Introduction

There is a host of types of interacting random systems in probability theory, out of which perhaps the interacting stochastic particle systems are the most prominent ones, see also the Mathematical Subjects Interacting stochastic particle systems, Large deviations, Spectral theory of random operators as well as the Application Fields Particle-based modeling in the Sciences, Diffusion Models in Statistical Physics, Coagulation, Mobile Communication Networks and Stochastic biologic evolution. Beyond these, at WIAS also a number of other systems are studied, some of which have intersections with the above mentioned areas. E.g., percolation and triangulation questions from stochastic geometry are studied. Below, we briefly highlight some of the questions that are considered at WIAS.

Contributions of WIAS:

In works by Wolfgang Wagner, the solution to the time-dependent complex Schrödinger equation and closely related equations are considered in the discrete-space setting. The main objective is a representation of the solution in terms of marked spatial branching processes. These formulas are in the spirit of Feynman-Kac formulas for parabolic equations and are quite explicit. They open up future possibilities for the analysis of the equations with the help of methods from branching process theory.

Hamilton systems of spatial particle movements are considered in works by Robert Patterson and Wolfgang Wagner. In particular, they aim to study the question how the initial joint state of the particles is propagated in short time if they are started from a product state.

Gibbs measure models, which are a priori static, are considered under standard dynamics in works by Benedikt Jahnel; in particular he studies the question if the Gibbsian property that the systems starts with is propagated under the dynamics in any way.

Discrete static spin models with smoothing interactions are considered by Alessandra Cipriani in the thermodynamic limit, in particular with the goal to show an exponential decay of the correlation between spins that are far away from each other.

Random walks in random environment are considered in works by Chiranjib Mukherjee and Renato dos Santos, respectively, in particular on percolation clusters in view of large deviations of the random walks, respectively in time-dependent environments with the goal to clarify the question of transience.

The long-time behaviour of a family of one-dimensional random paths (Browian motions respectively random walks) are studied in works by Wolfgang König and co-authors under the constraint that they never intersect or at least until a late time. Particular issues were considered, e.g, that the random walks can make arbitrarily large steps or the additional constraint that all the motions must not leave a certain box.


Publications

  Monographs

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

  • W. König, ed., Mathematics and Society, European Mathematical Society Publishing House, Zurich, 2016, 314 pages, (Collection Published).
    Abstract
    The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are f inance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society ? important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines.

  • P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

  • J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, 512 pages, (Collection Published).

  • S. Rjasanow, W. Wagner, Stochastic Numerics for the Boltzmann Equation, 37 of Springer Series in Computational Mathematics, Springer, Berlin, 2005, xiii+256 pages, (Monograph Published).

  Articles in Refereed Journals

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

  • K.F. Lee, M. Dosta, A. caps">Mc Guire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with discrete element method, Comput. Chem. Engng., 99 (2017) pp. 171--184.

  • O. Gün, A. Yilmaz, The stochastic encounter-mating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017) pp. 71--102.

  • W. König, (Book review:) Firas Rassoul-Agha and Timo Seppäläinen: A Course on Large Deviations with an Introduction to Gibbs Measures, Jahresbericht der Deutschen Mathematiker-Vereinigung, 119 (2017) pp. 63--67.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017) pp. 1562--1585, DOI 10.20347/WIAS.PREPRINT.2165 .
    Abstract
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • R.I.A. Patterson, S. Simonella, W. Wagner, A Kinetic Equation for the Distribution of Interaction Clusters in Rarefied Gases, Journal of Statistical Physics, 169 (2017) pp. 126--167.

  • A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individual-based mathematical model for the Lenski experiment, and the deceleration of the relative fitness, Stochastic Processes and their Applications, 126 (2016) pp. 2211--2252.

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

  • CH. Hirsch, On the absence of percolation in a line-segment based lilypond model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 52 (2016) pp. 127--145.

  • B. Jahnel, Ch. Külske, Attractor properties of non-reversible dynamics w.r.t. invariant Gibbs measures on the lattice, Markov Processes and Related Fields, 22 (2016) pp. 507--535.

  • P. Keeler, N. Ross, A. Xia, B. Błaszczyszyn, Stronger wireless signals appear more Poisson, IEEE Wireless Communications Letters, 5 (2016) pp. 572--575.
    Abstract
    Keeler, Ross and Xia [1] recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects. The aim of this note is to apply some of the main results of [1] in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.
    [1] P. Keeler, N. Ross, and A. Xia:“When do wireless network signals appear Poisson?? ”

  • CH. Mukherjee, A. Shamov, O. Zeitouni, Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $dgeq 3$, Electronic Communications in Probability, 21 (2016) pp. 1--12.

  • CH. Mukherjee, S.R.S. Varadhan, Brownian occupation measures, compactness and large deviations, The Annals of Probability, 44 (2016) pp. 3934--3964.
    Abstract
    In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of sub-probability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.

  • H. Döring, G. Faraud, W. König, Connection times in large ad-hoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016) pp. 2143--2176.
    Abstract
    We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation.

  • V. Gayrard, O. Gün, Aging in the GREM-like trap model, Markov Processes and Related Fields, 22 (2016) pp. 165--202.
    Abstract
    The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume L-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.

  • M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians, SIAM Journal on Mathematical Analysis, 48 (2016) pp. 2674--2700.
    Abstract
    We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.

  • M. Biskup, W. König, Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails, Communications in Mathematical Physics, 341 (2016) pp. 179--218.

  • J. Blath, A. González Casanova Soberón, N. Kurt, M. Wilke-Berenguer, A new coalescent for seed-bank models, The Annals of Applied Probability, 26 (2016) pp. 857--891.

  • E. Bouchet , Ch. Sabot, R. Soares Dos Santos, A quenched functional central limit theorem for random walks in random environments under (T)_gamma, Stochastic Processes and their Applications, 126 (2016) pp. 1206--1225.

  • A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of some Gaussian random interfaces, Journal of Statistical Physics, 165 (2016) pp. 521--544.
    Abstract
    In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in citeAGG. We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

  • A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of the supercritical Gaussian free field, ALEA. Latin American Journal of Probability and Mathematical Statistics, 13 (2016) pp. 711--724.
    Abstract
    We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).

  • T. Orenshtein, R. Soares Dos Santos, Zero-one law for directional transience of one-dimensional random walks in dynamic random environments, Electronic Communications in Probability, 21 (2016) pp. 15/1--15/11.
    Abstract
    We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time.

  • E.K.Y. Yapp, R.I.A. Patterson, J. Akroyd, S. Mosbach, E.M. Adkins, J.H. Miller, M. Kraft, Numerical simulation and parametric sensitivity study of optical band gap in a laminar co-flow ethylene diffusion flame, Combustion and Flame, 167 (2016) pp. 320--334.

  • A. Cipriani, A. Feidt, Rates of convergence for extremes of geometric random variables and marked point processes, Extremes. Statistical Theory and Applications in Science, Engineering and Economics, 19 (2016) pp. 105--138.
    Abstract
    We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author's PhD thesis under the supervision of Andrew D. Barbour. The thesis is available at http://arxiv.org/abs/1310.2564.

  • R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Physica D. Nonlinear Phenomena, 335 (2016) pp. 26--32.
    Abstract
    In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, de ned as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simpli ed context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

  • W. Wagner, O. Muscato, A class of stochastic algorithms for the Wigner equation, SIAM Journal on Scientific Computing, 38 (2016) pp. A1483--A1507.
    Abstract
    A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.

  • W. Wagner, A random cloud model for the Wigner equation, Kinetic and Related Models, 9 (2016) pp. 217--235.
    Abstract
    A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.

  • CH. Hirsch, G.W. Delaney, V. Schmidt, Stationary Apollonian packings, Journal of Statistical Physics, 161 (2015) pp. 35--72.

  • CH. Hirsch, G. Gaiselmann, V. Schmidt, Asymptotic properties of collective-rearrangement algorithms, ESAIM. Probability and Statistics, 19 (2015) pp. 236--250.

  • CH. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt, Asymptotic properties of Euclidean shortest-path trees in random geometric graphs, Statistics & Probability Letters, 107 (2015) pp. 122--130.

  • P. Keeler, P.G. Taylor, Discussion on ``On the Laplace transform of the aggregate discounted claims with Markovian arrivals'' by Jiandong Ren, Volume 12 (2), North American Actuarial Journal, 19 (2015) pp. 73--77.

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

  • B. Blaszczyszyn, P. Keeler, Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures, Institute of Electrical and Electronics Engineers. Transactions on Information Theory, 61 (2015) pp. 6774--6794.

  • B. Blaszczyszyn, M. Karray, P. Keeler, Wireless networks appear Poissonian due to strong shadowing, IEEE Transactions on Wireless Communications, 14 (2015) pp. 4379--4390.

  • A. Chiarini, A. Cipriani, R.S. Hazra, A note on the extremal process of the supercritical Gaussian free field, Electronic Communications in Probability, 20 (2015) pp. 74/1--74/10.
    Abstract
    We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite boxin dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process. The result follows from an application of the Stein-Chen method from Arratia et al. (1989).

  • D. Neuhäuser, Ch. Hirsch, C. Gloaguen, V. Schmidt, Parametric modeling of sparse random trees using 3D copulas, Stochastic Models, 31 (2015) pp. 226--260.

  • K.F. Lee, S. Mosbach, M. Kraft, W. Wagner, A multi-compartment population balance model for high shear granulation, Comput. Chem. Engng., 75 (2015) pp. 1--13.
    Abstract
    This work extends the granulation model published by Braumann et al. (2007) to include multiple compartments in order to account for mixture heterogeneity encountered in powder mixing processes. A stochastic weighted algorithm is adapted to solve the granulation model which includes simultaneous coalescence and breakage. Then, a new numerical method to solve stochastic reactor networks is devised. The numerical behaviour of the adapted stochastic weighted algorithm is compared against the existing direct simulation algorithm. Lastly, the performance of the new compartmental model is then investigated by comparing the predicted particle size distribution against an experimentally measured size distribution. It is found that the adapted stochastic weighted algorithm exhibits superior performance compared to the direct simulation algorithm and the multi-compartment model produces results with better agreement with the experimental results compared to the original single-compartment model.

  • K.F. Lee, R.I.A. Patterson, W. Wagner, M. Kraft, Stochastic weighted particle methods for population balance equations with coagulation, fragmentation and spatial inhomogeneity, Journal of Computational Physics, 303 (2015) pp. 1--18.

  • A. Cipriani, D. Zeindler, The limit shape of random permutations with polynomially growing cycle weights, ALEA. Latin American Journal of Probability and Mathematical Statistics, 12 (2015) pp. 971--999.
    Abstract
    In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set {1, ... n} under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.

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

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

  • W. Wagner, A class of probabilistic models for the Schrödinger equation, Monte Carlo Methods and Applications, 21 (2015) pp. 121--137.

  • R. Soares Dos Santos, Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process, Electronic Journal of Probability, 19 (2014) pp. 1--18.
    Abstract
    Linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalisation methods of Kesten and Sidoravicius [11].

  • R.I.A. Patterson, W. Wagner, Cell size error in stochastic particle methods for coagulation equations with advection, SIAM Journal on Numerical Analysis, 52 (2014) pp. 424--442.
    Abstract
    The paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.

  • W. Wagner, A random cloud model for the Schrödinger equation, Kinetic and Related Models, 7 (2014) pp. 361--379.

  • L. Avena, R. caps">Dos Santos, F. Völlering, Transient random walk in symmetric exclusion: Limit theorems and an Einstein relation, ALEA. Latin American Journal of Probability and Mathematical Statistics, 10 (2013) pp. 693--709.

  • W.J. Menz, R.I.A. Patterson, W. Wagner, M. Kraft, Application of stochastic weighted algorithms to a multidimensional silica particle model, Journal of Computational Physics, 248 (2013) pp. 221--234.
    Abstract
    This paper presents a detailed study of the numerical behaviour of stochastic weighted algorithms (SWAs) using the transition regime coagulation kernel and a multidimensional silica particle model. The implementation in the SWAs of the transition regime coagulation kernel and associated majorant rates is described. The silica particle model of Shekar et al. [S. Shekar, A.J. Smith, W.J. Menz, M. Sander, M. Kraft, A multidimensional population balance model to describe the aerosol synthesis of silica nanoparticles, Journal of Aerosol Science 44 (2012) 83?98] was used in conjunction with this coagulation kernel to study the convergence properties of SWAs with a multidimensional particle model. High precision solutions were calculated with two SWAs and also with the established direct simulation algorithm. These solutions, which were generated using large number of computational particles, showed close agreement. It was thus demonstrated that SWAs can be successfully used with complex coagulation kernels and high dimensional particle models to simulate real-world systems.

  • O. Muscato, V. Di Stefano, W. Wagner, A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation, Computers & Mathematics with Applications. An International Journal, 65 (2013) pp. 520--527.
    Abstract
    This paper is concerned with electron transport and heat generation in semiconductor devices. An improved version of the electrothermal Monte Carlo method is presented. This modification has better approximation properties due to reduced statistical fluctuations. The corresponding transport equations are provided and results of numerical experiments are presented.

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

  • O. Gün, W. König, O. Sekulovic, Moment asymptotics for branching random walks in random environment, Electronic Journal of Probability, 18 (2013) pp. 1--18.

  • R.I.A. Patterson, Convergence of stochastic particle systems undergoing advection and coagulation, Stochastic Analysis and Applications, 31 (2013) pp. 800--829.
    Abstract
    The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction.

  • W. Wagner, Some properties of the kinetic equation for electron transport in semiconductors, Kinetic and Related Models, 6 (2013) pp. 955--967.
    Abstract
    The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.

  • S. Jansen, W. König, Ideal mixture approximation of cluster size distributions at low density, Journal of Statistical Physics, 147 (2012) pp. 963--980.
    Abstract
    We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture.

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

  • G. Benarous, O. Gün, Universality and extremal aging for the dynamics of spin glasses on subexponential time scales, Communications on Pure and Applied Mathematics, 65 (2012) pp. 77--127.

  • S. Shekar, W.J. Menz, A.J. Smith, M. Kraft, W. Wagner, On a multivariate population balance model to describe the structure and composition of silica nanoparticles, Comput. Chem. Engng., 43 (2012) pp. 130--147.

  • G. Faraud, Y. Hu, Z. Shi, Almost sure convergence for stochastically biased random walk on trees, Probability Theory and Related Fields, 154 (2012) pp. 621--660.

  • W. König, M. Salvi, T. Wolff, Large deviations for the local times of a random walk among random conductances, Electronic Communications in Probability, 17 (2012) pp. 1--11.
    Abstract
    We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.

  • R.I.A. Patterson, W. Wagner, A stochastic weighted particle method for coagulation-advection problems, SIAM Journal on Scientific Computing, 34 (2012) pp. B290--B311.
    Abstract
    A spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two Direct Simulation Algorithms which develop instabilities.

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

  • W. Kirsch, B. Metzger, P. Müller, Random block operators, Journal of Statistical Physics, 143 (2011) pp. 1035--1054.
    Abstract
    We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.

  • G.A. Radtke, N.G. Hadjiconstantinou, W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Physics of Fluids, 23 (2011) pp. 030606/1--030606/12.

  • M. Sander, R.I.A. Patterson, A. Braumann, A. Raj, M. Kraft, Developing the PAH-PP soot particle model using process informatics and uncertainty propagation, Proceedings of the Combustion Institute, 33 (2011) pp. 675--683.
    Abstract
    n this work we present the new PAH-PP soot model and use a data collaboration approach to determine some of its parameters. The model describes the formation, growth and oxidation of soot in laminar premixed flames. Soot particles are modelled as aggregates containing primary particles, which are built from polycyclic aromatic hydrocarbons (PAHs), the main building blocks of a primary particle (PP). The connectivity of the primary particles is stored and used to determine the rounding of the soot particles due to surface growth and condensation processes. Two neighbouring primary particles are replaced by one if the coalescence level between the two primary particles reaches a threshold. The model contains, like most of the other models, free parameters that are unknown a priori. The experimental premixed flame data from Zhao et al. [B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang, Proc. Combust. Inst. 30 (2) (2005) 1441?1448] have been used to estimate the smoothing factor of soot particles, the growth factor of PAHs within particles and the soot density using a low discrepancy series method with a subsequent response surface optimisation. The optimised particle size distributions show good agreement with the experimental ones. The importance of a standardised data mining system in order to optimise models is underlined.

  • O. Muscato, W. Wagner, V. Di Stefano, Properties of the steady state distribution of electrons in semiconductors, Kinetic and Related Models, 4 (2011) pp. 809--829.
    Abstract
    This paper studies a Boltzmann transport equation with several electron-phonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasi-parabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wave-vector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate possible directions for future studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods.

  • G. Faraud, A central limit theorem for random walk in random environment on a marked Galton--Watson tree, Electronic Journal of Probability, 16 (2011) pp. 174--215.

  • G. Faraud, Estimates on the speedup and slowdown for a diffusion in a drifted Brownian potential, Journal of Theoretical Probability, 24 (2011) pp. 194-239.

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

  • R.I.A. Patterson, W. Wagner, M. Kraft, Stochastic weighted particle methods for population balance equations, Journal of Computational Physics, 230 (2011) pp. 7456--7472.
    Abstract
    A class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.

  • W. Wagner, Stochastic models in kinetic theory, Physics of Fluids, 23 (2011) pp. 030602/1--030602/14.
    Abstract
    The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent.

  • M. Aizenman, S. Jansen, P. Jung, Symmetry breaking in quasi-1D Coulomb systems, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 11 (2010) pp. 1453--1485.
    Abstract
    Quasi one-dimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called “jellium”, at any temperature and at any finite-strip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, two-dimens The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (1980) for a similar statement concerning symmetry breaking at all temperatures in strictly one-dimensional Coulomb systems. The extension is enabled through bounds which establish tightness of finite-volume charge fluctuations.

  • A. Braumann, M. Kraft, W. Wagner, Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation, Journal of Computational Physics, 229 (2010) pp. 7672--7691.

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

  • W. König, P. Schmid, Random walks conditioned to stay in Weyl chambers of type C and D, Electronic Communications in Probability, (2010) pp. 286--295.

  • W. Wagner, Random and deterministic fragmentation models, Monte Carlo Methods and Applications, 16 (2010) pp. 399--420.

  • A. Weiss, Escaping the Brownian stalkers, Electronic Journal of Probability, 14 (2009) pp. 139-160.
    Abstract
    We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance

  • M. Becker, W. König, Moments and distribution of the local times of a transient random walk on $Bbb Zsp d$, Journal of Theoretical Probability, 22 (2009) pp. 365--374.

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

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

  • R. Siegmund-Schultze, W. Wagner, Induced gelation in a two-site spatial coagulation model, The Annals of Applied Probability, 16 (2006) pp. 370-402.

  • W. Wagner, Post-gelation behavior of a spatial coagulation model, Electronic Journal of Probability, 11 (2006) pp. 893-933.

  • W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, The Annals of Applied Probability, 15 (2005) pp. 2081-2112.

  • A. Eibeck, W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, The Annals of Applied Probability, 13 (2003) pp. 845-889.

  Contributions to Collected Editions

  • J. Blath, E. Bjarki, A. González Casanova Soberón, N. Kurt, Genealogy of a Wright--Fisher model with strong seedbank component, in: XI Symposium of Probability and Stochastic Processes, R.H. Mena, J.C. Pardo, V. Rivero, G. Uribe Bravo, eds., 69 of Birkhäuser Progress in Probability, Springer International Publishing, Switzerland, 2015, pp. 81--100.

  • F. Castell, O. Gün, G. Maillard, Parabolic Anderson model with finite number of moving catalysts, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 91--116.

  • A. Schnitzler, T. Wolff, Precise asymptotics for the parabolic Anderson model with a moving catalyst or trap, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 69--88.

  • W. König, S. Schmidt, The parabolic Anderson model with acceleration and deceleration, in: Probability in Complex Physical Systems, in Honour of Erwin Bolthausen and Jürgen Gärtner, J.-D. Deuschel, B. Gentz, W. König, M. VON Renesse, M. Scheutzow, U. Schmock, eds., 11 of Springer Proceedings in Mathematics, Springer, Berlin Heidelberg, 2012, pp. 225--245.

  • G.A. Radtke, N.G. Hadjiconstantinou, W. Wagner, Low variance particle simulations of the Boltzmann transport equation for the variable hard sphere collision model, in: 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove, California, July 10-15, 2010, Part One, D.A. Levin, I.J. Wysong, A.L. Garcia, eds., 1333 of AIP Conference Proceedings, AIP Publishing Center, New York, 2011, pp. 307--312.

  • W. König, Upper tails of self-intersection local times of random walks: Survey of proof techniques, in: Excess Self-Intersections & Related Topics, 2 of Actes des Rencontres du CIRM, Centre International de Rencontres Mathématiques, Marseille, 2010, pp. 15--24.

  Preprints, Reports, Technical Reports

  • M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials, Preprint no. 2439, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2439 .
    Abstract, PDF (264 kByte)
    We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials. endabstract

  • O. Blondel, M.R. Hilário, R. Soares Dos Santos, V. Sidoravicius, A. Teixeira, Random walk on random walks: Higher dimensions, Preprint no. 2435, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2435 .
    Abstract, PDF (389 kByte)
    We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

  • O. Blondel, M.R. Hilário, R. Soares Dos Santos, V. Sidoravicius, A. Teixeira, Random walk on random walks: Low densities, Preprint no. 2434, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2434 .
    Abstract, PDF (356 kByte)
    We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.

  • S. Muirhead, R. Pymar, R. Soares Dos Santos, The Bouchaud--Anderson model with double-exponential potential, Preprint no. 2433, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2433 .
    Abstract, PDF (459 kByte)
    The Bouchaud--Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour.

  • O. Muscato, W. Wagner, A stochastic algorithm without time discretization error for the Wigner equation, Preprint no. 2415, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2415 .
    Abstract, PDF (400 kByte)
    Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

  • B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Preprint no. 2414, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2414 .
    Abstract, PDF (288 kByte)
    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 routing in highly dense multi-hop network, Preprint no. 2392, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2392 .
    Abstract, PDF (468 kByte)
    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.

  • A. González Casanova Soberón, D. Spanò, Duality and fixation in $Xi$-Wright--Fisher processes with frequency-dependent selection, Preprint no. 2390, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2390 .
    Abstract, PDF (347 kByte)
    A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of emphpotential parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types Ξ--Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

  • D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Preprint no. 2375, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2375 .
    Abstract, PDF (424 kByte)
    We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Preprint no. 2371, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2371 .
    Abstract, PDF (598 kByte)
    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

  • R.I.A. Patterson, S. Simonella, W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, Preprint no. 2365, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2365 .
    Abstract, PDF (978 kByte)
    We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.

  • N. Berger, Ch. Mukherjee, K. Okamura, Quenched large deviations for simple random walks on percolation clusters including long-range correlations, Preprint no. 2360, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2360 .
    Abstract, PDF (395 kByte)
    We prove a quenched large deviation principle (LDP)for a simple random walk on a supercritical percolation cluster (SRWPC) on the lattice.The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and its vacant set and the level sets of the Gaussian free field. Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures if the environment Markov chain in the non-elliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main approach of our proofs are based on exploiting coercivity properties of the relative entropy in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the percolation cluster under supercriticality.

  • A. Stivala, P. Keeler, Another phase transition in the Axelrod model, Preprint no. 2352, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2352 .
    Abstract, PDF (715 kByte)
    Axelrod's model of cultural dissemination, despite its apparent simplicity, demonstrates complex behavior that has been of much interest in statistical physics. Despite the many variations and extensions of the model that have been investigated, a systematic investigation of the effects of changing the size of the neighborhood on the lattice in which interactions can occur has not been made. Here we investigate the effect of varying the radius R of the von Neumann neighborhood in which agents can interact. We show, in addition to the well-known phase transition at the critical value of q, the number of traits, another phase transition at a critical value of R, and draw a q - R phase diagram for the Axelrod model on a square lattice. In addition, we present a mean-field approximation of the model in which behavior on an infinite lattice can be analyzed.

  • A. Cipriani, R.S. Hazra, W.M. Ruszel, The divisible sandpile with heavy-tailed variables, Preprint no. 2328, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2328 .
    Abstract, PDF (347 kByte)
    This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an α-stable random distribution.

  • L. Avena, O. Gün, M. Hesse, The parabolic Anderson model on the hypercube, Preprint no. 2319, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2319 .
    Abstract, PDF (240 kByte)
    We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and k-th largest potential. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.

  • CH. Hirsch, B. Jahnel, R.I.A. Patterson, Space-time large deviations in capacity-constrained relay networks, Preprint no. 2308, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2308 .
    Abstract, PDF (311 kByte)
    We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.

  • E. Bolthausen, A. Cipriani, N. Kurt, Exponential decay of covariances for the supercritical membrane model, Preprint no. 2301, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2301 .
    Abstract, PDF (291 kByte)
    We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δ-pinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.

  • B. Jahnel, Ch. Külske, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Preprint no. 2297, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2297 .
    Abstract, PDF (603 kByte)
    We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG> 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > tG , we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

  • M. Biskup, W. König, R. Soares Dos Santos, Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails, Preprint no. 2295, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2295 .
    Abstract, PDF (810 kByte)
    We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.

  • F. Flegel, Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model, Preprint no. 2290, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2290 .
    Abstract, PDF (567 kByte)
    We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

  • A. Chiarini, A. Cipriani, A note on the Green's function for the transient random walk without killing on the half lattice, orthant and strip, Preprint no. 2289, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2289 .
    Abstract, PDF (765 kByte)
    In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $Z^d$, $dge 3$.

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

  • A. Cipriani, R.S. Hazra, W.M. Ruszel, Scaling limit of the odometer in divisible sandpiles, Preprint no. 2268, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2268 .
    Abstract, PDF (377 kByte)
    In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

  • D. Coupier, Ch. Hirsch, Coalescence of Euclidean geodesics on the Poisson--Delaunay triangulation, Preprint no. 2243, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2243 .
    Abstract, PDF (320 kByte)
    Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an adapted Burton-Keane argument and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Preprint no. 2221, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2221 .
    Abstract, PDF (476 kByte)
    We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

  • E. Bolthausen, A. Cipriani, N. Kurt, Fast decay of covariances under delta-pinning in the critical and supercritical membrane model, Preprint no. 2220, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2220 .
    Abstract, PDF (271 kByte)
    We consider the membrane model, that is the centered Gaussian field on Z^d whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a delta-pinning condition, giving a reward of strength epsilon for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions larger than 4 covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in dimensions larger than 5 and logarithmic in 4 dimensions. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.

  • W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional, Preprint no. 2199, WIAS, Berlin, 2015.
    Abstract, PDF (262 kByte)
    We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large-deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

  • CH. Mukherjee, Brownian occupation measures, compactness and large deviations: Pair interaction, Preprint no. 2197, WIAS, Berlin, 2015.
    Abstract, PDF (211 kByte)
    Continuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Preprint no. 2173, WIAS, Berlin, 2015.
    Abstract, PDF (2647 kByte)
    We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, Preprint no. 2165, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2165 .
    Abstract, PDF (363 kByte)
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Preprint no. 2145, WIAS, Berlin, 2015.
    Abstract, PDF (262 kByte)
    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.

  • W. Wagner, A random walk model for the Schrödinger equation, Preprint no. 2109, WIAS, Berlin, 2015.
    Abstract, PDF (143 kByte)
    A random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.

  Talks, Poster

  • A. González Casanova Soberón, Branching processes with interactions and their relation to population genetics, The 3rd Workshop on branching processes and related topics, May 8 - 12, 2017, Beijing Normal University, School of Mathematical Sciences, Beijing, China, May 8, 2017.

  • A. González Casanova Soberón, Modeling selection via multiple parents, Annual Colloquium SPP 1590, October 4 - 6, 2017, Albert-Ludwigs-Universität Freiburg, Fakultät für Mathematik und Physik, Bath, UK, October 6, 2017.

  • A. González Casanova Soberón, Modelling selection via multiple parents, Probability Seminar, University of Oxford, Mathematical Institute, UK, January 24, 2017.

  • A. González Casanova Soberón, Modelling the Lenski experiment, 19th ÖMG Congress and Annual DMV- Meeting, Paris-Lodron University of Salzburg, Salzburg, Austria, September 14, 2017.

  • A. González Casanova Soberón, The ancestral efficiency graph, Spatial models in population genetics, September 6 - 8, 2017, University of Bath, Department of Mathematical Sciences, Bath, UK, September 6, 2017.

  • A. González Casanova Soberón, The discrete ancestral selection graph, Seminar, Center for I nterdisciplinary Research in Biology, Stochastic Models for the Inference of Life Evolution SMILE, Paris, France, October 23, 2017.

  • A. González Casanova Soberón, Seminar Probability, National Autonomous University of Mexico, UNAM, Mexico city, Mexico, February 23, 2017.

  • B. Jahnel, The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Université du Luxembour, Faculté des Sciences, de la Technologie et de la Communication (FSTC), Luxembourg, March 3, 2017.

  • B. Jahnel, The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, January 18, 2017.

  • B. Jahnel, The Widom-Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality, Oberseminar Wahrscheinlichkeitstheorie, Ludwig-Maximilians-Universität München, Fakultät für Mathematik, Informatik und Statistik, February 13, 2017.

  • B. Jahnel, The Widom-Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality, Oberseminar Stochastik, Johannes Gutenberg Universiät Mainz, Institut für Mathematik, April 25, 2017.

  • P. Keeler, Optimizing spatial throughput in device-to-device networks, Applied Probability @ The Rock An international workshop celebrating Phil Pollett's 60th birthday, April 17 - 21, 2017, University of Adelaide, School of Mathematical Sciences, Uluru, Australia, April 20, 2017.

  • CH. Mukherjee, Asymptotic behavior of the mean-field polaron, Probability and Mathematical Physics Seminar, Courant Institute of Mathematical Sciences, Department of Mathematics, New York, USA, March 20, 2017.

  • R.I.A. Patterson, Confidence Intervals for Coagulation? Advection Simulations, Clausthal-Göttingen-International Workshop ``Simulation Science'', April 27 - 28, 2017, Georg-August-Universität Göttingen, Institut für Informatik, April 28, 2017.

  • D.R.M. Renger, Banach-valued functions of bounded variation, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, July 28, 2017.

  • D.R.M. Renger, Was sind und was sollen die Zahlen, Tag der Mathematik, Universität Regensburg, Fakultät für Mathematik, July 28, 2017.

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

  • R. Soares Dos Santos, Complete localisation in the Bouchaud-Anderson model, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 10, 2017.

  • R. Soares Dos Santos, Concentration de masse dans le modèle parabolique d'Anderson, Séminaire de Probabilités,, April 10 - 12, 2017, Université de Grenoble, Institut Fourier, Laboratoire des Mathematiques, Grenoble, France, April 11, 2017.

  • R. Soares Dos Santos, Eigenvalue order statistics of random Schrödinger operators and ap- plications to the parabolic Anderson model, 19th ÖMG Congress and Annual DMV- Meeting, Paris-Lodron University of Salzburg, Salzburg, Austria, September 13, 2017.

  • W. van Zuijlen, Mean-field Gibbs-non-Gibbs transitions, Mark Kac Seminar, Utrecht University, Mathematical Institute, Netherlands, February 3, 2017.

  • B. Jahnel, Large deviations in relay-augmented wireless networks, Sharif University of Technology Tehran, Mathematical Sciences Department, Teheran, Iran, Islamic Republic Of, September 17, 2017.

  • B. Jahnel, Stochastic geometry in telecommunications, Summer School 2017: Probabilistic and statistical methods for networks, August 21 - September 1, 2017, Technische Universität Berlin, Berlin Mathematical School, Berlin.

  • W. König, A variational formula for an interacting many-body system, Probability Seminar, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, January 19, 2017.

  • W. König, Connectivity in large mobile ad-hoc networks, Summer School 2017: Probabilistic and statistical methods for networks, August 21 - September 1, 2017, Technische Universität Berlin, Berlin Mathematical School, Berlin, August 29, 2017.

  • W. König, Moment asymptotics of branching random walks in random environment, Modern perspective of branching in probability, September 26 - 29, 2017, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, Münster.

  • W. König, The principal part of the spectrum of a random Schrödinger operator in a large box, Mathematisches Kolloquium der Technischen Universität Dormund, Oberseminar Stochastik und Analysis, May 15 - 16, 2017, Technische Universität Dormund, May 15, 2017.

  • A. Pandey, Meshfree method for fluctuating hydrodynamics, International Conference on Advances in Scientific Computing, November 28 - 30, 2016, Indian Institute of Technology, Department of Mathematics, Madras, November 30, 2016.

  • A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, January 27 - 29, 2016, WIAS Berlin, Berlin, January 27, 2016.

  • A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, WIAS Berlin, Berlin, January 27, 2016.

  • A. González Casanova Soberón, Fixation in a Xi coalescent model with selection, Probability seminar, University of Warwick, Mathematics Institute, Warwick, UK, November 30, 2016.

  • A. González Casanova Soberón, Fixation in a Xi coalescent with selection, Miniworkshop on Probabilistic Models in Evolutionary Biology, November 24 - 25, 2016, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, November 25, 2016.

  • A. González Casanova Soberón, Modeling the Lenski experiment, Mathematical and Computational Evolutionary Biology, June 12 - 16, 2016, Le Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM), Hameau de l'Etoile, France, June 14, 2016.

  • A. González Casanova Soberón, The seed bank model, VIII School on Probability and Stochastic Processes, September 12 - 16, 2016, Centro de Investigación en Matemáticas (CIMAT), Department of Probability and Statistics, Guanajuato, Mexico.

  • A. González Casanova Soberón, The seedbank coalescent, World Congress in Probability and Statistics, Invited Session ``Stochastic Models of Evolution'', July 11 - 15, 2016, Fields Institute, Toronto, Canada, July 5, 2016.

  • CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, Workshop on Continuum Percolation, January 26 - 29, 2016, University Lille 1, Science et Technologies, France, January 28, 2016.

  • CH. Hirsch, Large deviations in relay-augmented wireless networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 22, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • CH. Hirsch, Large deviations in relay-augmented wireless networks, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

  • CH. Hirsch, On maximal hard-core thinnings of stationary particle processes, Oberseminar Wahrscheinlichkeitstheorie, Ludwig-Maximilians-Universität München, Fakultät für Mathematik, April 18, 2016.

  • B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

  • B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Romanian Academy of Sciences, Institute of Mathematical Statistics and Applied Mathematics, Bucharest, February 22, 2016.

  • B. Jahnel, GnG transitions for the continuum Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Transformations in Statistical Mechanics: Pathologies and Remedies, October 9 - 14, 2016, Lorentz Center -- International Center for Scientific Workshops, Leiden, Netherlands, October 11, 2016.

