Publications
Monographs

P. Friz, W. König, Ch. Mukherjee, S. Olla, eds., Probability and Analysis in Interacting Physical Systems. In Honor of S.R.S. Varadhan, Berlin, August, 2016, 283 of Springer Proceedings in Mathematics & Statistics book series, Springer, 2019, pp. 1294, (Monograph Published), DOI https://doi.org/10.1007/9783030153380 .
Articles in Refereed Journals

A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twinscrew granulation model, , 203 (2019), pp. 358379 ( published online on 03.04.2019), DOI https://doi.org/10.1016/j.ces.2019.03.078 .

L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 590606.
Abstract
We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space. 
F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unboundedrange jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 12261257, DOI 10.1214/18AIHP917 .
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almostsure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearestneighbor 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 longrange 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 twoscale convergence 
C. Bartsch, V. John, R.I.A. Patterson, Simulations of an ASA flow crystallizer with a coupled stochasticdeterministic approach, Comput. Chem. Engng., 124 (2019), pp. 350363, DOI 10.1016/j.compchemeng.2019.01.012 .
Abstract
A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a socalled kinetic MonteCarlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data. 
M. Heida, R.I.A. Patterson, D.R.M. Renger, Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space, Journal of Evolution Equations, 19 (2019), pp. 111152 (published online on 14.09.2018), DOI 10.1007/s0002801804711 .
Abstract
We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical AubinLions theorem. We finally provide some useful applications to stochastic processes. 
D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Journal of Statistical Physics, 172 (2018), pp. 12911326, DOI 10.1007/s1095501820830 .
Abstract
We consider a system of independent particles on a finite state space, and prove a dynamic largedeviation principle for the empirical measureempirical 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 largedeviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finitespace setting. 
D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Journal of NonNewtonian Fluid Mechanics, 28 (2018), pp. 19151957, DOI 10.1007/s0033201894710 .
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium 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 timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
D.R.M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 596/1596/26, DOI 10.3390/e20080596 .
Abstract
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the OnsagerMachlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well. 
C. Cotar, B. Jahnel, Ch. Külske, Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures, Electronic Communications in Probability, 23 (2018), pp. 112, DOI 10.1214/18ECP200 .
Abstract
The concept of metastate measures on the states of a random spin system was introduced to be able to treat the largevolume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strongcoupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter. 
M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials, Interdisciplinary Information Sciences, 24 (2018), pp. 5976.
Abstract
We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowestlying 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 
G. Botirov, B. Jahnel, Phase transitions for a model with uncountable spin space on the Cayley tree: The general case, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 23 (2019), pp. 291301 (published online on 17.08.2018), DOI 10.1007/s1111701806061 .
Abstract
In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14, Bo17]. The potential is of nearestneighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value θ_{ c } such that for θ≤θ _{ c } there is a unique translationinvariant splitting Gibbs measure. For θ _{ c } < θ there is a phase transition with exactly three translationinvariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated nonlinear Hammerstein integral operator for the boundary laws. 
CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Stochastic Processes and their Applications, published online on 20.11.2018, urlhttps://doi.org/10.1016/j.spa.2018.11.002, DOI 10.1016/j.spa.2018.11.002 .
Abstract
We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of nontrivial sub and supercritical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, highdensity and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives. 
CH. Hirsch, B. Jahnel, R.I.A. Patterson, Spacetime large deviations in capacityconstrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587615, DOI 10.30757/ALEA.v1524 .
Abstract
We consider a singlecell 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 spacetime evolution of frustrated transmitters in the highdensity regime. 
L. Andreis, P. Dai Pra, M. Fischer, McKeanVlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications, 36 (2018), pp. 960995, DOI 10.1080/07362994.2018.1486202 .
Abstract
Motivated by several applications, including neuronal models, we consider the McKeanVlasov limit for meanfield systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0,T],R^d). 
F. Flegel, Localization of the principal Dirichlet eigenvector in the heavytailed random conductance model, Electronic Journal of Probability, 23 (2018), pp. 68/168/43, DOI doi:10.1214/18EJP160 .
Abstract
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, BorelCantelli arguments, the RayleighRitz formula, results from percolation theory, and path arguments. 
W. Wagner, A random walk model for the Schrödinger equation, Mathematics and Computers in Simulation, 143 (2018), pp. 138148, DOI 10.1016/j.matcom.2016.07.012 .
Abstract
A random walk model for the spatially discretized timedependent 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 complexvalued 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.
Contributions to Collected Editions

