Publications

Monographs

  • 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, Springer, 2020, (Monograph Published).
    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

  • P. Friz, W. König, Ch. Mukherjee, S. Olla, eds., Probability and Analysis in Interacting Physical Systems. In Honor of S.R.S. Varadhan, Berlin, August, 2016, 283 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham, 2019, 294 pages, (Collection Published), DOI https://doi.org/10.1007/978-3-030-15338-0 .

Articles in Refereed Journals

  • 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.

  • D.R.M. Renger, J. Zimmer, Orthogonality of fluxes in general nonlinear reaction networks, Discrete and Continuous Dynamical Systems -- Series S, published online on 19.05.2020, DOI 10.3934/dcdss.2020346 .
    Abstract
    We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

  • 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

  • F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/1--8/39 (published online on 28.11.2019), DOI 10.1007/s00526-019-1663-4 .
    Abstract
    We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

  • C. Bartsch, V. Wiedmeyer, Z. Lakdawala, R.I.A. Patterson, A. Voigt, K. Sundmacher, V. John, Stochastic-deterministic population balance modeling and simulation of a fluidized bed crystallizer experiment, Chemical Engineering Sciences, 208 (2019), pp. 115102/1--115102/14, DOI 10.1016/j.ces.2019.07.020 .

  • V. Betz, L. Taggi, Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations, Electronic Journal of Probability, 24 (2019), pp. 74/1--74/33, DOI 10.1214/19-EJP328 .

  • 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.

  • B. Lees, L. Taggi, Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N, Communications in Mathematical Physics, pp. published online on 07.12.2019, urlhttps://doi.org/10.1007/s00220-019-03647-6, DOI 10.1007/s00220-019-03647-6 .

  • A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twin-screw granulation model, , 203 (2019), pp. 358--379, DOI https://doi.org/10.1016/j.ces.2019.03.078 .

  • L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 590--606.
    Abstract
    We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.

  • C. Bartsch, V. John, R.I.A. Patterson, Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach, Comput. Chem. Engng., 124 (2019), pp. 350--363, DOI 10.1016/j.compchemeng.2019.01.012 .
    Abstract
    A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.

  • B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Communications in Mathematical Physics, 366 (2019), pp. 139--172, DOI 10.1007/s00220-019-03352-4 .
    Abstract
    We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

  • B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Journal of Theoretical Probability, pp. published online on 03.11.2019, urlhttps://doi.org/10.1007/s10959-019-00960-7, DOI 10.1007/s10959-019-00960-7 .
    Abstract
    We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

  • W. König, A. Tóbiás, A Gibbsian model for message routeing in highly dense multihop networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 16 (2019), pp. 211--258, DOI 10.30757/ALEA.v16-08 .
    Abstract
    We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure.

  • W. König, A. Tóbiás, Routeing properties in a Gibbsian model for highly dense multihop networks, IEEE Transactions on Information Theory, 65 (2019), pp. 6875--6897, DOI 10.1109/TIT.2019.2924187 .
    Abstract
    We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized

  • R.I.A. Patterson, D.R.M. Renger, Large deviations of jump process fluxes, Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics, 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.

Contributions to Collected Editions

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, IEEE 18th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), Institut of Electrical and Electronics Engineer (IEEE), 2020, pp. 1--6.
    Abstract
    We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

  • CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, IEEE 17th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOPT), Avignon, France, June 3 - 7, 2019, Institut of Electrical and Electronics Engineer (IEEE), 2019, pp. 1--6, DOI 10.23919/WiOPT47501.2019.9144122 .
    Abstract
    The typical cell is a key concept for stochastic-geometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattan-type systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks.

  • 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.

Preprints, Reports, Technical Reports

  • D. Gabrielli, D.R.M. Renger, Dynamical phase transitions for flows on finite graphs, Preprint no. 2746, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2746 .
    Abstract, PDF (304 kByte)
    We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

  • B. Lees, L. Taggi, Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field, Preprint no. 2730, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2730 .
    Abstract, PDF (381 kByte)
    We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.

  • L. Taggi, Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity, Preprint no. 2722, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2722 .
    Abstract, PDF (311 kByte)
    We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events, extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks.

  • B. Lees, L. Taggi, Site-monotonicity properties for reflection positive measures with applications to quantum spin systems, Preprint no. 2713, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2713 .
    Abstract, PDF (298 kByte)
    We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model, lattice permutations, thus extending the previous results of Lees and Taggi (2019).

  • B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Preprint no. 2704, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2704 .
    Abstract, PDF (389 kByte)
    We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

  • J.-D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, Preprint no. 2685, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2685 .
    Abstract, PDF (335 kByte)
    We consider additive functionals of stationary Markov processes and show that under Kipnis--Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a non-reversible Ornstein-Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale.

  • F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a Galton--Watson tree, Preprint no. 2675, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2675 .
    Abstract, PDF (428 kByte)
    We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, Preprint no. 2674, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2674 .
    Abstract, PDF (3133 kByte)
    We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

  • B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Preprint no. 2659, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2659 .
    Abstract, PDF (356 kByte)
    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.

  • L. Taggi, Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $mathbbZ^d, d > 2$, via reflection positivity, Preprint no. 2647, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2647 .
    Abstract, PDF (580 kByte)
    Our first main result is that correlations between monomers in the dimer model in d do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Gilbert graphs, Preprint no. 2642, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2642 .
    Abstract, PDF (219 kByte)
    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.

  • D. Heydecker, R.I.A. Patterson, Bilinear coagulation equations, Preprint no. 2637, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2637 .
    Abstract, PDF (453 kByte)
    We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.

  • K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, Preprint no. 2606, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2606 .
    Abstract, PDF (588 kByte)
    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.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Surface energy and boundary layers for a chain of atoms at low temperature, Preprint no. 2589, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2589 .
    Abstract, PDF (529 kByte)
    We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As the temperature goes to zero and at fixed positive pressure, the Gibbs measures  for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature.

  • L. Andreis, W. König, R.I.A. Patterson, A large-deviations approach to gelation, Preprint no. 2568, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2568 .
    Abstract, PDF (338 kByte)
    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.

Talks, Poster

  • A. Hinsen, Malware propagation in urban D2D networks., The 14th Workshop on Spatial Stochastic Models for Wireless Networks (SPASWIN), June 19, 2020, online event, Greece, June 19, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastik Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, Israel, February 25, 2020.

  • T. Orenshtein, Functional Inequalities on sub-Riemannian manifolds via QCD, Geometric Functional Analysis and Probability Seminar, The Weizmann Institute of Science, Faculty of Mathematics and Computer Science, Rechovot, Israel, March 5, 2020.

  • W. van Zuijlen, Spectral asymptotics of the Anderson Hamiltonian, Forschungsseminar ''Functional Analysis``, Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Analysis, January 21, 2020.

  • L. Andreis, A large deviations approach to sparse random graphs, Bernoulli--IMS One World Symposium 2020, August 24 - 28, 2020, online Vortrag, August 25, 2020.

  • L. Andreis, A large-deviations approach to the phase transitionin inhomogeneous random graphs: part II, Spring School: Complex Networks, March 2 - 6, 2020, Technische Universität Darmstadt, Fachbereich Mathematik, March 2, 2020.

  • L. Andreis, Phase transitions in inhomogeneous random graphs and coagulation processes, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastik Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, Israel, February 25, 2020.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes, Bernoulli-IMS One World Symposium 2020, August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics, August 27, 2020.

  • W. König, Titel folgt (kein Archiv auf HP), Oberseminar Mathematische Stochastik, Westfälischen Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, January 22, 2020.

  • L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-correlations for the dimer model in $Z^d, d > 2$, Probability Seminar, University of Warwick, Mathematics Institute, UK, January 16, 2020.

  • L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-monomer correlations in the dimer model in $Z^d, d > 2$, Probability Seminar, University of Bristol, School of Mathematics Research, France, April 7, 2020.

  • L. Taggi, Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-monomer correlations in the dimer model in $Z^d, d > 2$, Probability Seminar, University of Bristol, School of Mathematics, UK, February 7, 2020.

  • A. Hinsen, Data mobility in ad-hoc networks: Vulnerability and security, KEIN öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

  • A. Hinsen, IPS in telecommunication I, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 4, 2019.

  • A. Hinsen, IPS in telecommunication II, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 4, 2019.

  • A. Hinsen, Introduction to interacting particles systems (IPS), Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra, October 2, 2019.

  • A. Hinsen, The White Knight model --- An epidemic on a spatial random network, Bocconi Summer School in Advanced Statistics and Probability, Lake Como School of Advanced Studies, Lake Como, Italy, July 8 - 19, 2019.

  • A. Hinsen, Typical Voronoi cells for Cox point processes on Manhattan grids, The International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt 2019) [The 13th Workshop on Spatial Stochastic Models for Wireless Networks (SpasWin 2019 )], June 7, 2019, Avignon, France, June 7, 2019.

  • T. Orenshtein, Random walks in random environment as rough paths, Probability Seminar, New York University Shanghai, Institute of Mathematical Sciences, Shanghai, China, November 26, 2019.

  • D.R.M. Renger, A generic formulation of a chemical reaction network from Onsager--Machlup theory, Conference to Celebrate 80th Jubilee of Miroslav Grmela, May 18 - 19, 2019, Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Prague, May 19, 2019.

  • D.R.M. Renger, Macroscopic fluctuation theory of chemical reaction networks, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ``G. L. Lagrange``, Italy, July 2, 2019.

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

  • W. van Zuijlen, Bochner integrals in ordered vector spaces, Analysis Seminar, University of Canterbury, Department of Mathematics and Statistics, UK, March 1, 2019.

  • W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, 12th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, December 4 - 6, 2019, University of Oxford, Mathematical Institute, UK, December 6, 2019.

  • W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, Seminar Forschergruppe 2402: Research Unit --- Rough paths, stochastic partial differential equations and related topics, Technische Universität Berlin, Institut für Mathematik, December 12, 2019.

  • W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

  • W. van Zuijlen, Mini-course on Besov spaces I--III, Junior Trimester Program: Randomness, PDEs and Nonlinear Fluctuations (Sept. 2 to Dec. 19, 2019), October 16 - November 6, 2019, Hausdorff Research Institute for Mathematics (HIM), Bonn.

  • W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Stochastic Analysis Seminar, University of Oxford, Mathematical Institute, UK, February 4, 2019.

  • W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Analysis and Probability Seminar, Imperial College London, Department of Mathematics, UK, February 5, 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, Coagulation processes and gelation from a large deviation point of view, BMS -- BGSMath Junior Meeting 2019, June 26 - 28, 2019, Berlin Mathematical School (BMS), Barcelona Graduate School of Mathematics (BGSMath), Technische Universität Berlin, June 26, 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.

  • L. Andreis, Multiplicative coagulation process and random graphs in sparse regime: a large-deviations approach, STAR Workshop on Random Graphs 2019, April 10 - 12, 2019, University of Groningen, Department of Mathematics and Natural Sciences, Netherlands, April 12, 2019.

  • L. Andreis, Phase transitions in coagulation processes and random graphs, Workshop ``Welcome Home 2019'', December 19 - 20, 2019, Università di Torino, Dipartimento di Matematica ``G. Peano'', Italy, December 19, 2019.

  • B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Quaid-i-Azam University Islamabad, Department of Mathematics, Pakistan, November 19, 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.

  • B. Jahnel, Coverage and mobility in device-to-device networks, Kein öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

  • B. Jahnel, Dynamical Gibbs-non-Gibbs transitions for the continuum Widom--Rowlinson model, The 41st Conference on Stochastic Processes and their Applications 2019 (SPA 2019), July 8 - 12, 2019, Northwestern University Evanston, USA, July 9, 2019.

  • B. Jahnel, Is the Mathern process Gibbs?, Workshop on Stochastic Modeling of Complex Systems, GWOT '19, April 8 - 12, 2019, Universität Mannheim, Institut für Mathematik, April 9, 2019.

  • B. Jahnel, Three models for data propagation in mobile ad-hoc networks, kein öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

  • W. König, A large-deviations approach to coagulation, Workshop on Stochastic Modeling of Complex Systems, GWOT '19, April 8 - 12, 2019, Universität Mannheim, Institut für Mathematik, April 10, 2019.

  • W. König, A large-deviations approach to coagulation, Maxwell Analysis Seminar, Harriot Watt University, The Maxwell Institute for Mathematical Sciences, Edinburgh, UK, December 27, 2019.

  • W. König, A large-deviations approach to the multiplicative coalescent, Math Probability Seminar Series, New York University Shanghai, Institute of Mathematical Sciences, China, February 19, 2019.

  • W. König, A large-deviations approach to the multiplicative coalescent, 18. Erlanger-Münchner Tag der Stochastik / Probability Day 2019, Friedrich-Alexander-Universität Erlangen--Nürnberg, Department Mathematik, May 10, 2019.

