Publications

Articles in Refereed Journals

  • B. Jahnel, Ch. Külske, Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 23 (2017) pp. 2808--2827.
    Abstract
    We investigate the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments

  • 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?? ''

Preprints, Reports, Technical Reports

  • 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.

  • 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.

  • 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.

  • 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.

Talks, Poster

  • B. Jahnel, Fabrics of dreams, Seminar am Ökonomischen Institut, Johannes Gutenberg Universiät Mainz, Ökonomisches Institut.

  • 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.

  • 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.

  • B. Jahnel, Large deviations in relay-augmented wireless networks, Workshop on Probabilistic Methods in Telecommunication, November 14 - 16, 2016, WIAS Berlin, November 16, 2016.

  • P. Keeler, A random walk through the history of random terms, Weekly Seminar, The University of Melbourne, Department of Mathematics and Statistics, Australia, October 3, 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.