Contact (Karlsruhe)
Prof. Dr. Günter Last
Karlsruhe Institute of Technology
Institut für Stochastik
Kaiserstraße 89
76133 Karlsruhe
Phone: +49-721-608 43265
Fax: +49-721-608 46691
Contact (Erlangen)
Prof. Dr. Klaus Mecke
Universität Erlangen-Nürnberg
Institut für Theoretische Physik
Staudtstraße 7
91058 Erlangen
Phone: +49-9131-85 28442
Fax: +49-9131-85 28444
Contact (Aarhus)
Department of Mathematical Sciences
Ny Munkegade 118
building 1530
8000 Aarhus C


2. Tessellations and hard-core particle systems

image1.jpg 2010_EllipsoidsVoronoiHardSpheres.jpg toc-2012-evans-ipp.jpg toc-2012-kapfer-jammed-altern.jpg

Hard-core particle systems and random tessellations are classical models of spatial stochastics. Hard-core processes model the spatial structure of assemblies of particles which only interact by collisions, such as the grains in a granular material; random tessellations divide the space into cells with disjoint interiors, such as the cells in solid or liquid random foams. These models are important for the coarsening behaviour of foams and for the understanding of random packings. The mathematical focus is on properties of Poisson driven tessellations, especially Voronoi and hyperplane tessellations, and related models such as the lilypond model and flat processes.

Project Members

Cooperating partners



  • Evans, Myfanwy E and Schröder-Turk, Gerd E and Kraynik, Andrew M
  • A geometric exploration of stress in deformed liquid foams
  • Journal of Physics: Condensed Matter 29(12), 124004 (2017)


  • Károly Böröczky and Daniel Hug
  • Isotropic measures and stronger forms of the reverse isoperimetric inequality
  • Trans. Amer. Math. Soc. (to appear) (2016)
  • Károly Böröczky, Ferenc Fodor and Daniel Hug
  • Strengthened volume inequalities for L_p zonoids of even isotropic measures
  • Preprint (2016)
  • Daniel Hug and Rolf Schneider
  • Random conical tessellations
  • Discrete Comput. Geom. 52, 395–426 (2016)
  • Daniel Hug and Matthias Reitzner
  • Introduction to Stochastic Geometry
  • to appear in: Stochastic analysis for Poisson point processes: Malliavin calculus, Wiener-Ito chaos expansions and stochastic geometry (edited by Giovanni Peccati and Matthias Reitzner, Springer & Bocconi Series (2016)) (2016)
  • Daniel Hug and Markus Kiderlen and Anne Marie Svane
  • Voronoi-based estimation of Minkowski tensors from finite point samples
  • Preprint (2016)
  • Klatt, Michael A.
  • Morphometry of random spatial structures in physics
  • FAU University Press, Erlangen 2016
    note: PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg
  • Michael A. Klatt, Günter Last, Klaus Mecke, Claudia Redenbach, Fabian M. Schaller and Gerd E.Schröder-Turk
  • Cell shape analysis of random tessellations based on Minkowski tensors
  • arXiv:1606.07653 [cond-mat] (2016)
    note: arXiv: 1606.07653
  • Michael A. Klatt and Salvatore Torquato
  • Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations
  • Phys. Rev. E 94(2) (2016)
  • Sabrina Kombrink, Erin P. J. Pearse and Steffen Winter
  • Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable
  • Math. Z. 283(3-4), 1049–1070 (2016)
  • Benedikt Krüger and Klaus Mecke
  • Genus dependence of the number of (non-)orientable surface triangulations
  • Phys. Rev. D 93, 085018 (2016)
  • Benedikt Krüger, Ella M. Schmidt, and Klaus Mecke
  • Spectral Properties of Unimodular Lattice Triangulations
  • Journal of Statistical Physics 163(3), 514–543 (2016)
  • Günter Last, Giovanni Peccati and Matthias Schulte
  • Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization
  • Probab. Theory Related Fields 165(3-4), 667–723 (2016)
  • Holger Mitschke and Fabian Schury and Klaus Mecke and Fabian Wein and Michael Stingl and Gerd E. Schröder-Turk
  • Geometry: The leading parameter for the Poisson's ratio of bending-dominated cellular solids
  • International Journal of Solids and Structures 100–101, 1 - 10 (2016)
  • René Wittmann and Matthieu Marechal and Klaus Mecke
  • Fundamental measure theory for non-spherical hard particles: predicting liquid crystal properties from the particle shape
  • Journal of Physics: Condensed Matter 28(24), 244003 (2016)