E. Cali, N.N. Gafur, Ch. Hirsch, B. Jahnel, T. EnNajjari, R.I.A. Patterson, Percolation for D2D networks on street systems, in: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789106/117789106/6, DOI 10.23919/WIOPT.2018.8362866 .
Abstract
We study fundamental characteristics for the connectivity of multihop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as PoissonVoronoi or PoissonDelaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical deviceintensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold. 
P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in highdensity cellular networks, in: 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), IEEE Xplore digital library, 2018, pp. 17789136/117789136/8, DOI 10.23919/WIOPT.2018.8362867 .
Abstract
Stochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signaltointerference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the nearfar effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.
Preprints, Reports, Technical Reports

CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Preprint no. 2632, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2632 .
Abstract, PDF (1268 kByte)
This work develops a methodology for analyzing largedeviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of PoissonVoronoi cells, as well as powerweighted edge lengths in the random geometric, κnearest neighbor and relative neighborhood graph. 
D.R.M. Renger, J. Zimmer, Orthogonality of fluxes in general nonlinear reaction networks, Preprint no. 2609, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2609 .
Abstract, PDF (261 kByte)
We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional. 
K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimension, Preprint no. 2606, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2606 .
Abstract, PDF (588 kByte)
In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula. 
M. Kantner, Generalized ScharfetterGummel schemes for electrothermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient, Preprint no. 2605, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2605 .
Abstract, PDF (7347 kByte)
Many challenges faced in today's semiconductor devices are related to selfheating phenomena. The optimization of device designs can be assisted by numerical simulations using the nonisothermal driftdiffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the socalled Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known selfheating effects, namely Joule, recombination and ThomsonPeltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the (nonlinear) diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume ScharfetterGummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two nonisothermal generalizations of the ScharfetterGummel scheme for degenerate semiconductors (FermiDirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme. 
S. Jansen, W. König, B. Schmidt, F. Theil, Surface energy and boundary layers for a chain of atoms at low temperature, Preprint no. 2589, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2589 .
Abstract, PDF (529 kByte)
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of LennardJones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As the temperature goes to zero and at fixed positive pressure, the Gibbs measures for infinite chains and semiinfinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature. 
A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Preprint no. 2572, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2572 .
Abstract, PDF (276 kByte)
In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk. 
L. Andreis, W. König, R.I.A. Patterson, A largedeviations approach to gelation, Preprint no. 2568, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2568 .
Abstract, PDF (338 kByte)
A largedeviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a timedependent version of the ErdősRényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller ErdősRényi graphs are connected. 
CH. Hirsch, B. Jahnel, A. Hinsen, E. Cali, The typical cell in anisotropic tessellations, Preprint no. 2557, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2557 .
Abstract, PDF (311 kByte)
The typical cell is a key concept for stochasticgeometry based modeling in communication networks, as it provides a rigorous framework for describing properties of a serving zone associated with a component selected at random in a large network. We consider a setting where network components are located on a large street network. While earlier investigations were restricted to street systems without preferred directions, in this paper we derive the distribution of the typical cell in Manhattantype systems characterized by a pattern of horizontal and vertical streets. We explain how the mathematical description can be turned into a simulation algorithm and provide numerical results uncovering novel effects when compared to classical isotropic networks. 
F. Flegel, M. Heida, The fractional $p$Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unboundedrange jumps, Preprint no. 2541, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2541 .
Abstract, PDF (633 kByte)
We study a general class of discrete pLaplace operators in the random conductance model with longrange jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional pLaplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator. 
D.R.M. Renger, Gradient and Generic systems in the space of fluxes, applied to reacting particle systems, Preprint no. 2516, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2516 .
Abstract, PDF (392 kByte)
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the OnsagerMachlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well. 
R.I.A. Patterson, D.R.M. Renger, Large deviations of reaction fluxes, Preprint no. 2491, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2491 .
Abstract, PDF (304 kByte)
We study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic largedeviation principle for the reaction fluxes under general assumptions that include massaction kinetics. This result immediately implies the dynamic large deviations for the empirical concentration. 
P. Nelson, R. Soares Dos Santos, Brownian motion in attenuated or renormalized inversesquare Poisson potential, Preprint no. 2482, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2482 .
Abstract, PDF (461 kByte)
We consider the parabolic Anderson problem with random potentials having inversesquare singularities around the points of a standard Poisson point process in ℝ ^{d}, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 𝔎 behaving as 𝔎 (x)≈ Θ x ^{2} near the origin, where Θ ∈(0,(d2)^{2}/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that 𝔎 is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and largetime asymptotics of nonnegative solutions via FeynmanKac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9]. 
F. Flegel, Eigenvector localization in the heavytailed random conductance model, Preprint no. 2472, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2472 .
Abstract, PDF (292 kByte)
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavytailed random conductance Laplacian to the first k eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the BauerFike theorem to show that the kth eigenvector is close to the principal eigenvector of an auxiliary spectral problem.
Talks, Poster

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

A. Hinsen, Introduction to Interacting Particles Systems (IPS), Probability, Analysis and Applications (PAA) Workshop, September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana.

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

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

D.R.M. Renger, A generic formulation of a chemical reaction network from OnsagerMachlup theory, Conference to celebrate 80th jubilee of Miroslav Grmela, May 18  19, 2019, Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Prag, Czech Republic, May 19, 2019.

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

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

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

W. van Zuijlen, Massasymptotics for the parabolic Anderson model in 2 d, BerlinLeipzig workshop in analysis and stochastics, January 16  18, 2019, Max Planck Institute ür Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

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

W. van Zuijlen, The parabolic Anderson model in 2 d, mass and Eigenvalue asymptotics, Analysis and Probability Seminar, Imperial College London, Department of Mathematics, London, UK, February 5, 2019.

L. Andreis, A largedeviations approach to the multiplicative coagulation process, Workshop ``Woman in Probability'', May 31  June 1, 2019, Technische Universität München, Fakulät für Mathematik, München, May 31, 2019.

L. Andreis, Coagulation processes and gelation from a large deviation pointof view, Berlin Mathematical School  BGSMath Junior Meeting 2019, June 26  28, 2019, Berlin Mathematical School (BMS), June 26, 2019.

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

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

L. Andreis, Stochastic processes with interaction: random environment and particle systems, Second Italian Meeting on Probability and Mathematical Statistics, June 17  20, 2019, Vietri sul Mare, Italy, June 19, 2019.

B. Jahnel, Continuum Percolation in Random Environment I  IV, Probability, Analysis and Applications (PAA) Workshop, September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana.

B. Jahnel, Dynamical GibbnonGibbs transitions for the continuum WidomRowlinson model, The 41st Conference on Stochastic Processes and their Applications 2019  SPA 2019, July 8  August 12, 2019, Northwestern University Evanston, Evanston, USA, July 9, 2019.

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

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

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

W. König, A largedeviations approach to the multiplicative coalescent, 18. ErlangerMünchner Tag der Stochastik / Probability Day 2019, FriedrichAlexanderUniversität ErlangenNürnberg, Department Mathematik, May 10, 2019.

W. König, Cluster size distribution in a classical manybody system, Deutschen Zentrum für Luft und Raumfahrt (DLR), Institut für Materialphysik im Weltraum, June 18, 2019.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, BerlinLeipzig workshop in analysis and stochastics, January 16  18, 2019, Max Planck Institute ür Mathematik in den Naturwissenschaften, Leipzig, January 17, 2019.

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

W. König, Micromacro phase transitions in coagulating particle systems I  IV, Probability, Analysis and Applications (PAA) Workshop, September 23  October 4, 2019, African Institute for Mathematical Sciences  Ghana (AIMS Ghana), Accra, Ghana.

I.A. Patterson, Fluctuations and Confidence Intervals for Stochastic Particle Simulations, First Berlin  Leipzig Workshop on Fluctuating Hydrodynamics, August 26  30, 2019, Max Planck Institute for Mathematics in the Sciences, Leipzig, August 29, 2019.

R.I.A. Patterson, Interaction Clusters for the Kac Process, BerlinLeipzig workshop in analysis and stochastics, January 16  18, 2019, Max Planck Institute ür Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

R.I.A. Patterson, A Novel Simulation Method for Stochastic Particle Systems, Seminar Department of Chemical Engineering and Biotechnology, University of Cambridge, Faculty of Mathematics, Cambridge, UK, May 9, 2019.

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

R.I.A. Patterson, Flux Large Deviations, Seminar Statistical Laboratory, University of Cambridge, Faculty of Mathematics, Cambridge, UK, May 7, 2019.

R.I.A. Patterson, Interaction Clusters for the Kac Process, Workshopon Effective equations: frontiers in classical and quantum systems, June 24  28, 2019, Hausdorff Research Institute for Mathematics, Bonn, June 28, 2019.

R.I.A. Patterson, Kinetic Interaction Clusters, Oberseminar, MartinLutherUniversität HalleWittenberg, Naturwissenschaftliche Fakultät II  Chemie, Physik und Mathematik, April 17, 2019.

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

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

L. Taggi, Nondecay of correlations in the dimer model and phase transition in lattice permutations in Z d , d > 2, via reflection positivity, Meeting of the Swiss Mathematical Society: Recent advances in loop models and height functions, September 2  4, 2019, Université Fribourg, Fribourg, Switzerland, September 3, 2019.

L. Taggi, Nondecay of correlations in the dimer model and phase transition in lattice permutations in Z d , d > 2, via reflection positivity, Colloquium, Università degli Studi "La Sapienza" di Roma, Rom, Italy, September 27, 2019.

A. Hinsen, Random Malware Propagation, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

A. Hinsen, The White Knight Model  Propagation of Malware on a D2D Network, 14. Doktorand*innentreffen Stochastik, Essen 2018, August 1  3, 2018, Universität DuisburgEssen, August 3, 2018.

A. Hinsen, Vulnerability and security in adhoc networks, Universität Osnabrück, Fachbereich Mathematik/Informatik, December 11, 2018.

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

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

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

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

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

R. Soares Dos Santos, Random walk on random walks, University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science, Netherlands, February 14, 2018.

R. Soares Dos Santos, Random walk on random walks, Oberseminar Mathematische Stochastik, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, Münster, July 4, 2018.

R. Soares Dos Santos, The parabolic Anderson model with renormalized inversesquare Poisson potential, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 2, 2018.

W. van Zuijlen, A HamiltonJacobi point of view on meanfield GibbsnonGibbs transitions, Workshop on Transformations and Phase Transitions, January 29  31, 2018, RuhrUniversität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

W. van Zuijlen, Eigenvalues of the Anderson Hamiltonian with white noise potential in 2D, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 1, 2018.

W. van Zuijlen, Massasymptotics for the parabolic Anderson model in 2D, 10th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis, November 29  December 1, 2018, University of Oxford, Mathematical Institute, Oxford, UK, November 29, 2018.

W. van Zuijlen, Massasymptotics for the parabolic Anderson model in 2D, Statistical Mechanics Seminar, University of Warwick, Department of Statistics, Coventry, UK, December 6, 2018.

W. van Zuijlen, Meanfield GibbsnonGibbs transitions, Spring School, Spin Systems: Discrete and Continuous, March 19  23, 2018, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt.

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 28, 2018.

L. Andreis, A largedeviations approach to the multiplicative coagulation process, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica ``Tullio LeviCivita'', Italy, October 12, 2018.

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

L. Andreis, Ergodicity of a system of interacting random walks with asymmetric interaction, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 1, 2018.

L. Andreis, Networks of interacting components with macroscopic selfsustained periodic behavior, Neural Coding 2018, September 9  14, 2018, University of Torino, Department of Mathematics, Italy, September 10, 2018.

L. Andreis, Selfsustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26  29, 2018, LudwigMaximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 26, 2018.

L. Andreis, System of interacting random walks with asymmetric interaction, 48th Probability Summer School, July 8  20, 2018, Clermont Auvergne University, Saint Flour, France, July 17, 2018.

F. Flegel, Localization vs. homogenization in the random conductance model, Forschungsseminar Analysis, Technische Universität Chemnitz, June 6, 2018.

F. Flegel, Spectral homogenization vs. localization in the barrier model, Symposium on the occasion of the 60th birthday of Igor Sokolov, Bernstein Center for Computational Neuroscience Berlin, HumboldtUniversität zu Berlin, February 26, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11  17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 12, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Seminar Angewandte Analysis, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, March 9, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Seminar of SFB/CRC 1060 Bonn, Rheinische FriedrichWilhelmsUniversität Bonn, June 12, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Stochastik, Universität zu Köln, Mathematisches Institut, June 14, 2018.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Wahrscheinlichkeitstheorie, LudwigsMaximiliansUniversität München, July 9, 2018.

C. Bartsch, V. John, R.I.A. Patterson, A new mixed stochasticdeterministic simulation approach to particle populations in fluid flows, 6th International Conference on Population Balance Modelling (PBM2018), Ghent University, Belgium, May 7  9, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, ICM 2018 Satellite Conference: Topics in Mathematical Physics, July 26  31, 2018, Institute of Mathematics and Statistics, University of São Paulo, Institute of Physics, Brazil, July 27, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Random Structures in Neuroscience and Biology, March 26  29, 2018, LudwigMaximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 29, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Ibn Zohr University, Agadir, Morocco, September 28, 2018.

B. Jahnel, Continuum percolation for Cox point processes, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 28, 2018.

B. Jahnel, Continuum percolation for Cox point processes, Seminar, Universität Potsdam, Institut für Mathematik, April 13, 2018.

B. Jahnel, Continuum percolation for Cox point processes, Universität Osnabrück, Fachbereich Mathematik/Informatik, February 1, 2018.

B. Jahnel, Dynamical GibbsnonGibbs transitions for continuous spin models, DFGAIMS Workshop on Evolutionary Processes on Networks, March 20  24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

B. Jahnel, Dynamical GibbsnonGibbs transitions for the continuum WidomRowlinson model, Seminar der AG Stochastik, Technische Universität Darmstadt, Fachbereich Mathematik, September 21, 2018.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Workshop on Transformations and Phase Transitions, January 29  31, 2018, RuhrUniversität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Universität Potsdam, Institut für Mathematik, October 10, 2018.

B. Jahnel, Influence of mobility on connectivity, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

B. Jahnel, Percolation for Cox point processes, Geometry and Scaling of Random Structures, July 16  27, 2018, Centre International de Mathématiques Pures et Appliquées (CIMPA), School and X Escuela Santaló, ICM Rio Satellite Workshop, Buenos Aires, Argentina, July 18, 2018.

B. Jahnel, Spatial stochastic models with applications in telecommunications, Summer School 2018 ``Combinatorial Structures in Geometry'', September 24  27, 2018, Universität Osnabrück, Institut für Mathematik (DFG GK 1916).

B. Jahnel, Telecommunication models in random environments, BIMoS Day : The Mathematics of Quantum Information, May 23, 2018, Technische Universität Berlin, Berlin, May 23, 2018.

B. Jahnel, Telecommunication models in random environments, BIMoS Day, Berlin International Graduate School in Model and Simulation based Research, Technische Universität Berlin, May 23, 2018.

W. König, A largedeviations approach to the multiplicative coalescent, Workshop on Highdimensional Phenomena in Probability  Fluctuations and Discontinuity (Research Training Group 2131), September 24  28, 2018, RuhrUniversität Bochum, September 28, 2018.

W. König, Large deviations theory and applications, Classical and Quantum Dynamics of Interacting Particle Systems, June 15, 2018, Universität zu Köln, Mathematisches Institut, Köln.

W. König, Probabilistic Methods in Telecommunication, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

W. König, Random message routing in highly dense multihop networks, DFGAIMS Workshop on Evolutionary Processes on Networks, March 20  24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

W. König, The principal part of the spectrum of random Schrödinger operators in large boxes, RheinMain Kolloquium Stochastik, GoetheUniversität Frankfurt am Main, January 26, 2018.

R.I.A. Patterson, Large deviations for reaction fluxes, Séminaire EDP, Modélisation et Calcul Scientifique (commun ICJ & UMPA), École Normale Supérieure de Lyon (CNRS), France, July 12, 2018.

R.I.A. Patterson, Large deviations for reaction fluxes, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11  17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 15, 2018.
External Preprints

E. Candellero, A. Stauffer, L. Taggi, Abelian oil and water dynamics does not have an absorbingstate phase transition, Preprint no. arXiv:1901.08425v1, , 2019.

V. Betz, H. Schäfer, L. Taggi, Interacting selfavoiding polygons, Preprint no. arXiv:1902.08517, , 2019.
Abstract
We consider a system of selfavoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a subregion of the phase diagram where the selfavoiding polygons are space filling and we provide a nontrivial characterization of the regime where the polygon length admits uniformly bounded exponential moments. 
J.D.D. Deuschel, H.E. Altman , T. Orenshtein, Über die mit Benetzungsmodellen verbundene Gradientendynamik, Preprint no. arXiv: 1908.08850, , 2019.

D. Heydecker , R.I.A. Patterson, Kac interaction clusters: A bilinear coagulation equation and phase transition, Preprint no. arXiv:1902.07686, , 2019.
Abstract
We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing wellposedness, and a closed form for the time of the gelation phase transition tg when a macroscopic cluster suddenly emerges. We further prove that the second moment of the cluster size distribution diverges exactly at tg. Our methods apply immediately to coagulating particle systems with other bilinear coagulation kernels. 
B. Lees, L. Taggi, Site monotonicity and uniform positivity for interacting random walks and the spin O(N) model with arbitrary N, Preprint no. arXiv:1902.07252, , 2019.

L. Taggi, Uniformly positive correlations in the dimer model and phase transition in lattice permutations on Zd, d>2, via reflection positivity, Preprint no. arXiv:1909.06558, , 2019.
Abstract
Our first main result is that correlations between monomers in the dimer model on Zd do not decay to zero when d>2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero algebraically. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a selfavoiding walk interacting with lattice permutations and we prove that, in the regime of fullypacked loops, such a walk is `long' and the distance between its endpoints grows linearly with the diameter of the box. These results follow from the derivation of an Infraredultraviolet bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertice 
A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twinscrew granulation model, Technical report no. 195, University of Cambridge, c4ePreprint Series, 2018.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations