential attachment trees, 44th Conference on Stochastic Processes and their Applications, July 14 - 18, 2025, Wroclaw University of Science and Technology and University of Wrocław, Faculty of Mathematics and Computer Science, Wrocław, July 17, 2025.

• T. Iyer, Persistent hubs in generalised preferential attachment trees, Workshop ``Recent Advances in Evolving and Spatial Random Graphs'', June 2 - 4, 2025, Universität Augsburg, Institut für Mathematik, Bayrischzell, June 2, 2025.

• T. Iyer, Persistent hubs in preferential attachment trees, Kombinatorik Seminar, Universität Heidelberg, Fakultät für Mathematik und Informatik, May 22, 2025.

• T. Iyer, What is...?? Geometric inhomogeneous random graphs, Random Geometric Systems, Fourth Annual Conference of SPP2265, June 23 - 26, 2025, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, June 23, 2025.

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

• E. Magnanini, Convergence of subgraph densities in Exponential Random Graphs, Women in Random Geometric Systems, March 5 - 7, 2025, Technische Universität München, TUM School of Computation, Information and Technology, March 5, 2025.

• E. Magnanini, Gelation in a spatial coagulation process, YEP Yorkshop: Interacting Particle Systems on Random Structures, June 23 - 27, 2025, EURANDOM, Eindhoven, Netherlands, June 27, 2025.

• E. Magnanini, Recent results on gelation-type phase transitions in spatial coagulation process, Particle Systems and PDE's XIII, December 1 - 5, 2025, Università di Modena e Reggio Emilia, Modena, Italy, December 4, 2025.

• E. Magnanini, What is...?? Geometric inhomogeneous random graphs, Random Geometric Systems, Fourth Annual Conference of SPP2265, June 23 - 26, 2025, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, June 23, 2025.

• W. König, Spatial particle processes with coagulation: Gibbs-measure approach, gelation, and Smoluchowski equation, Particle Systems and PDE's XIII, December 1 - 5, 2025, Università di Modena e Reggio Emilia, Modena, Italy, December 4, 2025.

• W. König, Spatial particle processes with coagulation: Gibbs-measure approach, gelation, and Smoluchowski equation, Critical behaviour in spatial particle systems, January 29 - 31, 2025, WIAS Berlin, January 30, 2025.

• W. König, Spatial particle processes with coagulation: Gibbs-measure approach, limits, and gelation, Seminar Stochastic group, Aarhus University, Department of Mathematics, Aarhus, Denmark, June 19, 2025.

• W. König, The free energy of the interacting Bose gas, loops and interlacements, 17th German Probability and Statistics Days (GPSD), March 11 - 14, 2025, Technische Universität Dresden, March 11, 2025.

• W. König, The free energy of the interacting Bose gas, loops and interlacements, Workshop on Stochastic Processes on Random Geometries, February 17 - 21, 2025, Technische Universiät Braunschweig, Institut für Mathematische Stochastik, February 21, 2025.

• W. König, The free energy of the interacting Bose gas, loops and interlacements, Random Geometric Structures and Statistical Physics, June 30 - July 4, 2025, University of Rome La Sapienza, Department of Mathematicsit, Rome, Italy, July 4, 2025.

• A. Zass, A dynamical model of interacting colloids, Recent Developments in Spatial Interacting Random Systems, October 15 - 17, 2025, WIAS Berlin.

• A. Zass, A dynamical model of interacting colloids, Seminar Laboratoire Jean Kuntzmann, September 29 - October 6, 2025, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble, France, October 2, 2025.

• A. Zass, Infinite-dimensional diffusions and depletion interaction for a model of colloids, Critical behaviour in spatial particle systems, January 29 - 31, 2025, WIAS Berlin.

• A. Zass, Gibbs measures and infinite-dimensional diffusions for interacting colloids, Seminar on Applied Probability and Statistics, Delft University of Technology, Delft Institute of Applied Mathematics, Delft, Netherlands, November 6, 2025.

• W. van Zuijlen, Anderson Models, from Schrödinger operators to singular SPDEs, Seminar on Applied Probability and Statistics, Delft University of Technology, Delft Institute of Applied Mathematics, Delft, Netherlands, October 27, 2025.

• W. van Zuijlen, The quenched Edwards--Wilkinson equation with Gaussian disorder, 17th German Probability and Statistics Days (GPSD), March 11 - 14, 2025, Technische Universität Dresden, March 12, 2025.

• W. van Zuijlen, The quenched Edwards--Wilkinson equation with Gaussian disorder, Oberseminar Analysis, Universität Augsburg, Institut für Mathematik, November 6, 2025.

• W. van Zuijlen, The quenched Edwards--Wilkinson equation with Gaussian disorder, Berliner Kolloquium Wahrscheinlichkeitstheorie, Wias Berlin, HUB, TUB, December 17, 2025.

External Preprints

• Y. Steenbeck, A. Zass, J. Köppl, B. Jahnel, Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties, Preprint no. arXiv:2508.21196, Cornell University Library, arXiv.org, 2025, DOI 10.48550/arXiv.2508.21196 .
  Abstract
  We consider continuous-time birth-and-death dynamics in Rd that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure