Stochastische Systeme mit Wechselwirkung

Publikationen

Artikel in Referierten Journalen

• A. Quitmann, L. Taggi, Macroscopic loops in the Bose gas, Spin O(N) and related models, Communications in Mathematical Physics, published online on 25.02.2023, 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.

• 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, published online on 11.01.2023, 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].

• D.R.M. Renger, Anisothermal chemical reactions: Onsager--Machlup and macroscopic fluctuation theory, Journal of Physics A: Mathematical and Theoretical, 55 (2022), pp. 315001/1--315001/24, DOI 10.1088/1751-8121/ac7c47 .
  Abstract
  We study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.

• A. Zass, Gibbs point processes on path space: Existence, cluster expansion and uniqueness, Markov Processes and Related Fields, 28 (2022), pp. 329--364.
  Abstract
  We study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: The starting points belong to R^d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.

• A. Bianchi, F. Collet, E. Magnanini, The GHS and other inequalities for the two-star model, ALEA. Latin American Journal of Probability and Mathematical Statistics, 19 (2022), pp. 1679--1695, DOI 10.30757/ALEA.v19-64 .
  Abstract
  We consider the two-star model, a family of exponential random graphs indexed by two real parameters, h and ?, that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, under a mild hypothesis on the parameters, we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, the average edge density turns out to be an increasing and concave function of the parameter h, at any fixed size of the graph

• A. Agazzi, L. Andreis, R.I.A. Patterson, D.R.M. Renger, Large deviations for Markov jump processes with uniformly diminishing rates, Stochastic Processes and their Applications, 152 (2022), pp. 533--559, DOI 10.1016/j.spa.2022.06.017 .
  Abstract
  We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics.

• N. Fountoulakis, T. Iyer, C. Mailler, H. Sulzbach, Dynamical models for random simplicial complexes, The Annals of Applied Probability, 32 (2022), pp. 2860--2913, DOI 10.1214/21-AAP1752 .
  Abstract
  We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of Complex Quantum Network Manifolds in dimensions d>2, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [Sci.Rep.5, 13979 (2015) and Phys.Rev.E93, 032315 (2016)].

• N. Fountoulakis, T. Iyer, Condensation phenomena in preferential attachment trees with neighbourhood influence, Electronic Journal of Probability, 27 (2022), pp. 76/1--76/49, DOI 10.1214/22-EJP787 .
  Abstract
  We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of edges with endpoints having certain weights, and the distribution of degrees of vertices having a given weight. We show that the former exhibits a condensation phenomenon under a certain critical condition, whereas the latter converges almost surely to a distribution that resembles a power law distribution. Moreover, in the absence of condensation, we prove almost-sure setwise convergence of the related quantities. This generalises existing results on the Bianconi-Barabási tree as well as on an evolving tree model introduced by the second author.

• P. Houdebert, A. Zass, An explicit Dobrushin uniqueness region for Gibbs point process with repulsive interactions, Journal of Applied Probability, 59 (2022), pp. 541--555, DOI 10.1017/jpr.2021.70 .
  Abstract
  We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature ?. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions

• Z. Mokhtari, R.I.A. Patterson, F. Höfling, Spontaneous trail formation in populations of auto-chemotactic walkers, New Journal of Physics, 24 (2022), pp. 013012/1--013012/11, DOI 10.1088/1367-2630/ac43ec .
  Abstract
  We study the formation of trails in populations of self-propelled agents that make oriented deposits of pheromones and also sense such deposits to which they then respond with gradual changes of their direction of motion. Based on extensive off-lattice computer simulations aiming at the scale of insects, e.g., ants, we identify a number of emerging stationary patterns and obtain qualitatively the non-equilibrium state diagram of the model, spanned by the strength of the agent--pheromone interaction and the number density of the population. In particular, we demonstrate the spontaneous formation of persistent, macroscopic trails, and highlight some behaviour that is consistent with a dynamic phase transition. This includes a characterisation of the mass of system-spanning trails as a potential order parameter. We also propose a dynamic model for a few macroscopic observables, including the sub-population size of trail-following agents, which captures the early phase of trail formation.

• N. Perkowski, W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift, Potential Analysis, published online on 27.01.2022, DOI 10.1007/s11118-021-09984-3 .
  Abstract
  We consider the stochastic differential equation on ℝ ^d given by d X _t = b(t,X_t ) d t + d B_t, where B is a Brownian motion and b is considered to be a distribution of regularity > - 1/2. We show that the martingale solution of the SDE has a transition kernel Γ_t and prove upper and lower heat kernel bounds for Γ_t with explicit dependence on t and the norm of b.

• D.R.M. Renger, S. Schindler, Gradient flows for bounded linear evolution equations, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 41 (2022), pp. 229--238, DOI 10.4171/ZAA/1706 .
  Abstract
  We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real diagonalisable. The proof is constructive, from which we also derive geodesic lambda-convexity.

• B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Random Structures and Algorithms, 62 (2022), pp. 240--255, DOI 10.1002/rsa.21084 .
  Abstract
  We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

• W. König, N. Perkowski, W. van Zuijlen, Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 58 (2022), pp. 1351--1384, DOI 10.1214/21-AIHP1215 .
  Abstract
  We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.

Preprints, Reports, Technical Reports

• 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.

• W. König, H. Shafigh, Multi-channel ALOHA and CSMA medium-access protocols: Markovian description and large deviations, Preprint no. 2985, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2985 .
  Abstract, PDF (314 kByte)
  We consider a multi-channel communication system under ALOHA and CSMA protocols, resepc- tively, in continuous time. We derive probabilistic formulas for the most important quantities: the numbers of sending attempts and the number of successfully delivered messages in a given time interval. We derive (1) explicit formulas for the large-time limiting throughput, (2) introduce an explicit and ergodic Markov chain for a deeper probabilistic analysis, and use this to (3) derive exponential asymptotics for rare events for these quantities in the limit of large time, via large-deviation principles.

• T. Matsuda, W. van Zuijlen, Anderson Hamiltonians with singular potentials, Preprint no. 2976, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2976 .
  Abstract, PDF (732 kByte)
  We construct random Schrödinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated density of states of these Anderson Hamiltonians, and we relate the Lifschitz tails (the asymptotics of the left tails of the integrated density of states) to the left tails of the principal eigenvalues.

• A. Quitmann, L. Taggi, Macroscopic loops in the 3D double-dimer model, Preprint no. 2944, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2944 .
  Abstract, PDF (265 kByte)
  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.

• D.R.M. Renger, U. Sharma, Untangling dissipative and Hamiltonian effects in bulk and boundary driven systems, Preprint no. 2936, WIAS, Berlin, 2022.
  Abstract, PDF (330 kByte)
  Using the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behaviour of non-equilibrium dynamics and steady states in emphdiffusive systems. We extend this framework to a minimal model of non-equilibrium emphnon-diffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and non-dissipative bulk and boundary forces that drives the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the large-deviation cost into dissipative and non-dissipative terms. We establish that the purely non-dissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples.

• O. Collin, B. Jahnel, W. König, The free energy of a box-version of the interacting Bose gas, Preprint no. 2914, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2914 .
  Abstract, PDF (1441 kByte)
  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.

Vorträge, Poster

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

• A. Quitmann, Macroscopic loops in a random walk loop soup, Spring School on Random Geometric Graphs, March 28 - April 1, 2022, Technische Universität Darmstadt, Fachbereich Mathematik, March 31, 2022.

• A. Quitmann, Macroscopic loops in a random walk loop soup, Mathematics of Random Systems Summer School 2022, September 26 - 30, 2022, Centre for Doctoral Trainig, Mathematics of Random Systems, Oxford, UK, September 27, 2022.

• A. Quitmann, Macroscopic loops in an interacting random walk loop soup, Focused Research: Graphical Representations of Spin Systems, September 5 - 7, 2022, University of Bristol, Heilbronn Institute for Mathematical Research, Bristol, UK, September 6, 2022.

• A. Quitmann, Macroscopic loops in random walk loop soups, Workshop and Summerschool on Random Graphs (RandNET), August 22 - 30, 2022, Workshop Centre in the area of Stochastics (EURANDOM), Eindhoven, Netherlands, August 24, 2022.

• A. Quitmann, Macroscopic loops in the Bose gas and related models, Random Geometric Systems, First Annual Conference of SPP2265, April 11 - 14, 2022, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, April 14, 2022.

• A. Quitmann, Macroscopic loops in the Bose gas and spin O(N) model, SPP2265 Workshop on Random spatial networks, March 14 - 17, 2022, Universität zu Köln, Mathematisches Institut, Bonn.

• A. Quitmann, Macroscopic loops in the Spin O(N) and related models (online talk), Percolation Today (Online Event), ETH Zürich, Italy, February 15, 2022.

• T. Iyer, Preferential attachment trees with neighbourhood influence, Summer School: Mathematics of Large Networks, May 30 - June 3, 2022, Erdős Center, Budapest, Hungary, May 31, 2022.

• T. Iyer, Preferential attachment trees with neighbourhood influence, Probability Seminar, Universität zu Köln, Department Mathematik/Informatik, April 29, 2022.

• T. Iyer, Spatial coagulation and gelation, Random Geometric Systems, First Annual Conference of SPP2265, April 11 - 14, 2022, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, April 13, 2022.

• T. Iyer, The influence of competition on genealogical trees associated with explosive age-dependent branching processes, Oberseminar Stochastik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, November 16, 2022.

• D.R.M. Renger, Variational formulations beyond gradient flows (online talk), British Applied Mathematics Colloquium BAMC 2022 (HybridEvent), April 11 - 13, 2022, Loughborough University, Loughborough, UK, April 13, 2022.

• M. Renger, Variational structures beyond gradient flows --- Part II (online talk), Seminar on Variational Evolutionary Problems and Related Problems (Online Event), Technische Universität Dresden, Fakulät für Mathematik, January 19, 2022.