  • Imre Barany, Daniel Hug, Matthias Reitzner, and Rolf Schneider
  • Random points in halfspheres
  • Random Structures Algorithms (to appear) (2015)
  • Sven Ebert and Günter Last
  • On a class of growth-maximal hard-core processes
  • Stoch. Models 31, 153–184 (2015)
  • Myfanwy E. Evans and Gerd E. Schröder-Turk
  • In a material world: hyperbolic geometry in biological materials
  • Asia Pacific Mathematics Newsletter 5, 21 (2015)
  • Julia Hörrmann, Daniel Hug, Matthias Reitzner, and Christoph Thäle
  • Poisson polyhedra in high dimensions
  • Adv. Math 281, 1–39 (2015)
  • Johannes F. Knauf, Benedikt Krüger and Klaus Mecke
  • Entropy of unimodular lattice triangulations
  • EPL (Europhysics Letters) 109(4), 40011 (2015)
  • Benedikt Krüger, Ella M. Schmidt and Klaus Mecke
  • Unimodular lattice triangulations as small-world and scale-free random graphs
  • New Journal of Physics 17(2), 023013 (2015)
  • Fabian M. Schaller, Sebastian C. Kapfer, James E. Hilton, Paul W. Cleary, Klaus Mecke, Cristiano De Michele, Tanja Schilling, Mohammad Saadatfar, Matthias Schröter, Gary W. Delaney and Gerd E. Schröder-Turk
  • Non-universal Voronoi cell shapes in amorphous ellipsoid packs
  • EPL (Europhysics Letters) 111(2), 24002 (2015)
  • Fabian M. Schaller, Max Neudecker, Mohammad Saadatfar, Gary W. Delaney, Gerd E. Schröder-Turk, and Matthias Schröter
  • Local Origin of Global Contact Numbers in Frictional Ellipsoid Packings
  • Phys. Rev. Lett. 114, 158001 (2015)
  • Benjamin Winter, Benjamin Butz, Christel Dieker, Gerd E. Schröder-Turk, Klaus Mecke and Erdmann Spiecker
  • Coexistence of both gyroid chiralities in individual butterfly wing scales of Callophrys rubi
  • Proceedings of the National Academy of Sciences 112(42), 12911-12916 (2015)


  • D.J. Daley, Sven Ebert, and Günter Last
  • Two lilypond systems of finite line-segments
  • Preprint (2014)
    note: to appear in Probab. Math. Statist.
  • Myfanwy E. Evans and Gerd E. Schröder-Turk
  • In a Material World. Hyperbolische Geometrie in Biologischen Materialien
  • Mitt. Dtsch. Math.-Ver. 22(3), 158–166 (2014)
  • Ellen Fischermeier, Matthieu Marechal, and Klaus Mecke
  • Dynamical States in Driven Colloidal Liquid Crystals
  • Journal of Chemical Physics 141, 1–7 (2014)
  • Ellen Fischermeier, Dominik Bartuschat, Tobias Preclik, Matthieu Marechal, and Klaus Mecke
  • Simulation of a hard-spherocylinder liquid crystal with the pe
  • Computer Physics Communications 185, 3156–3161 (2014)
  • Julia Hörrmann and Daniel Hug
  • On the volume of the zero cell of a class of isotropic Poisson hyperplane tessellations
  • Adv. in Appl. Probab. 46, 622–642 (2014)
  • Daniel Hug and Rolf Schneider
  • Approximation properties of random polytopes associated with Poisson hyperplane processes
  • Adv. in Appl. Probab. 46, 919–936 (2014)
  • Michael A. Klatt and Salvatore Torquato
  • Characterization of maximally random jammed sphere packings: Voronoi correlation functions
  • Phys. Rev. E 90, 052120 (2014)
  • Günter Last, Mathew Penrose, Matthias Schulte, and Christoph Thäle
  • Moments and central limit theorems for some multivariate Poisson functionals
  • Adv. in Appl. Probab. 46, 348-364 (2014)
  • René Wittmann, Matthieu Marechal, and Klaus Mecke
  • Fundamental measure theory for smectic phases: Scaling behavior and higher order terms
  • The Journal of Chemical Physics 141, 064103 (2014)
  • René Wittmann and Klaus Mecke
  • Surface tension of isotropic-nematic interfaces: Fundamental measure theory for hard spherocylinders
  • The Journal of Chemical Physics 140, 104703 (2014)


  • Myfanwy E. Evans, Andrew M. Kraynik, Douglas A. Reinelt, Klaus Mecke and Gerd E. Schröder-Turk
  • Networklike propagation of cell-level stress in sheared random foams
  • Physical Review Letters 111, 138301 (2013)
  • Daniel Hug
  • Random polytopes
  • pages 205–238 in: Lecture Notes in Mathematics 2068: Stochastic Geometry, Spatial Statistics and Random Fields (editor(s): Evgeny Spodarev), 2013
  • Markus Kiderlen and Mario Hörig
  • Matérn's hard core models of types I and II with arbitrary compact grains
  • Preprint (2013)
  • Walter Mickel, Sebastian C. Kapfer, Gerd E. Schröder-Turk, and Klaus Mecke
  • Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter
  • The Journal of Chemical Physics 138(4), 044501:1–7 (2013)
  • Fabian M. Schaller, Sebastian C. Kapfer, Myfanwy E. Evans, Matthias J.F. Hoffmann, Tomaso Aste, Mohammad Saadatfar, Klaus Mecke, Gary W. Delaney, Gerd E. Schröder-Turk
  • Set Voronoi diagrams of 3D assemblies of aspherical particles
  • Philosophical Magazine (2013)
  • Fabian M. Schaller, Max Neudecker, Mohammad Saadatfar, Gary W. Delaney, Klaus Mecke, Gerd E. Schröder-Turk, and Matthias Schröter
  • Tomographic analysis of jammed ellipsoid packings
  • AIP Conf. Proc. 1542, 377 (2013)
  • Gerd E. Schröder-Turk, Richard Schienlein, Sebastian C. Kapfer, Fabian M. Schaller, Gary W. Delaney, Timothy Senden, Mohammad Saadatfar, Tomaso Aste and Klaus Mecke
  • Minkowski tensors and local structure metrics: amorphous and crystalline sphere packings
  • AIP Conf. Proc. 1542, 349-352 (2013)


  • Myfanwy E. Evans, Johannes Zirkelbach, Gerd E. Schröder-Turk, Andrew M. Kraynik, and Klaus Mecke
  • Deformation of Platonic foam cells: Effect on growth rate
  • Physical Review E 85(6), 061401:1–9 (2012)
  • Mario Hörig and Claudia Redenbach
  • The maximum volume hard subset model for Poisson processes: simulation aspects
  • Journal of Statistical Computation and Simulation 82, 107–121 (2012)
  • Sebastian C. Kapfer, Walter Mickel, Klaus Mecke, and Gerd E. Schröder-Turk
  • Jammed spheres: Minkowski tensors reveal onset of local crystallinity
  • Physical Review E 85(3), 030301:1–4 (2012)
  • Erin Pearse and Steffen Winter
  • Geometry of canonical self-similar tilings
  • Rocky Mountain J. Math. 42, 1327–1357 (2012)


  • Daniel Hug and Rolf Schneider
  • Faces with given directions in anisotropic Poisson hyperplane mosaics
  • Adv. in Appl. Probab. 43(2), 308–321 (2011)
  • Daniel Hug and Rolf Schneider
  • Faces of Poisson-Voronoi mosaics
  • Probab. Theory and Related Fields 151, 125–151 (2011)
  • Susan Nachtrab, Sebastian C. Kapfer, Dominik Rietzel, Dietmar Drummer, Mahyar Madadi, Christoph H. Arns, Andrew M. Kraynik, Gerd E. Schröder-Turk, and Klaus Mecke
  • Tuning elasticity of open-cell solid foams and bone scaffolds via randomized vertex connectivity
  • Advanced Engineering Materials 14(1–2), 120–124 (2011)