Die Theorie der Großen Abweichungen, ein Zweig der Wahrscheinlichkeitstheorie, stellt Mittel bereit zur Beschreibung der asymptotischen exponentiellen Abfallrate von sehr kleinen Wahrscheinlichkeiten für sehr große oder sehr kleine Werte eines Parameters. Beispiele für solche Parameter sind große Zeiten, große Anzahlen von Zufallsgrößen, der Radius großer Boxen, tiefe Temperaturen oder Approximationsparameter. Diese probabilistische Theorie ist auch unverzichtbar bei der Behandlung etlicher Modelle der statistischen Physik, denn sie macht sie einer Analyse mit Hilfe von Variationstechniken zugänglich. Am WIAS werden sowohl Theorie als auch diffizile Anwendungen in Physik und Chemie vorangetrieben.

Ausführlichere Darstellungen der WIAS-Forschungsthemen finden sich auf der jeweils zugehörigen englischen Seite.

Publikationen

  Monografien

  • B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/978-3-030-36090-0 .
    Abstract
    This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They may feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance.

  • W. König, Große Abweichungen, Techniken und Anwendungen, M. Brokate, A. Heinze , K.-H. Hoffmann , M. Kang , G. Götz , M. Kerz , S. Otmar, eds., Mathematik Kompakt, Birkhäuser Basel, 2020, VIII, 167 pages, (Monograph Published), DOI 10.1007/978-3-030-52778-5 .
    Abstract
    Die Lehrbuchreihe Mathematik Kompakt ist eine Reaktion auf die Umstellung der Diplomstudiengänge in Mathematik zu Bachelor- und Masterabschlüssen. Inhaltlich werden unter Berücksichtigung der neuen Studienstrukturen die aktuellen Entwicklungen des Faches aufgegriffen und kompakt dargestellt. Die modular aufgebaute Reihe richtet sich an Dozenten und ihre Studierenden in Bachelor- und Masterstudiengängen und alle, die einen kompakten Einstieg in aktuelle Themenfelder der Mathematik suchen. Zahlreiche Beispiele und Übungsaufgaben stehen zur Verfügung, um die Anwendung der Inhalte zu veranschaulichen. Kompakt: relevantes Wissen auf 150 Seiten Lernen leicht gemacht: Beispiele und Übungsaufgaben veranschaulichen die Anwendung der Inhalte Praktisch für Dozenten: jeder Band dient als Vorlage für eine 2-stündige Lehrveranstaltung

  • W. König, The Parabolic Anderson Model -- Random Walks in Random Potential, Pathways in Mathematics, Birkhäuser, Basel, 2016, xi+192 pages, (Monograph Published).

  Artikel in Referierten Journalen

  • B. Jahnel, U. Rozikov, Gibbs measures for hardcore-SOS models on Cayley trees, Journal of Statistical Mechanics: Theory and Experiment, (2024), 073202, DOI 10.1088/1742-5468/ad5433 .
    Abstract
    We investigate the finite-state p-solid-on-solid model, for p=∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1→3 →5 →6 →7. This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p>0 and, in the case of k=2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous.

  • 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.

  • 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, GN, 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 GN 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 GN. 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.

  • B. Jahnel, Ch. Külske, Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 29 (2023), pp. 3013--3032, DOI 10.3150/22-BEJ1572 .
    Abstract
    We consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [0,1]. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large p, as it changes only a small fraction p(1-p)2d of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.

  • 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. 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.

  • B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Adv. Appl. Math., 54 (2022), pp. 227--253, DOI 10.1017/apr.2021.25 .
    Abstract
    Signal-to-interference plus noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

  • C. Giardinà, C. Giberti, E. Magnanini, Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph, Journal of Statistical Physics, 182 (2021), pp. 23/1--23/22, DOI 10.1007/s10955-021-02707-3 .
    Abstract
    We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös--Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to t he one of an equi-bipartite graph.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Distribution of cracks in a chain of atoms at low temperature, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 22 (2021), pp. 4131--4172, DOI 10.1007/s00023-021-01076-7 .
    Abstract
    We consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy.

  • K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, The Annals of Probability, 49 (2021), pp. 1917--1964, DOI 10.1214/20-AOP1497 .
    Abstract
    In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula.

  • L. Andreis, W. König, R.I.A. Patterson, A large-deviations principle for all the cluster sizes of a sparse Erdős--Rényi random graph, Random Structures and Algorithms, 59 (2021), pp. 522--553, DOI 10.1002/rsa.21007 .
    Abstract
    A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

  • A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energy-dissipation landscapes via tilting: Three types of EDP convergence, Continuum Mechanics and Thermodynamics, 33 (2021), pp. 611--637, DOI 10.1007/s00161-020-00932-x .
    Abstract
    This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Poisson--Gilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/1--25/14, DOI 10.1214/20-ECP306 .
    Abstract
    We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

  • CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Electronic Communications in Probability, 25 (2020), pp. 41/1--41/12, DOI 10.1214/20-ECP322 .
    Abstract
    This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

  • A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Journal of Statistical Physics, 179 (2020), pp. 90--109, DOI 10.1007/s10955-020-02521-3 .
    Abstract
    In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the Poisson--Voronoi, Poisson--Delaunay and Poisson line tessellation, we also treat the Johnson--Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .
    Abstract
    We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

  • CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Markov Processes and Related Fields, 25 (2019), pp. 33--73.
    Abstract
    We derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), pp. 779--798, DOI 10.1080/14697688.2018.1529420 .
    Abstract
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • R.I.A. Patterson, D.R.M. Renger, Large deviations of jump process fluxes, Mathematical Physics, Analysis and Geometry, 22 (2019), pp. 21/1--21/32, DOI 10.1007/s11040-019-9318-4 .
    Abstract
    We study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration.

  • M. Heida, M. Röger, Large deviation principle for a stochastic Allen--Cahn equation, Journal of Theoretical Probability, 31 (2018), pp. 364--401, DOI 10.1007/s10959-016-0711-7 .
    Abstract
    The Allen-Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction-diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen-Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber [Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013)]. We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder-continuous in time, which extends results by Budhiraja, Dupuis and Maroulas [Ann. Probab. 36 (2008)]. From this result and a continuity argument we deduce a large deviation principle for the Allen-Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.

  • D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Journal of Non-Newtonian Fluid Mechanics, 28 (2018), pp. 1915--1957, DOI 10.1007/s00332-018-9471-0 .
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Journal of Statistical Physics, 172 (2018), pp. 1291--1326, DOI 10.1007/s10955-018-2083-0 .
    Abstract
    We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.

  • D.R.M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 596/1--596/26, DOI 10.3390/e20080596 .
    Abstract
    In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.

  • CH. Hirsch, B. Jahnel, R.I.A. Patterson, Space-time large deviations in capacity-constrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587--615, DOI 10.30757/ALEA.v15-24 .
    Abstract
    We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), pp. 779--798 (published online on 13.11.2018), DOI 10.1080/14697688.2018.1529420 .
    Abstract
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .
    Abstract
    Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

  • W. van Zuijlen, Large deviations of continuous regular conditional probabilities, Journal of Theoretical Probability, published online on 27.12.2016., DOI 10.1007/s10959-016-0733-1 .

  • E. Bolthausen, W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017), pp. 1598--1629.
    Abstract
    We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the “mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

  • J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Stochastic Processes and their Applications, 128 (2018), pp. 2228--2269 (published online on 21.09.2017), DOI 10.1016/j.spa.2017.09.010 .
    Abstract
    We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean?Vlasov type limit, as shown in two corollaries.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Queueing Systems. Theory and Applications, 88 (2018), pp. 349--387 (published online on 28.10.2017).
    Abstract
    We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

  • W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 2214--2228, DOI 10.1214/16-AIHP788 .
    Abstract
    We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large-deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .
    Abstract
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large-deviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 1061--1094.
    Abstract
    We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

  • S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, The Annals of Applied Probability, 25 (2015), pp. 930--973.
    Abstract
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

  • D. Belomestny, M. Ladkau, J.G.M. Schoenmakers, Simulation based policy iteration for American style derivatives -- A multilevel approach, SIAM ASA J. Uncertainty Quantification, 3 (2015), pp. 460--483.
    Abstract
    This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration.

  • M. Erbar, J. Maas, D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electronic Communications in Probability, 20 (2015), pp. 1--12.
    Abstract
    We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gamma-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.

  • O. Gün, W. König, O. Sekulović, Moment asymptotics for multitype branching random walks in random environment, Journal of Theoretical Probability, 28 (2015), pp. 1726--1742.
    Abstract
    We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/ρij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

  • W. König, T. Wolff, Large deviations for the local times of a random walk among random conductances in a growing box, Special issue for Pastur's 75th birthday, Markov Processes and Related Fields, 21 (2015), pp. 591--638.
    Abstract
    We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Zd. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.
    An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p=2 to these values.
    As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

  • A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 1293--1325.
    Abstract
    Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(zt,żt)=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

  • M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 1166--1188.

  • M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the Fokker--Planck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/1--1350017/43.

  • W. König, Ch. Mukherjee, Large deviations for Brownian intersection measures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 263--306.
    Abstract
    We consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)<d$. Let $ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $R^d$ that assigns to any measurable set $Asubset R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen citeCh09 derived the logarithmic asymptotics of the upper tails of the total mass $ell_t(R^d)$ as $ttoinfty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^-pell_t$ on the set of positive measures on some open bounded set $BsubsetR^d$ as $ttoinfty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $Usubset B$. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.

  • M. Becker, W. König, Self-intersection local times of random walks: Exponential moments in subcritical dimensions, Probability Theory and Related Fields, 154 (2012), pp. 585--605.
    Abstract
    Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the it Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for $ ell_t _p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[ ell_t _p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^1/d$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

  • S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting many-particle system, The Annals of Probability, 39 (2011), pp. 683--728.
    Abstract
    We consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^-beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.

  • W. König, P. Schmid, Brownian motion in a truncated Weyl chamber, Markov Processes and Related Fields, 17 (2011), pp. 499--522.
    Abstract
    We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

  • A. Collevecchio, W. König, P. Mörters, N. Sidorova, Phase transitions for dilute particle systems with Lennard--Jones potential, Communications in Mathematical Physics, 299 (2010), pp. 603--630.

  • G. Grüninger, W. König, Potential confinement property in the parabolic Anderson model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 45 (2009), pp. 840--863.

  • W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347--392.

  Beiträge zu Sammelwerken

  • W. König, Branching random walks in random environment, in: Probabilistic Structures in Evolution, E. Baake, A. Wakolbinger, eds., Probabilistic Structures in Evolution, EMS Series of Congress Reports, European Mathematical Society Publishing House, 2021, pp. 23--41, DOI 10.4171/ECR/17-1/2 .

  • B. Jahnel, W. König, Probabilistic methods for spatial multihop communication systems, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 239--268.

  • TH. Dickhaus, H. Finner, Asymptotic density crossing points of self-normalized sums and normal (electronic only), in: Proceedings of the 3rd Annual International Conference on Computational Mathematics, Computational Geometry and Statistics, CMCGS 2014, Singapore, February 3--4, 2014, Global Science and Technology Forum (GSTF), Singapore, pp. 84--88.

  Preprints, Reports, Technical Reports

  • 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.

  • B. Jahnel, L. Lüchtrath, M. Ortgiese, Cluster sizes in subcritical soft Boolean models, Preprint no. 3106, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3106 .
    Abstract, PDF (435 kByte)
    We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected.

  • P.P. Ghosh, B. Jahnel, S.K. Jhawar, Large and moderate deviations in Poisson navigations, Preprint no. 3096, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3096 .
    Abstract, PDF (318 kByte)
    We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewal-process arguments.

  • B. Jahnel, J. Köppl, Time-periodic behaviour in one- and two-dimensional interacting particle systems, Preprint no. 3092, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3092 .
    Abstract, PDF (311 kByte)
    We provide a class of examples of interacting particle systems on $Z^d$, for $din1,2$, that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.

  • L. Lüchtrath, Ch. Mönch, The directed age-dependent random connection model with arc reciprocity, Preprint no. 3090, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3090 .
    Abstract, PDF (304 kByte)
    We introduce a directed spatial random graph model aimed at modelling certain aspects of social media networks. We provide two variants of the model: an infinite version and an increasing sequence of finite graphs that locally converge to the infinite model. Both variants have in common that each vertex is placed into Euclidean space and carries a birth time. Given locations and birth times of two vertices, an arc is formed from younger to older vertex with a probability depending on both birth times and the spatial distance of the vertices. If such an arc is formed, a reverse arc is formed with probability depending on the ratio of the endpoints' birth times. Aside from the local limit result connecting the models, we investigate degree distributions, two different clustering metrics and directed percolation.

  • 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

  • CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixed-degree nodes in weighted random connection models, Preprint no. 3057, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3057 .
    Abstract, PDF (474 kByte)
    We present a process-level Poisson-approximation result for the degree-$k$ vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.

  • 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.

  • B. Jahnel, J. Köppl, B. Lodewijks, A. Tóbiás, Percolation in lattice k-neighbor graphs, Preprint no. 3028, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3028 .
    Abstract, PDF (437 kByte)
    We define a random graph obtained via connecting each point of ℤ d independently to a fixed number 1 ≤ k ≤ 2d of its nearest neighbors via a directed edge. We call this graph the emphdirected k-neighbor graph. Two natural associated undirected graphs are the emphundirected and the emphbidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k=1 even the undirected k-neighbor graph never percolates, but the directed one percolates whenever k≥ d+1, k≥ 3 and d ≥5, or k ≥4 and d=4. We also show that the undirected 2-neighbor graph percolates for d=2, the undirected 3-neighbor graph percolates for d=3, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

  • 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.

  • B. Jahnel, J. Köppl, On the long-time behaviour of reversible interacting particle systems in one and two dimensions, Preprint no. 3004, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3004 .
    Abstract, PDF (287 kByte)
    By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly non-translation-invariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits time-periodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of non-attractive stochastic Ising models this answers a question due to Liggett.

  • 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.

  • M. Mittnenzweig, Hydrodynamic limit and large deviations of reaction-diffusion master equations, Preprint no. 2521, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2521 .
    Abstract, PDF (389 kByte)
    We derive the hydrodynamic limit of a reaction-diffusion master equation, that combines an exclusion process with a reversible chemical master equation expression for the reaction rates. The crucial assumption is that the associated macroscopic reaction network has a detailed balance equilibrium. The hydrodynamic limit is given by a system of reaction-diffusion equations with a modified mass action law for the reaction rates. We provide the upper bound for large deviations of the empirical measure from the hydrodynamic limit.

  • R.I.A. Patterson, D.R.M. Renger, Dynamical large deviations of countable reaction networks under a weak reversibility condition, Preprint no. 2273, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2273 .
    Abstract, PDF (343 kByte)
    A dynamic large deviations principle for a countable reaction network including coagulation--fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.

  • P. Dupuis, V. Laschos, K. Ramanan, Large deviations for empirical measures generated by Gibbs measures with singular energy functionals, Preprint no. 2203, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2203 .
    Abstract, PDF (448 kByte)
    We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on n-particle configurations, each of which is defined in terms of an inverse temperature bn and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (bn)/(n) ® ¥, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed bn =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in “A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in “First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.

  Vorträge, Poster

  • B. Jahnel, Time-periodic behavior in one- and two-dimensional interacting particle systems (online talk), International Scientific Conference on Gibbs Measures and the Theory of Dynamical Systems (online event), May 20 - 21, 2024, Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan, Romanovskiy Institut of Mathematics and University of Exact and Social Sciences, Tashkent, Uzbekistan, May 20, 2024.

  • J. Köppl, The long-time behaviour of interacting particle systems, Stochastic Processes and Related Fields, Kyoto, September 4 - 8, 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, Spatial coagulation and gelation, SPP2265-Reviewer-Kolloquium, Köln, August 29, 2023.

  • B. Jahnel, Continuum percolation in random environment, Oberseminar zur Stochastik, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, June 22, 2023.

  • B. Jahnel, Percolation, Oberseminar, Technische Universität Braunschweig, Institut für Mathematische Stochastik, November 8, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Seminar der Fakultät Informatik, Hochschule Reutlingen, October 6, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, November 2, 2023.

  • B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, January 30, 2023.

  • B. Jahnel, Subcritical percolation phases for generalized weight-dependent random connection models, 21st INFORMS Applied Probability Society Conference, June 28 - 30, 2023, Centre Prouvé, Nancy, France, June 29, 2023.

  • B. Jahnel, Subcritical percolation phases for generalized weight-dependent random connection models, DMV Annual Meeting 2023, Minisymposium MS 12 ``Random Graphs and Statistical Network Analysis'', September 25 - 28, 2023, Technische Universität Ilmenau, September 25, 2023.

  • B. Jahnel, 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 29, 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 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.

  • 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.

  • E. Magnanini, Limit theorems for the edge density in exponential random graphs, Workshop ``Junior Female Researchers in Probability'', October 4 - 6, 2021, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, October 5, 2021.

  • E. Magnanini, Limit theorems for the edge density in exponential random graphs, Probability Seminar, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Italy, November 17, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), DYOGENE Seminar (Online Event), INRIA Paris, France, January 11, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), Probability Seminar Bath (Online Event), University of Bath, Department of Mathematical Sciences, UK, October 18, 2021.

  • L. Andreis, A large deviations approach to sparse random graphs (online talk), Bernoulli--IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics. Bernoulli-IMS One World Symposium 2020, August 25, 2020.

  • L. Andreis, Sparse inhomogeneous random graphs from a large deviation point of view (online talk), Probability Seminar (Online Event), University of Bath, Department of Mathematical Sciences, UK, June 1, 2020.

  • L. Andreis, The phase transition in random graphs and coagulation processes: A large deviation approach (online talk), Seminar of DISMA (Online Event), Politecnico di Torino, Department of Mathematical Sciences (DISMA), Italy, July 14, 2020.

  • R.I.A. Patterson, Interpreting LDPs without detailed balance, Workshop ``Variational Methods for Evolution'', September 13 - 19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

  • D.R.M. Renger, Reaction fluxes, Applied Mathematics Seminar, University of Birmingham, School of Mathematics, UK, April 4, 2019.

  • L. Andreis, A large-deviations approach to the multiplicative coagulation process, Workshop ``Woman in Probability'', May 31 - June 1, 2019, Technische Universität München, Fakultät für Mathematik, May 31, 2019.

  • L. Andreis, Coagulating particles and gelation phase transition: A large-deviation approach, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Vietri sul Mare, Italy, June 19, 2019.

  • L. Andreis, Large-deviation approach to coagulation processes and gelation, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ''G. L. Lagrange``, Italy, July 2, 2019.

  • B. Jahnel, Continuum percolation in random environment, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra.

  • R.I.A. Patterson, Fluctuations and confidence intervals for stochastic particle simulations, First Berlin--Leipzig Workshop on Fluctuating Hydrodynamics, August 26 - 30, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, August 29, 2019.

  • R.I.A. Patterson, Flux large deviations, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ``G. L. Lagrange``, Italy, July 2, 2019.

  • R.I.A. Patterson, Flux large deviations, Seminar, Statistical Laboratory, University of Cambridge, Faculty of Mathematics, UK, May 7, 2019.

  • R.I.A. Patterson, The role of fluctuating hydrodynamics in the CRC 1114, CRC 1114 School 2019: Fluctuating Hydrodynamics, Zuse Institute Berlin (ZIB), October 28, 2019.

  • M. Maurelli, Sanov theorem for Brownian rough paths and an application to interacting particles, Università di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Italy, January 18, 2018.

  • D.R.M. Renger, Gradient and GENERIC structures from flux large deviations, POLYPHYS Seminar, Eidgenössische Technische Hochschule Zürich, Department of Materials, Zürich, Switzerland, March 28, 2018.

  • D.R.M. Renger, Gradient and GENERIC structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

  • D.R.M. Renger, Gradient and Generic structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

  • D.R.M. Renger, Large deviations for reaction fluxes, Workshop on Transformations and Phase Transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, January 29, 2018.

  • D.R.M. Renger, Large deviations for reaction fluxes, Università degli Studi dell'Aquila, Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, L'Aquila, Italy, January 10, 2018.

  • W. van Zuijlen, A Hamilton--Jacobi point of view on mean-field Gibbs-non-Gibbs transitions, Workshop on Transformations and Phase Transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

  • L. Andreis, A large-deviations approach to the multiplicative coagulation process, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica ``Tullio Levi--Civita'', Italy, October 12, 2018.

  • L. Andreis, A large-deviations approach to the multiplicative coagulation process, Seminar ''Theory of Complex Systems and Neurophysics --- Theory of Statistical Physics and Nonlinear Dynamics``, Humboldt-Universität zu Berlin, Institut für Physik, October 30, 2018.

  • CH. Bayer, Short-time near-the-money skew in rough fractional volatility models, 9-th International Workshop on Applied Probability, June 18 - 21, 2018, Eörvös Loránd University (ELU), Budapest, Hungary, June 19, 2018.

  • A. Mielke, Construction of effective gradient systems via EDP convergence, Workshop on Mathematical Aspects of Non-Equilibrium Thermodynamics, March 5 - 7, 2018, Rheinisch-Westfälische Technische Hochschule Aachen, March 6, 2018.

  • D.R.M. Renger, Gradient flows and GENERIC in flux space, Workshop ``Variational Methods for Evolution'', November 12 - 18, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 16, 2017.

  • D.R.M. Renger, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27 - March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, March 1, 2017.

  • A. Mielke, Perspectives for gradient flows, GAMM-Workshop on Analysis of Partial Differential Equations, September 27 - 29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

  • M. Heida, Large deviation principle for a stochastic Allen--Cahn equation, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 25, 2016.

  • M. Maurelli, Enhanced Sanov theorem and large deviations for interacting particles, Workshop ``Rough Paths, Regularity Structures and Related Topics'', May 1 - 7, 2016, Mathematisches Forschungsinstitut Oberwolfach, May 5, 2016.

  • R.I.A. Patterson, Pathwise LDPs for chemical reaction networks, 12th German Probability and Statistics Days 2016 --- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 4, 2016.

  • D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 30, 2015.

  • M. Maurelli, A large deviation principle for enhanced Brownian empirical measure, 4th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, December 7 - 9, 2015, WIAS Berlin, December 8, 2015.

  • M. Maurelli, A large deviation principle for interacting particle SDEs via rough paths, 38th Conference on Stochastic Processes and their Applications, July 13 - 17, 2015, University of Oxford, Oxford-Man Institute of Quantitative Finance, UK, July 14, 2015.

  • M. Maurelli, Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles, Seminar Stochastic Analysis, Imperial College London, UK, October 20, 2015.

  • D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Workshop on Gradient Flows, Large Deviations and Applications, November 22 - 29, 2015, EURANDOM, Mathematics and Computer Science Department, Eindhoven, Netherlands, November 23, 2015.

  • D.R.M. Renger, The empirical process of reacting particles: Large deviations and thermodynamic principles, Minisymposium ``Real World Phenomena Explained by Microscopic Particle Models'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 8 - 22, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 2015.

  • D.R.M. Renger, The inverse problem: From gradient flows to large deviations, Workshop ``Analytic Approaches to Scaling Limits for Random System'', January 26 - 30, 2015, Universität Bonn, Hausdorff Research Institute for Mathematics, January 26, 2015.

  • A. Mielke, Chemical Master Equation: Coarse graining via gradient structures, Kolloquium des SFB 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, Fachbereich Mathematik, Berlin, June 4, 2015.

  • A. Mielke, Evolutionary $Gamma$-convergence for generalized gradient systems, Workshop ``Gradient Flows'', June 22 - 23, 2015, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France, June 22, 2015.

  • A. Mielke, The Chemical Master Equation as a discretization of the Fokker--Planck and Liouville equation for chemical reactions, Colloquium of Collaborative Research Center/Transregio ``Discretization in Geometry and Dynamics'', Technische Universität Berlin, Institut für Mathematik, Berlin, February 10, 2015.

  • A. Mielke, The Chemical Master Equation as entropic gradient flow, Conference ``New Trends in Optimal Transport'', March 2 - 6, 2015, Universität Bonn, Institut für Angewandte Mathematik, March 2, 2015.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Conference ``Stochastic Processes and High Dimensional Probability Distributions'', June 16 - 20, 2014, Euler International Mathematical Institute, Saint-Petersburg, Russian Federation, June 20, 2014.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Conference on Nonlinearity, Transport, Physics, and Patterns, October 6 - 10, 2014, Fields Institute for Research in Mathematical Sciences, Toronto, Canada, October 9, 2014.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Oberseminar ``Stochastische und Geometrische Analysis'', Universität Bonn, Institut für Angewandte Mathematik, May 28, 2014.

  • TH. Dickhaus, H. Finner, Asymptotic density crossing points of self-normalized sums and normal, 3rd Annual International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2014), February 3 - 4, 2014, Singapore, February 3, 2014.

  • T. Wolff, Annealed asymptotics for occupation time measures of a random walk among random conductances, ``Young European Probabilists 2013 (YEP X)'' and ``School on Random Polymers'', January 8 - 12, 2013, EURANDOM, Eindhoven, Netherlands, January 10, 2013.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, SFB/TR12 Workshop, November 4 - 8, 2012, Universität zu Köln, SFB TR12 ``Symmetries and Universality in Mesoscopic Systems'', Langeoog, November 7, 2012.

  • W. König, Large deviations for the cluster size distributions in a classical interacting many-particle system with Lennard--Jones potential, Mark Kac Seminar, Eindhoven University of Technology, Netherlands, November 9, 2012.

  • M. Becker , Random walks and self-intersections, Evolving Complex Networks (ECONS) Phd-Student Meeting, WIAS, August 24, 2010.

  • S. Jansen, M. Aizenman, P. Jung, Symmetry breaking in quasi 1D Coulomb systems, Combinatorics and Analysis in Spatial Probability --- ESF Mathematics Conference in Partnership with EMS and ERCOM, Eindhoven, Netherlands, December 12 - 18, 2010.

  • B. Metzger, The parabolic Anderson model: The asymptotics of the statistical moments and Lifshitz tails revisited, EURANDOM, Eindhoven, Netherlands, December 1, 2010.

  • W. König, Die Universalitätsklassen im parabolischen Anderson-Modell, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, July 7, 2010.

  • W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, University of Bath, Department of Mathematical Sciences, UK, April 14, 2010.

  • W. König, The parabolic Anderson model, XIV Escola Brasileira de Probabilidade, August 2 - 7, 2010, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil.

  Preprints im Fremdverlag

  • A. Bianchi, F. Collet, E. Magnanini, Limit theorems for exponential random graphs, Preprint no. arXiv:2105.06312, Cornell University Library, arXiv.org, 2021.
    Abstract
    We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs

  • A. Bianchi, F. Collet, E. Magnanini, The GHS and other inequalities for the two-star model, Preprint no. arXiv:2107.08889, Cornell University Library, arXiv.org, 2021.
    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

  • CH. Bayer, P. Friz, P. Gassiat, J. Martin, B. Stemper , A regularity structure for rough volatility, Preprint no. arXiv:1710.07481, Cornell University Library, arXiv.org, 2017.
    Abstract
    A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.

  • J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Preprint no. arxiv:1602.08043, Cornell University Library, arXiv.org, 2016.