  • P. Keeler, Signal-to-interference ratio in wireless communication networks, Workshop on Dynamical Networks and Network Dynamics, January 17 - 24, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • P. Keeler, Wireless network models: Geometry OR signal-to-interference ratio (SIR), Paris, Paris, France, June 3, 2016.

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

  • CH. Mukherjee, Compactness and large deviations, Probability Seminar, Stanford University, Department of Mathematics and Statistics, USA, November 14, 2016.

  • CH. Mukherjee, Compactness and large deviations, Mathematisches Kolloquium, Universität Konstanz, Fachbereich Mathematik und Statistik, May 18, 2016.

  • CH. Mukherjee, Compactness and large deviations, Probability Seminar, University of California at Berkeley, Department of Statistics, USA, October 19, 2016.

  • CH. Mukherjee, Compactness, large deviations and statistical mechanics, Seminar des Fachbereichs Mathematik und Statistik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, October 17, 2016.

  • CH. Mukherjee, Compactness, large deviations and the polaron, Probability Seminar, University of Washington, Department of Mathematics, Seattle, USA, October 31, 2016.

  • CH. Mukherjee, Compactness, large deviations and the polaron, The University of Arizona, Department of Mathematics, USA, November 2, 2016.

  • CH. Mukherjee, Compactness, large deviations, and the polaron problem, 12th German Probability and Statistics Days 2016 -- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 3, 2016.

  • CH. Mukherjee, Occupation measures, compactness and large deviations, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6 - 11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

  • CH. Mukherjee, On some aspects of large deviations, Mathematics Colloquium, West Virginia University, Department of Mathematics, Morgantown, USA, March 17, 2016.

  • CH. Mukherjee, Polaron problem, Probability Seminar, University of California at Irvine, Department of Mathematics, USA, October 25, 2016.

  • CH. Mukherjee, Quenched large deviations for random walks on supercritical percolation clusters, Probability and Mathematical Physics Seminar, Courant Institute, New York, Department of Mathematics, USA, November 4, 2016.

  • CH. Mukherjee, The polaron problem, Rutgers University, Department of Mathematics, New Brunswick, USA, November 17, 2016.

  • CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Analysis Seminar, University of California at Berkeley, Department of Mathematics, USA, October 21, 2016.

  • CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Probability and Mathematical Physics Seminar, University of California at Los Angeles, Department of Mathematics, USA, October 27, 2016.

  • CH. Mukherjee, Weak/strong disorder for stochastic heat equation, City University of New York, Department of Mathematics, USA, November 8, 2016.

  • D.R.M. Renger, Functions of bounded variation with an infinite-dimensional codomain, Meeting in Applied Mathematics and Calculus of Variations, September 13 - 16, 2016, Università di Roma ``La Sapienza'', Dipartimento di Matematica ``Guido Castelnuovo'', Italy, September 16, 2016.

  • D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble process, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6 - 11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

  • R. Soares Dos Santos, Random walk on random walks, University College London, Department of Mathematics, London, UK, June 15, 2016.

  • R. Soares Dos Santos, Random walk on random walks, Rhein-Main Kolloquium Stochastik, Johannes-Gutenberg Universität, Institut für Mathematik, Mainz, May 13, 2016.

  • W. van Zuijlen, Mean field Gibbs-non-Gibbs transitions, 6th Berlin--Oxford Meeting, December 8 - 10, 2016, University of Oxford, Mathematics Department, UK, December 9, 2016.

  • A. Cipriani, Extremes of some Gaussian random interfaces, Seminar Series in Probability and Statistics, Delft University of Technology, Department of Applied Probability, Netherlands, January 21, 2016.

  • A. Cipriani, The membrane model, seminar, Delft University of Technology, Department of Applied Probability, Netherlands, September 5, 2016.

  • A. Cipriani, The membrane model, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, September 29, 2016.

  • F. Flegel, Spectral localization in the random conductance model, 2nd Berlin Dresden Prague Würzburg Workshop on Mathematics of Continuum Mechanics, Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Summer School 2016, August 21 - 26, 2016, Research Training Group (RTG) 1845 ``Stochastic Analysis with Applications in Biology, Finance and Physics'', Hejnice, Czech Republic, August 22, 2016.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Probability Seminar at UCLA, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, October 13, 2016.

  • O. Gün, Fixation times for the mutation-selection model on random fitness landscapes, Joint Meeting of the SPP 1590 and 1819, September 28 - 29, 2016, Universität zu Köln, Köln, September 29, 2016.

  • W. König, A variational formula for the free energy of an interacting many-body system, Workshop ``Variational Structures and Large Deviations for Interacting Particle Systems and Partial Differential Equations'', March 15 - 18, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 17, 2016.

  • W. König, Connection times in large ad-hoc mobile networks, Workshop on Dynamical Networks and Network Dynamics, January 18 - 21, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

  • W. König, The mean-field polaron model, Workshop on Stochastic Processes in honour of Erwin Bolthausen's 70th birthday, September 14 - 16, 2016, Universität Zürich, Institut für Mathematik, Switzerland, September 15, 2016.

  • W. König, The spatially discrete parabolic Anderson model with time-dependent potential, ``Guided Tour: Random Media'' --- Special occasion to celebrate the 60th birthday of Frank den Hollander, December 14 - 16, 2016, EURANDOM, Eindhoven, Netherlands, December 16, 2016.

  • R.I.A. Patterson, Monte Carlo simulation of soot, King Abdullah University of Science and Technology (KAUST), Clean Combustion Research Center, Thuwal, Saudi Arabia, January 11, 2016.

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

  • R.I.A. Patterson, Population balance simulation, University of Cambridge, Department for Chemical Engineering and Biotechnology, UK, May 5, 2016.

  • R.I.A. Patterson, Simulations of flame generated particles, Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016), January 5 - 10, 2016, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, January 5, 2016.

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

  • A. González Casanova Soberón, An individual-based model for the Lenski experiment, and the deceleration of the relative fitness, Workshop on Probabilistic Models in Biology, October 24 - 30, 2015, Playa del Carmen, Mexico, October 28, 2015.

  • A. González Casanova Soberón, Modeling the Lenski experiment, Genealogies in Evolution: Looking Backward and Forward, Workshop of the Priority Program (SPP) 1590 ``Probabilistic Structures in Evolution'', October 5 - 6, 2015, Goethe-Universität Frankfurt, October 6, 2015.

  • A. González Casanova Soberón, Modeling the Lenski experiment, 11. Doktorandentreffen Stochastik, Humboldt Universität zu Berlin, Institut für Mathematik (gemeinsam mit der TU Berlin), Berlin, September 2, 2015.

  • CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, 18. Workshop on Stochastic Geometry, Stereology and Image Analysis, March 22 - 27, 2015, Universität Osnabrück, Lingen, March 23, 2015.

  • CH. Hirsch, Asymptotic properties of collective-rearrangement algorithms, International Conference on Geometry and Physics of Spatial Random Systems, September 6 - 11, 2015, Karlsruher Institut für Technology (KIT), Bad Herrenalb, September 7, 2015.

  • CH. Hirsch, Large-deviation principles in SINR-based wireless network models, Simons Conference on Networks and Stochastic Geometry, May 18 - 21, 2015, University of Texas, Austin, USA, May 18, 2015.

  • B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, November 16, 2015.

  • B. Jahnel, Classes of non-ergodic interacting particle systems with unique invariant measure, Kac-Seminar, April 30 - May 3, 2015, Utrecht University, Department of Mathematics, Netherlands, May 1, 2015.

  • B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Workshop ``Recent Trends in Stochastic Analysis and Related Topics'', September 20 - 21, 2015, Universität Hamburg, September 21, 2015.

  • B. Jahnel, Classes of non-ergodic interacting particle systems with unique invariant measure, Workshop ``Interacting Particles Systems and Non-Equilibrium Dynamics'', Institut Henri Poincaré, Paris, France, March 9 - 13, 2015.

  • P. Keeler, Large-deviation theory and coverage in mobile phone networks, Seminar ``Applied Probability'', The University of Melbourne, Department of Mathematics and Statistics, Australia, August 17, 2015.

  • P. Keeler, The Poisson--Dirichlet process and coverage in mobile phone networks, Stochastic Processes and Special Functions Workshop, August 13 - 14, 2015, The University of Melbourne, Melbourne, Australia, August 14, 2015.

  • P. Keeler, When do wireless network signals appear Poisson?, Simons Conference on Networks and Stochastic Geometry, May 18 - 21, 2015, University of Texas, Austin, USA, May 20, 2015.

  • P. Keeler, When do wireless network signals appear Poisson?, 18th Workshop on Stochastic Geometry, Stereology and Image Analysis, March 22 - 27, 2015, Universität Osnabrück, Lingen, March 24, 2015.

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

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

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

  • M. Maurelli, Stochastic 2D Euler equations: A poorly correlated multiplicative noise regularizes the two-point motion, Universität Augsburg, Institut für Mathematik, March 24, 2015.

  • CH. Mukherjee, Compactness, large deviations and the mean-field polaron problem, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 27 - 31, 2015, Mathematisches Forschungsinstitut Oberwolfach, July 28, 2015.

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

  • R. Soares Dos Santos, Mass concentration in the parabolic Anderson model, Oberseminar Stochastik, Johannes-Gutenberg-Universität, Institut für Mathematik, Mainz, November 17, 2015.

  • R. Soares Dos Santos, Random walk on a dynamic random environment consisting of a system of independent simple symmetric random walks, Oberseminar Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt, January 22, 2015.

  • R. Soares Dos Santos, Random walk on random walks, YEP XII: Workshop on Random Walk in Random Environment, March 23 - 27, 2015, Technical University of Eindhoven, EURANDOM, Netherlands, March 24, 2015.

  • R. Soares Dos Santos, Random walk on random walks, Mathematical Physics Seminar, Université de Genève, Section de Mathématiques, Genève, Switzerland, April 27, 2015.

  • A. Cipriani, Extremes of the super critical Gaussian free field, Workshop ``Women in Probability 2015'', July 3 - 4, 2015, Technische Universität München, July 3, 2015.

  • A. Cipriani, Extremes of the supercritical Gaussian free field, Seminar Series in Probability and Statistics, Technical University of Delft, Applied Mathematics, Netherlands, June 11, 2015.

  • A. Cipriani, Extremes of the supercritical Gaussian free field, Probability Seminar, Leiden University, Netherlands, June 18, 2015.

  • A. Cipriani, Rates of convergence for extremes of geometric random variables and marked point processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 28, 2015.

  • A. Cipriani, Rates of convergence for extremes of geometric random variables and marked point processes, Università degli Studi di Milano-Bicocca, Dipartimento di Matematica Applicazioni, Milano, Italy, March 30, 2015.

  • F. Flegel, Localization of the first Dirichlet-eigenvector in the heavy-tailed random conductance model, Summer School 2015 of the RTG 1845 Berlin-Potsdam ``Stochastic Analysis with Applications in Biology, Finance and Physics'', September 28 - October 3, 2015, Levico Terme, Italy, October 1, 2015.

  • F. Flegel, Localization of the first Dirichlet-eigenvector in the heavy-tailed random conductance model, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, July 30, 2015.

  • O. Gün, Branching random walks in random environments on hypercubes, Workshop on Random Walk in Random Environment, March 22 - 27, 2015, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 27, 2015.

  • O. Gün, Fluid and diffusion limits for the Poisson encounter-mating model, Mini-Workshop on Population Dynamics, April 6 - 17, 2015, Bŏgaziçi University Istanbul, Department of Mathematics, Istanbul, Turkey, April 6, 2015.

  • O. Gün, Stochastic encounter-mating model, Mathematical Model in Ecology and Evolution (MMEE 2015), July 7 - 13, 2015, Collège de France, Paris, France, July 8, 2015.

  • W. König, Moment asymptotics for a branching random walk in random environment, Applied Mathematics Seminars, University of Warwick, Mathematics Institute, Coventry, UK, November 6, 2015.

  • R.I.A. Patterson, Approximation errors for Smoluchowski simulations, 10 th IMACS Seminar on Monte Carlo Methods, July 6 - 10, 2015, Johannes Kepler Universität Linz, Austria, July 7, 2015.

  • R.I.A. Patterson, Particle systems, kinetic equations and their simulation, 8th International Congress on Industrial and Applied Mathematics, ICIAM 2015, August 8 - 15, 2015, CNCC - China National Convention Center, Beijing, China.

  • W. Wagner, Probabilistic models for the Schrödinger equation, 6th Workshop ``Theory and Numerics of Kinetic Equations'', June 1 - 4, 2015, Universität Saarbrücken, June 2, 2015.

  • CH. Hirsch, From heavy-tailed Boolean models to scale-free Gilbert graphs, Karlsruher Institut für Technologie, Institut für Stochastik, November 14, 2014.

  • A. Cipriani, Thick points for generalized Gaussian fields with different cut-offs, Berlin-Padova Young Researchers Meeting, October 23 - 25, 2014, Technische Universität Berlin, October 24, 2014.

  • A. Cipriani, Thick points for massive Gaussian free fields on R^d, School and Workshop on Random Interacting Systems, June 23 - 27, 2014, University of Bath, UK, June 23, 2014.

  • O. Gün, Stochastic encounter-mating models, University of Leiden, Mathematical Institute, Netherlands, September 4, 2014.

  • W. König, A variational formula for the free energy of a many-Boson system, IKERBASQUE, Basque Foundation for Science, Bilbao, Spain, May 29, 2014.

  • W. König, Von der Binomialverteilung zur Normalverteilung, 11th German Probability and Statistics Days 2014, March 4 - 7, 2014, Ulm, March 6, 2014.

  • R.I.A. Patterson, Statistical error analysis for coagulation-advection simulations, Eleventh International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2014), April 6 - 11, 2014, KU Leuven, Belgium, April 8, 2014.

  • R.I.A. Patterson, Statistical error analysis for coagulation-advection simulations, University of Cambridge, Department of Chemical Engineering and Biotechnology, UK, May 1, 2014.

  • R.I.A. Patterson, Stochastic numerical methods for coagulating particles, Seminar ``Geophysical Fluid Dynamics'', Freie Universität Berlin, Institut für Mathematik, June 4, 2014.

  • W. Wagner, Heat generation in the electrothermal Monte Carlo method, The 18th European Conference on Mathematics for Industry 2014 (ECMI 2014), Minisymposium ``Semiclassical and Quantum Transport in Semiconductors and Low Dimensional Materials'', June 9 - 13, 2014, Taormina, Italy, June 11, 2014.

  • W. Wagner, Random cloud models for the Schrödinger equation, Sapienza -- Università di Roma, Dipartimento di Matematica, Italy, October 9, 2014.

  • L. Avena, A local CLT for some convolution equations with applications to self-avoiding walks, Università degli Studi di Roma ``La Sapienza'', Dipartimento di Matematica, Italy, December 17, 2013.

  • G. Faraud, Connection times in large ad-hoc networks, Ecole de Printemps ``Marches Aléatoires, Milieux Aléatoires, Renforcements'' (MEMEMO2), June 10 - 14, 2013, Aussois, France, June 13, 2013.

  • O. Gün, Aging for GREM-like trap models, CIRM-Conference "`Dynamical and Disordered Systems"', February 11 - 15, 2013, Centre International de Rencontres Mathématiques, Marseille, France, February 15, 2013.

  • O. Gün, Moment asymptotics for branching random walks in random environment, Workshop on Disordered Systems, June 24 - 28, 2013, Centre International de Rencontres Mathématiques, Marseille, France, June 24, 2013.

  • W. König, A variational formula for the free energy of a many-boson system, Random Combinatorial Structures and Statistical Mechanics, May 6 - 10, 2013, University of Warwick, Mathematics Institute, Warwick in Venice, Palazzo Pesaro-Papafava, Italy, May 10, 2013.

  • W. König, Geordnete Irrfahrten, Technische Universität Darmstadt, Fachbereich Mathematik, October 23, 2013.

  • W. König, Large deviations for the local times of a random walk among random conductances, Random Walks: Crossroads and Perspectives, Satellite Meeting of the Erdős Centennial Conference, June 24 - 28, 2013, Alfréd Rényi Institute of Mathematics, Budapest, Hungary, June 28, 2013.

  • W. König, Upper tails of self-intersection local times: Survey of proof techniques, 12. Erlanger-Münchner Tag der Stochastik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, July 12, 2013.

  • W. König, Upper tails of self-intersection local times: Survey of proof techniques, Kyoto University, Research Institute for Mathematical Sciences, Japan, April 12, 2013.

  • R.I.A. Patterson, Monte Carlo simulation of nano-particle formation, University of Technology Eindhoven, Institute for Complex Molecular Systems, Netherlands, September 5, 2013.

  • R.I.A. Patterson, Stochastic methods for particle coagulation problems in flows with boundaries, 5th Workshop ``Theory and Numerics of Kinetic Equations'', May 13 - 15, 2013, Universität des Saarlandes, Saarbrücken, May 14, 2013.

  • W. Wagner, Stochastic particle methods for population balance equations, 5th Workshop ``Theory and Numerics of Kinetic Equations'', May 13 - 15, 2013, Universität des Saarlandes, Saarbrücken, May 13, 2013.

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

  • S. Jansen, Cluster size distributions and approximate random partition model for Gibbs measures in continuous configuration space, 10th German Probability and Statistics Days (GPSD2012), March 6 - 9, 2012, Johannes Gutenberg-Universität Mainz, Mainz, March 8, 2012.

  • S. Jansen, Cluster size distributions at low temperature and low density, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 24, 2012.

  • S. Jansen, Fermionic and bosonic Laughlin state on thick cylinders, Mathematical Physics Seminar, University of Alabama, Department of Mathematics, Birmingham, USA, February 17, 2012.

  • S. Jansen, Random partitions and heavy-tailed variables in statistical mechanics, Probability Seminar, University of Alabama, Department of Mathematics, Birmingham, USA, February 9, 2012.

  • O. Gün, Moment asymptotics for branching random walks in random environment, Probability Laboratory at Bath, Prob-L@b Seminar, University of Bath, Department of Mathematical Sciences, UK, August 20, 2012.

  • W. König, Large deviations for cluster size distributions in a classical many-body system, Probability Laboratory at Bath, Prob-L@b Seminar, University of Bath, Department of Mathematical Sciences, UK, August 20, 2012.

  • W. König, Large deviations for the cluster size distribution in a classical interacting many-particle system, Warwick Mathematics Institute Seminars, University of Warwick, Mathematics Institute, Coventry, UK, August 1, 2012.

  • O. Gün, Dynamics of GREM-like trap models, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 27, 2012.

  • O. Gün, Multilevel trap models and aging for spin glasses, Seminar Wahrscheinlichkeitstheorie, Universität Wien, Fakultät für Mathematik, Austria, May 21, 2012.

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

  • W. König, Large deviations for cluster size distributions in a classical many-body system, Oberseminar Stochastik, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Angewandte Mathematik, January 12, 2012.

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

  • W. König, Large deviations for the local times of random walk among random conductances, EPSRC Symposium Workshop -- Large Scale Behaviour of Random Spatial Models, May 28 - June 1, 2012, University of Warwick, Mathematical Institute, Coventry, UK, June 1, 2012.

  • W. König, Moment asymptotics for branching random walks in random environment, Stochastik-Oberseminar, Westfälische Wilhelms-Universität Münster, Institut für Mathematische Statistik, December 6, 2012.

  • W. König, Ordered random walks, Stochastisches Kolloquium, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, June 6, 2012.

  • R.I.A. Patterson, Convergence of simulable processes for coagulation with transport, WIAS-workshop "From particle systems to dierential equations", February 21 - 23, 2012, WIAS Berlin, February 22, 2012.

  • R.I.A. Patterson, Soot as a boundary value problem, University of Cambridge, Department of Chemical Engineering, UK, May 8, 2012.

  • R.I.A. Patterson, Stochastic methods for particle populations in flows, 3rd European Seminar on Computing, June 25 - 29, 2012, Pilsen, Czech Republic.

  • W. Wagner, Coagulation equations and particle systems, WIAS-workshop "From Particle Systems to Differential Equations", February 21 - 23, 2012, WIAS Berlin, February 22, 2012.

  • W. Wagner, Stochastic particle methods for coagulation problems, 28th International Symposium on Rarefied Gas Dynamics, July 9 - 13, 2012, University of Zaragoza, Spain, July 9, 2012.

  • T. Wolff, Annealed asymptotics for occupation time measures of a random walk among random conductances, University of California at Los Angeles, Mathematics Department, USA, October 24, 2012.

  • T. Wolff, Non-exit probability from a time-dependent region of a random walk among random conductances, Interplay of Analysis and Probability in Physics, January 22 - 28, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 26, 2012.

  • S. Jansen, Cluster size distributions at low density and low temperature, The University of Arizona, Department of Mathematics, Tucson, USA, April 13, 2011.

  • S. Jansen, Fermionic and bosonic Laughlin state on thick cylinders, Venice 2011 --- Quantissima in the Serenissima, August 1 - 5, 2011, University of Warwick (VB), Warwick in Venice, Italy, August 4, 2011.

  • S. Jansen, Large deviations for interacting many-particle systems in the Saha regime, Berlin-Leipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

  • S. Jansen, Random partitions in statistical physics, 5th International Conference on Stochastic Analysis and its Applications, September 5 - 9, 2011, Hausdorff Center for Mathematics and Rheinische Friedrich-Wilhelms-Universität Bonn, September 8, 2011.

  • S. Jansen, Statistical mechanics at low density and low temperature: Cross-over transitions from small to large cluster sizes, 2011 School on Mathematical Statistical Physics, August 29 - September 4, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 8, 2011.

  • S. Jansen, Random partitions in statistical physics, Warwick Statistical Mechanics Seminar, University of Warwick, Department of Mathematics, UK, November 17, 2011.

  • M. Roberts, The unscaled paths of branching Brownian motion, Oberseminar "Biological Models and Statistical Mechanics", The University of Nottingham, School of Community Health Sciences, UK, January 17, 2011.

  • G. Faraud, Marche aléatoires en milieu aléatoire: Le cas des arbres, École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, Séminaire de Probabilités, France, February 3, 2011.

  • G. Faraud, Random walks in random environment on trees, 2011 School on Mathematical Statistical Physics, August 28 - September 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 2, 2011.

  • W. König, A variational formula for the free energy of a many-Boson system, University of California at Los Angeles, Department of Mathematics, USA, April 11, 2011.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Berlin-Leipzig Seminar on Analysis and Probability Theory, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

  • W. König, Eigenvalue order statistics for the heat equation with random potential, Extreme Value Statistics in Mathematics, Physics and Beyond, July 4 - 8, 2011, Lorentz Center, International Center for Workshops in the Sciences, Leiden, Netherlands, July 6, 2011.

  • W. König, Large deviations for cluster size distributions in a classical many-body system, Università Ca' Foscari Venezia, Dipartimento di Management, Italy, October 13, 2011.

  • W. König, Localisation of the parabolic Anderson model in one island, Jahrestagung der Deutschen Mathematiker-Vereinigung (DMV) 2011, September 20 - 22, 2011, Universität zu Köln, Mathematisches Institut, September 20, 2011.

  • W. König, Ordered random walks, University of California at Los Angeles, Department of Mathematics, USA, April 4, 2011.

  • W. König, Phase transitions for a dilute particle system with Lennard--Jones potential, Ludwig-Maximilians-Universität München, Mathematisches Institut, January 20, 2011.

  • W. König, The parabolic Anderson model, 2011 School on Mathematical Statistical Physics, September 4 - 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science.

  • W. König, The universality classes in the parabolic Anderson model, Technische Universität Dresden, Institut für Analysis, June 24, 2011.

  • W. König, Upper tails of self-intersection local times: Survey of proof techniques, University of Warwick, Mathematics Institute, Coventry, UK, February 17, 2011.

  • T. Wolff, Annealed behaviour of local times in the random conductance model, 2011 School on Mathematical Statistical Physics, September 4 - 9, 2011, Charles University of Prague, Center for Theoretical Study, and Academy of Sciences of the Czech Republic, Institute of Theoretical Computer Science, September 8, 2011.

  • T. Wolff, Random walk among random conductances, Spring Meeting Beijing/Bielefeld --- Berlin/Zurich of the International Research Group Stochastic Models of Complex Processes, March 30 - April 1, 2011, Technische Universität Berlin, March 31, 2011.

  • T. Wolff, The parabolic Anderson model from the perspective of a moving catalyst, 7th Cornell Probability Summer School, July 8 - 24, 2011, Cornell University, Ithaka, USA, July 18, 2011.

  • G. Faraud, Marche aléatoires en milieu aléatoire: Le cas des arbres, Université Paris VI ``Pierre et Marie Curie'', Laboratoire de Probabilités et Modèles Aléatoires, France, November 21, 2011.

  • O. Gün, Parabolic Anderson model with finite number of moving catalysts, Istanbul Center For Mathematical Sciences, Turkey, December 23, 2011.

  • O. Gün, Trap models and aging for spin glasses, Bogazici University, Department of Mathematics, Istanbul, Turkey, December 21, 2011.

  • O. Gün, Trap models and aging for spin glasses, Koc University, Istanbul, Turkey, December 27, 2011.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Technische Universität München, Fakultät für Mathematik, December 21, 2011.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Tokyo Institute of Technology, Department of Mathematics, Japan, December 9, 2011.

  • W. König, Large deviations for cluster size distributions in a classical many-body system, 10th Workshop ``Stochastic Analysis on Large Scale Interacting Systems'', December 5 - 7, 2011, Kochi University, Faculty of Science, Shikoku, Japan, December 6, 2011.

  • W. König, Large deviations for cluster size distributions in a classical many-body system, Universität Augsburg, Institut für Mathematik, December 22, 2011.

  • R.I.A. Patterson, Simulating coagulating particles in flows, University of Cambridge, Department of Chemical Engineering, UK, October 17, 2011.

  • R.I.A. Patterson, Simulating coagulating particles with advection, University of Cambridge, Department of Chemical Engineering, UK, May 3, 2011.

  • W. Wagner, Direct simulation Monte Carlo algorithms, Università di Catania, Dipartimento di Matematica e Informatica, Italy, October 6, 2011.

  • W. Wagner, Stochastic particle methods, 8th International Conference on Large-Scale Scientific Computations, June 6 - 10, 2011, Institute of Information and Communication Technologies, Bulgarian Academy of Sciences and Society for Industrial and Applied Mathematics (SIAM), Sozopol, Bulgaria, June 6, 2011.

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

  • S. Jansen, Combinatorics and Analysis in Spatial Probability, ESF Mathematics Conference in partnership with EMS and ERCOM:, December 12 - 18, 2010, EURANDOM, Eindhoven, Netherlands.

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

  • M. Sander, R.I.A. Patterson, A. Braumann, A. Rai, M. Kraft, Boundary value stochastic particle methods for population balance problems, 4th International Conference on Population Balance Modelling (PBM 2010), Berlin, September 15 - 17, 2010.

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

  • W. König, On random matrix theory, Introductory Course for the IRTG Summer School Pro*Doc/IRTG Berlin-Zürich ``Stochastic Models of Complex Processes'' (Disentis, Switzerland, July 26--30, 2010), July 21 - 22, 2010, Technische Universität Berlin, July 21, 2010.

  • W. König, Ordered random walks, Augsburger Mathematisches Kolloquium, Universität Augsburg, Institut für Mathematik, January 26, 2010.

  • W. König, Ordered random walks, Mathematisches Kolloquium der Universität Trier, Fachbereich Mathematik, April 29, 2010.

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

  • W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, Università di Roma ``Tor Vergata'', Dipartimento di Matematica, Italy, November 17, 2010.

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

  • W. König, Upper tails of self-intersection local times of random walks: Survey of proof techniques, Excess Self-Intersection Local Times and Related Topics, December 6 - 10, 2010, Université de Marseille, Centre International de Rencontres Mathématiques (CIRM), France, December 7, 2010.

  • W. Wagner, Kinetic equations and Markov jump processes, Isaac Newton Institute for Mathematical Sciences, Programme: Partial Differential Equations in Kinetic Theories, Cambridge, UK, November 29, 2010.

  • W. König, Modeling and understanding random Hamiltonians: Beyond monotonicity, linearity and independence, Miniworkshop ``Numerics for Kinetic Equations'', December 6 - 12, 2009, Mathematisches Forschungsinstitut Oberwolfach.

  • W. König, Phase transitions for dilute particle systems with Lennard--Jones potential, Workshop on Mathematics of Phase Transitions: Past, Present, Future, November 12 - 15, 2009, University of Warwick, Coventry, UK, November 15, 2009.

  • W. Wagner, Explosion properties of random fragmentation models, Workshop ``Coagulation et Fragmentation Stochastiques'', April 15 - 18, 2008, Université Paris VI, Laboratoire de Probabilités, France, April 16, 2008.

  • W. Wagner, Explosion properties of random fragmentation models, Università di Catania, Dipartimento di Matematica e Informatica, Italy, May 8, 2008.

  • W. Wagner, Introduction to Markov jump processes, Università di Catania, Dipartimento di Matematica e Informatica, Italy, May 7, 2008.

  • A. Weiss, Escaping the Brownian stalkers, BRG Workshop on Stochastic Models from Biology and Physics, October 9 - 10, 2006, Johann Wolfgang Goethe-Universität Frankfurt, October 10, 2006.

  • A. Weiss, Escaping the Brownian stalkers, 5th Prague Summer School 2006 "`Statistical Mathematical Mechanics"', September 10 - 23, 2006, Charles University, Center for Theoretical Study and Institute of Theoretical Computer Science, Prague, Czech Republic, September 20, 2006.

  • W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation model, October 17 - 24, 2006, Università di Catania, Dipartimento di Matematica e Informatica, Italy, October 19, 2006.

  • W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, University of Cambridge, Centre for Mathematical Sciences, UK, May 9, 2006.

  • W. Wagner, Gelation in stochastic models, Workshop ``Stochastic Methods in Coagulation and Fragmentation'', December 8 - 12, 2003, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, December 10, 2003.

  External Preprints

  • D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: coherence analysis for finite Lagrangian data, Preprint no. arXiv:1709.02352, Cornell University Library, arXiv.org, 2017.
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individual-based model for the Lenski experiment, and the deceleration of the relative fitness, Preprint no. arxiv.org:1505.01751, Cornell University Library, arXiv.org, 2016.

  • J. Blath, A. González Casanova Soberón, N. Kurt, M. Wilke-Berenguer, A new coalescent for seed-bank models, Preprint no. arxiv.org:1411.4747, Cornell University Library, arXiv.org, 2016.

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

  • M. Kraft, W. Wagner, A numerical scheme for the Random Cloud Model, Technical report no. 173, c4e-Preprint Series, 2016.

  • K.F. Lee, M. Dosta, A.D. Mcguire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with Discrete Element Method, Technical report no. 170, c4e-Preprint Series, 2016.
    Abstract
    This paper presents a multi-compartment population balance model for wet granulation coupled with DEM (Discrete Element Method) simulations. Methodologies are developed to extract relevant data from the DEM simulations to inform the population balance model. First, compartmental residence times are calculated for the population balance model from DEM. Then, a suitable collision kernel is chosen for the population balance model based on particle-particle collision frequencies extracted from DEM. It is found t hat the population balance model is able to predict the trends exhibited by the experimental size and porosity distributions by utilising the information provided by the DEM simulations.

  • R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Preprint no. arXiv: 1601.05838, Cornell University Libary, arXiv.org, 2016.
    Abstract
    In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, de ned as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simpli ed context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

  • E.K.Y. Yapp, R.I.A. Patterson, J. Akroyd, S. Mosbach, E.M. Adkins, J.H. Miller, M. Kraft, Numerical simulation and parametric sensitivity study of optical band gap in a laminar co-flow ethylene diffusion flame, Technical report no. 159, University of Cambridge, c4e-Preprint Series, 2015.
    Abstract
    A detailed population balance model is used to perform a parametric sensitivity study on the computed optical band gap (OBG) of polycyclic aromatic hydrocarbons (PAHs) in a laminar co-flow ethylene diffusion flame. OBG may be correlated with the number of aromatic rings in PAHs which allows insights into which are the key species involved in the formation of soot. PAH size distributions are computed along the centerline and in the wings of the flame. We compare our simulations with experimentally determined soot volume fraction and OBG (derived from extinction measurements) from the literature. It is shown that the model predicts reasonably well the soot volume fraction and OBG throughout the flame. We find that the computed OBG is most sensitive to the size of the smallest PAH which is assumed to contribute to the OBG. The best results are obtained accounting for PAH contribution in both gas and particle phases assuming a minimum size of ovalene (10 rings). This suggests that the extinction measurements show a significant absorption by PAHs in the gas phase at the visible wavelength that is used, which has been demonstrated by experiments in the literature. It is further shown that PAH size distributions along the centerline and in the wings are unimodal at larger heights above burner. Despite the different soot particle histories and residence times in the flame, the PAH size associated with both modes are similar which is consistent with the near-constant OBG that is observed experimentally. The simulations indicate that the transition from the gas phase to soot particles begins with PAHs with as few as 16 aromatic rings, which is consistent with experimental observations reported in the literature.

  • A. Cipriani, S.H. Rajat, Thick points for Gaussian free fields with different cut-offs, Preprint no. arXiv:1407.5840, Cornell University Library, arXiv.org, 2014.

  • W.J. Menz, R.I.A. Patterson, W. Wagner, M. Kraft, Application of stochastic weighted algorithms to a multidimensional silica particle model, Preprint no. 120, University of Cambridge, Cambridge Center for Computational Chemical Engineering, 2012.

  • S. Shekar, A.J. Smith, M. Kraft, W. Wagner, On a multivariate population balance model to describe the structure and composition of silica nanoparticles, Technical report no. 105, c4e-Preprint Series, Cambridge, 2011.

  • A. Braumann, M. Kraft, W. Wagner, Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation, Technical report no. 93, c4e-Preprint Series, Cambridge Centre for Computational Chemical Engineering, University of Cambridge, Department of Chemical Engineering, 2010.
    Abstract
    This paper is concerned with computational aspects of a multi-dimensional population balance model of a wet granulation process. Wet granulation is a manufacturing method to form composite particles, granules, from small particles and binders. A detailed numerical study of a stochastic particle algorithm for the solution of a five-dimensional population balance model for wet granulation is presented. Each particle consists of two types of solids (containing pores) and of external and internal liquid (located in the pores). Several transformations of particles are considered, including coalescence, compaction and breakage. A convergence study is performed with respect to the parameter that determines the number of numerical particles. Averaged properties of the system are computed. In addition, the ensemble is subdivided into practically relevant size classes and analysed with respect to the amount of mass and the particle porosity in each class. These results illustrate the importance of the multi-dimensional approach. Finally, the kinetic equation corresponding to the stochastic model is discussed.

  • W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, Preprint no. NI04006-IGS, Isaac Newton Institute for Mathematical Sciences, 2004.