• A. Zass, Existence of infinite-volume marked Gibbs point processes: a path space example, Third Italian Meeting on Probability and Mathematical Statistics, June 13 - 16, 2022, University of Bologna, Department of Mathematics, Department of Statistical Sciences of the Alma Mater Studiorum, Bologna, Italy, June 15, 2022.

• A. Zass, Gibbs point process on path space: existence, cluster, cluster expansion and uniqueness, Oberseminar zur Stochastik, Otto von Guericke Universität Magdeburg, Fakulät für Mathematik, January 20, 2022.

• A. Zass, Interacting diffusions as marked Gibbs point processes, Random Point Processes in Statistical Physics, June 29 - July 1, 2022, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, June 30, 2022.

• A. Zass, Interacting diffusions as marked Gibbs point processes, Seminar of Stochastic Geometry, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, October 11, 2022.

• A. Zass, Marked Gibbs point processes (crash course), Random Geometric Systems, First Annual Conference of SPP2265, April 11 - 14, 2022, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, April 12, 2022.

• A. Zass, Marked Gibbs point processes: a path space example, Workshop: New trends in point process theory, February 28 - March 2, 2022, Karlsruher Institut für Technologie (KIT), Fakultät für Mathematik, Karlsruhe.

• W. van Zuijlen, Anderson Hamiltonians with singular potentials, 15th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, May 12 - 14, 2022, WIAS Berlin, May 13, 2022.

• E. Magnanini, Gelation in a Spatial Marcus-Lushnikov Process, Conference for Junior Female Researchers in Probability 2022, October 5 - 7, 2022, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, October 4, 2022.

• E. Magnanini, Limit theorems for the edge density in exponential random graphs, Third Italian Meeting on Probability and Mathematical Statistics, June 13 - 16, 2022, University of Bologna, Department of Mathematics, Department of Statistical Sciences of the Alma Mater Studiorum, Bologna, Italy, June 15, 2022.

• E. Magnanini, Limit theorems for the edge density in exponential random graphs, Workshop and Summerschool on Random Graphs (RandNET), August 22 - 30, 2022, Workshop Centre in the area of Stochastics (EURANDOM), Eindhoven, Netherlands, August 24, 2022.

• A. Quitmann, Macroscopic loops in interacting random walk loop soups, Oberseminar AG Stochastik, Technische Universität Darmstatd, Fachbereich Mathematik, November 3, 2022.

• A. Quitmann, Macroscopic loops in interacting random walk loop soups, Oberseminar Wahrscheinlichkeitstheorie, Ludwig--Maximilians-Universität, Mathematisches Institut, November 14, 2022.

• H. Shafigh, Große Abweichungen für den Durchsatz zweier Kommunikationsmodelle, DIES Matematicus, November 25, 2022, Technische Universität Berlin, Institut für Mathematik, November 25, 2022.

• W. König, A large-deviations principle for all the components in a sparse inhomogeneous Erdős--Rényi graph, Workshop: Interacting Particle Systems and Hydrodynamic Limits, March 21 - 25, 2022, Université Montreal, Centre de Recherches de Mathématiques, Canada, March 22, 2022.

• W. König, Many-body Systems and the Interacting Bose Gas (minicourse), Random Point Processes in Statistical Physics, June 29 - July 1, 2022, Harnack House, Berlin.

• W. König, Self-repellent Brownian bridges in the interacting Bose gas, Oberseminar, Universität zu Köln, Mathematisches Institut, December 7, 2022.

• W. König, Spatial coagulation and gelation, Random Geometric Systems, First Annual Conference of SPP2265, April 11 - 14, 2022, Harnack-Haus, Tagungsstätte der Max--Planck-Gesellschaft, April 13, 2022.

• W. König, The free energy of a box version of the interacting Bose gas, Quantum Many Body System and Interacting Particles: In Honor of Herbert Spohn, June 20 - 24, 2022, Westfälische Wilhelms-Universität Münster, June 24, 2022.

• W. König, The free energy of a box-version of the interacting Bose gas, SPP 2265 Workshop on Random Spatial Networks, March 14 - 17, 2022, Universität zu Köln, Mathematisches Institut, March 17, 2022.

• R.I.A. Patterson, Large deviations with vanishing reactant concentrations, Workshop on Chemical Reaction Networks, July 6 - 8, 2022, Politecnico di Torino, Department of Mathematical Sciences ``G. L. Lagrange'', Torino, Italy, July 7, 2022.

• W. van Zuijlen, Weakly self avoiding walk in a random potential, Seminar d'UFR de mathématiques, Université Paris Cité, UFR de mathématiques, Paris, France, December 14, 2022.

• W. van Zuijlen, Weakly self-avoiding walk in a random potential, Forschungsseminar Wahrscheinlichkeitstheorie, Universität Potsdam, Institut für Mathematik, October 24, 2022.

Preprints im Fremdverlag

• J. Kern, The Skorokhod topologies: What they are and why we should care, Preprint no. arXiv:2210.16026, Cornell University, 2022, DOI 10.48550/arXiv.2210.16026 .
  Abstract
  This paper presents a gentle and informal introduction to the Skorokhod topologies. Focus is on motivating examples and concepts.