  • W. König, A large-deviations approach to the multiplicative coalescent, Oberseminar Wahrscheinlichkeitstheorie, Universität München, Mathematisches Institut, November 25, 2019.

  • W. König, Cluster size distribution in a classical many-body system, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Materialphysik im Weltraum, Köln, June 18, 2019.

  • W. König, EF4: Particles and Agents, 1st MATH+ Day, Berlin, December 13, 2019.

  • W. König, EF4: Particles and Agents, MATH+ Day, URANIA, December 13, 2019.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 17, 2019.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Workshop on Spectral Properties of Disordered Systems, January 7 - 11, 2019, Paris, France, January 11, 2019.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Workshop on Dynamics, Random Media and Universality of Complex Physical Systems, August 26 - 30, 2019, Westfälische Wilhelms-Universität Münster, Department of Mathematics and Computer Science, August 27, 2019.

  • W. König, Micro-macro phase transitions in coagulating particle systems, 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, A novel simulation method for stochastic particle systems, Seminar, Department of Chemical Engineering and Biotechnology, University of Cambridge, Faculty of Mathematics, UK, May 9, 2019.

  • 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, Interaction clusters for the Kac process, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

  • R.I.A. Patterson, Interaction clusters for the Kac process, Workshop on Effective Equations: Frontiers in Classical and Quantum Systems, June 24 - 28, 2019, Hausdorff Research Institute for Mathematics, Bonn, June 28, 2019.

  • R.I.A. Patterson, Kinetic interaction clusters, Oberseminar, Martin-Luther-Universität Halle-Wittenberg, Naturwissenschaftliche Fakultät II -- Chemie, Physik und Mathematik, April 17, 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.

  • L. Taggi, Absorbing-state phase transition in activated random walk and oil and water, Probability Seminars, Università degli Studi ``La Sapienza'' di Roma, Italy, October 1, 2019.

  • L. Taggi, Critical density in activated random walks, Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems, Tel Aviv University, School of Mathematical Sciences, Israel, May 20, 2019.

  • L. Taggi, Essential enhancements for activated random walks, Second Italian Meeting on Probability and Mathematical Statistics, June 17 - 20, 2019, Vietri sul Mare, Italy, June 19, 2019.

  • L. Taggi, Non-decay of correlations in the dimer model and phase transition in lattice permutations in $Z^d, d > 2$ via reflection positivity, Meeting of the Swiss Mathematical Society: Recent Advances in Loop Models and Height Functions, September 2 - 4, 2019, Université Fribourg, Switzerland, September 3, 2019.

  • L. Taggi, Phase transition in lattice permutations and uniformly positive correlations in the dimer model in $Z^d, d > 2$, via reflection positivity, Kolloquium des Fachbereichs Mathematik, Technische Universität Darmstadt, December 5, 2019.

  • L. Taggi, Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $Z^d, d > 2$, via reflection positivity, Seminaire Probabilités et Statistiques, Université Claude Bernard Lyon 1, Institut Camille Jordan (ICJ), France, November 14, 2019.

External Preprints

  • E. Candellero, A. Stauffer, L. Taggi, Abelian oil and water dynamics does not have an absorbing-state phase transition, Preprint no. arXiv:1901.08425v1, Cornell University Library, 2019.

  • V. Betz, H. Schäfer, L. Taggi, Interacting self-avoiding polygons, Preprint no. arXiv:1902.08517, Cornell University Library, 2019.
    Abstract
    We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.

  • J.-D.D. Deuschel, H.E. Altman , T. Orenshtein, Über die mit Benetzungsmodellen verbundene Gradientendynamik, Preprint no. arXiv: 1908.08850, Cornell University Library, arXiv.org, 2019.

  • D. Heydecker , R.I.A. Patterson, Kac interaction clusters: A bilinear coagulation equation and phase transition, Preprint no. arXiv:1902.07686, Cornell University Library, 2019.
    Abstract
    We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing well-posedness, and a closed form for the time of the gelation phase transition tg when a macroscopic cluster suddenly emerges. We further prove that the second moment of the cluster size distribution diverges exactly at tg. Our methods apply immediately to coagulating particle systems with other bilinear coagulation kernels.

  • B. Lees, L. Taggi, Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N, Preprint no. arXiv:1902.07252, Cornell University Library, 2019.