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

Artikel in Referierten Journalen

  • 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, (2021), published online on 05.04.2021, 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. 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.

  • T. Orenshtein, Ch. Sabot, Random walks in random hypergeometric environment, Electronic Journal of Probability, 25 (2020), pp. 33/1--33/21, DOI 10.1214/20-EJP429 .
    Abstract
    We consider one-dependent random walks on Zd in random hypergeometric environment for d ≥ 3. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is absolutely continuous with respect to the initial distribution on the environment in terms of a function ? of the initial weights. These results generalize [Sab11] and [Sab13] on random walks in Dirichlet environment. It turns out that ? coincides with the corresponding parameter in the Dirichlet case, and so in particular the existence of such invariant measures is independent of the weights on pairs of directed edges, and determined solely by the weights on directed edges.

  • D.R.M. Renger, J. Zimmer, Orthogonality of fluxes in general nonlinear reaction networks, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 205--217 (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.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Surface energy and boundary layers for a chain of atoms at low temperature, Archive for Rational Mechanics and Analysis, 239 (2021), pp. 915--980 (published online on 21.12.2020), DOI 10.1007/s00205-020-01587-3 .
    Abstract
    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.

  • V. Betz, H. Schäfer, L. Taggi, Interacting self-avoiding polygons, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 56 (2020), pp. 1321--1335, DOI 10.1214/19-AIHP1003 .
    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. Deuschel, T. Orenshtein, Scaling limit of wetting models in 1+1 dimensions pinned to a shrinking strip, Stochastic Processes and their Applications, 130 (2020), pp. 2778--2807, DOI 10.1016/j.spa.2019.08.001 .

  • D. Gabrielli, D.R.M. Renger, Dynamical phase transitions for flows on finite graphs, Journal of Statistical Physics, 181 (2020), pp. 2353--2371, DOI 10.1007/s10955-020-02667-0 .
    Abstract
    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.

  • 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.

  • CH. Kwofie, I. Akoto, K. Opoku-Ameyaw, Modelling the dependency between inflation and exchange rate using copula, Journal of Probability and Statistics, 2020 (2020), pp. 2345746/1--2345746/7, DOI 10.1155/2020/2345746 .
    Abstract
    n this paper, we propose a copula approach in measuring the dependency between inflation and exchange rate. In unveiling this dependency, we first estimated the best GARCH model for the two variables. Then, we derived the marginal distributions of the standardised residuals from the GARCH. The Laplace and generalised t distributions best modelled the residuals of the GARCH(1,1) models, respectively, for inflation and exchange rate. These marginals were then used to transform the standardised residuals into uniform random variables on a unit interval [0, 1] for estimating the copulas. Our results show that the dependency between inflation and exchange rate in Ghana is approximately 7%.

  • 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, 376 (2020), pp. 487--520, DOI https://doi.org/10.1007/s00220-019-03647-6 .

  • 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.

  • L. Andreis, M. Flora, F. Fontini, T. Vargiolu, Pricing reliability options under different electricity price regimes, Energy Economics, 87 (2020), pp. 104705/1--104705/25, DOI 10.1016/j.eneco.2020.104705 .
    Abstract
    Reliability Options are capacity remuneration mechanisms aimed at enhancing security of supply in electricity systems. They can be framed as call options on electricity sold by power producers to System Operators. This paper provides a comprehensive mathematical treatment of Reliability Options. Their value is first derived by means of closed-form pricing formulae, which are obtained under several assumptions about the dynamics of electricity prices and strike prices. Then, the value of the Reliability Option is simulated under a real-market calibration, using data of the Italian power market. We perform sensitivity analyses to highlight the role of the level and volatility of both power and strike price, of the mean reversion speeds and of the correlation coefficient on the Reliability Options' value. Finally, we calculate the parameter model risk to quantify the impact that a model misspecification has on the equilibrium value of the RO.

Beiträge zu Sammelwerken

  • F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a Galton--Watson tree, in: In and out of equilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., 77 of Progress in Probability, Birkhäuser, 2021, pp. XXIII, 820, DOI 10.1007/978-3-030-60754-8 .
    Abstract
    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, in: IEEE 18th International Symposium on on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks, (WiOpt), Volos, Greece, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 1--9.
    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.

  • A. Hinsen, Ch. Hirsch, B. Jahnel, E. Cali, Typical Voronoi cells for Cox point processes on Manhatten grids, in: 2019 International Symposium on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks (WiOPT), Avignon, France, 2019, Institute of Electrical and Electronics Engineers (IEEE), 2020, 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.

Preprints, Reports, Technical Reports

  • R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuation-theory perspective, Preprint no. 2826, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2826 .
    Abstract, PDF (522 kByte)
    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.

  • N. Nüsken, D.R.M. Renger, Stein variational gradient descent: Many-particle and long-time asymptotics, Preprint no. 2819, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2819 .
    Abstract, PDF (430 kByte)
    Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: emphvariational inference and emphMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the emphStein-Fisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

  • A. Agazzi, L. Andreis, R.I.A. Patterson, D.R.M. Renger, Large deviations for Markov jump processes with uniformly diminishing rates, Preprint no. 2816, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2816 .
    Abstract, PDF (358 kByte)
    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.

  • T. Orenshtein, Rough invariance principle for delayed regenerative processes, Preprint no. 2809, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2809 .
    Abstract, PDF (267 kByte)
    We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Distribution of cracks in a chain of atoms at low temperature, Preprint no. 2789, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2789 .
    Abstract, PDF (414 kByte)
    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.

  • W. König, Branching random walks in random environment: A survey, Preprint no. 2779, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2779 .
    Abstract, PDF (253 kByte)
    We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.

  • B. Jahnel, A. Tóbiás, Absence of percolation in graphs based on stationary point processes with degrees bounded by two, Preprint no. 2774, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2774 .
    Abstract, PDF (548 kByte)
    We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

  • N. Perkowski, W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift, Preprint no. 2768, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2768 .
    Abstract, PDF (308 kByte)
    We consider the stochastic differential equation on ℝ d given by d X t = b(t,Xt ) d t + d Bt, where B is a Brownian motion and b is considered to be a distribution of regularity > - 1/2. We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat kernel bounds for Γt with explicit dependence on t and the norm of b.

  • M.A. Peletier, D.R.M. Renger, Fast reaction limits via $Gamma$-convergence of the flux rate functional, Preprint no. 2766, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2766 .
    Abstract, PDF (492 kByte)
    We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

  • W. König, N. Perkowski, W. van Zuijlen, Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential, Preprint no. 2765, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2765 .
    Abstract, PDF (471 kByte)
    We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.

  • J.-D. Deuschel, T. Orenshtein, G.R. Moreno Flores, Aging for the stationary Kardar--Parisi--Zhang equation and related models, Preprint no. 2763, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2763 .
    Abstract, PDF (368 kByte)
    We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of space-time stationarity - covariance-to-variance reduction - which allows to deduce the asymptotic behavior of the correlations of two space-time points by the one of the variances at one point. We formulate several open problems.

  • 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.

  • 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.

Vorträge, Poster

  • L. Andreis, Introduction to large deviations and random graphs (online talk), Minicourse , cycle of doctoral seminars (a short 8-hour course) for the Ph D program of Turin University, January 14 - 25, 2021, Università degli Studi di Torino, Dipartimento di Matematica (Online Event), Italy.

  • W. König, tba (online talk), Spatial Networks and Percolation (Hybrid meeting), January 17 - 23, 2021, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach.

  • W. König, A grid version of the interacting Bose gas, Probability Seminar, University of Bath, Department of Mathematical Sciences, UK, February 15, 2021.

  • R.I.A. Patterson, Decomposing large deviations rate functions into reversible and irreversible parts (online talk), BMC-BAMC Glasgow 2021 (Online Event), April 6 - 9, 2021, The British Mathematical Colloquium (BMC) and the British Applied Mathematics Colloquium (BAMC), April 7, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Meeting (Online Event), University of Oxford, Department of Statistics, UK, February 10, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability and Statistical Physics Seminar( Online Event), The University of Chicago, Department of Mathematics, Statistics, and Computer Science, USA, February 12, 2021.

  • W. van Zuijlen, Quantitative heat kernel estimates for diffusions with distributional drift (online talk), 14th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis (Online Event), February 10 - 12, 2021, University of Oxford, Mathematical Institute, UK, February 12, 2021.

  • 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 O'Conell--Yor model in intermediate disorder (online talk), Joint Israeli Probability Seminar (Online Event), Israel, November 17, 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 24, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation (online talk), Bernoulli-IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics, August 27, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation (online talk), 13th Annual ERC Berlin--Oxford Young Researchers Meeting on Applied Stochastic Analysis (Online Event), June 8 - 10, 2020, Berlin, June 10, 2020.

  • T. Orenshtein, Aging in Edwards--Wilkinson and KPZ universality classes (online talk), Probability, Stochastic Analysis and Statistics Seminar (Online Event), University of Pisa, Italy, October 27, 2020.

  • T. Orenshtein, Rough walks (online talk), Mathematics Colloquium (Online Event), Bar Ilan University, Ramat Gan, Israel, November 8, 2020.

  • D.R.M. Renger, Dynamical Phase Transitions on Finite Graphs (online talk), DMV Jahrestagung 2020 (Online Event), September 14 - October 17, 2020, Technische Universität Chemnitz, Chemnitz, September 15, 2020.

  • D.R.M. Renger, Fast reaction limits via Γ-convergence of the Flux Rate Functional, Variational Methods for Evolution, September 13 - 19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

  • D.R.M. Renger, Variational structures and particle systems (online talk), Student-compact-course (Online Event), October 15 - 16, 2020, Technische Universität Berlin.

  • D.R.M. Renger, Fast reaction limits via Gamma-convergence of the Flux Rate Functional, CRC 1114: Scaling Cascades in Complex Systems (SCCS Days) (Online Event), December 2 - 4, 2020.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Seminar, Universidade Federal da Bahia, Instituto de Matematica Doutorado em Matematica (Online Event), Salvador, Brazil, October 21, 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 (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, A large-deviations approach to the phase transition inhomogeneous random graphs: part II, Spring School on 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.

  • 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.

  • H. Langhammer, A large deviation approach to the phase transition in inhomogeneous random graphs. Part 1, Spring School on Complex Networks, March 2 - 6, 2020, Technische Universität Darmstadt, Fachbereich Mathematik, March 5, 2020.

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

  • W. König, Probabilistic treatment of Bose--Einstein Condensation, Summer School 2020: Jahrestreffen des IRTG 2544, September 14 - 17, 2020, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, Döllnsee, September 16, 2020.

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

  • L. Taggi, Exponential decay of correlations in the spin and loop o(n) model (online talk), Percolation Today (Online Event), Eidgenössische Technische Hochschule Zürich (ETH), Switzerland, October 6, 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 $Z^d, d > 2$ (online talk, Probability Seminar (Online Event), University of Bristol, School of Mathematics Research, UK, 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.