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



  • Károly Böröczky, Ferenc Fodor and Daniel Hug
  • Strengthened volume inequalities for L_p zonoids of even isotropic measures
  • Transactions of the American Mathematical Society (2018)
  • Klatt, Michael A. and Torquato, Salvatore
  • Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions
  • Phys. Rev. E 97(1) (2018)
  • Hug, Daniel and Thaele, Christoph
  • Splitting tessellations in spherical spaces
  • Preprint (2018)
    note: arXiv: 1804.08740
  • Hug, Daniel and Kabluchko, Zakhar
  • An inclusion-exclusion identity for normal cones of polyhedral sets
  • Mathematika 64(1), 124–136 (2018)
  • Hirsch, Christian and Last, Günter
  • On Maximal Hard-Core Thinnings of Stationary Particle Processes
  • J. Stat. Phys. 170, 554–583 (2018)
  • Lachiéze-Rey, Raphaël and Schulte, Matthias and Yukich, J. E.
  • Normal approximation for stabilizing functionals
  • Annals of Applied Probability (to appear) (2018)
    note: arXiv: 1702.00726
  • Reitzner, Matthias and Schneider, Rolf
  • On the cells in a stationary Poisson hyperplane mosaic
  • Adv. Geom. (2018)
  • Schneider, Rolf
  • Small faces in stationary Poisson hyperplane tessellations
  • Preprint (2018)
    note: arXiv: 1808.05432
  • Schneider, Rolf
  • The polytopes in a Poisson hyperplane tessellation
  • Preprint (2018)
    note: arXiv: 1804.05622
  • Michael Klatt, Günter Last, Dhandapani Yogeshwaran
  • Hyperuniform and rigid stable matchings
  • Preprint (2018)
  • P. W. A. Schönhöfer AND M. Marechal AND G. Schröder-Turk
  • Density functional theory for hard uniaxial particles: Complex ordering of pear-shaped and spheroidal particles near a substrate
  • The Journal of Chemical Physics 148(12), 124104 (2018)
  • Philipp W A Schönhöfer and Douglas J Cleaver and Gerd E Schröder-Turk
  • Double diamond phase in pear-shaped nanoparticle systems with hard sphere solvent
  • Journal of Physics D: Applied Physics 51(46), 464003 (2018)
  • Simon Weis and Gerd E Schröder-Turk and Matthias Schröter
  • Structural similarity between dry and wet sphere packings
  • Preprint (2018)
    note: arXiv: 1808.04342
  • Giustiniani, Anaïs and Weis, Simon and Poulard, Christophe and Kamm, Paul H. and García-Moreno, Francisco and Schröter, Matthias and Drenckhan, Wiebke
  • Skinny emulsions take on granular matter
  • Soft Matter 14, 7310-7323 (2018)
  • Matthias C. Bott and Felix Winterhalter and Matthieu Marechal and Abhinav Sharma and Joseph M. Brader and René Wittmann
  • Isotropic-nematic transition of self-propelled rods in three dimensions
  • Phys. Rev. E 98, 012601 (2018)
  • Philipp W. A. Schönhöfer and Gerd E. Schröder-Turk and Matthieu Marechal
  • Density functional theory for hard uniaxial particles: Complex ordering of pear-shaped and spheroidal particles near a substrate
  • J. Chem. Phys. 148, 124104 (2018)


  • Imre Bárány, Daniel Hug, Matthias Reitzner, and Rolf Schneider
  • Random points in halfspheres
  • Random Structures & Algorithms 50(1), 3–22 (2017)
  • Károly Böröczky and Daniel Hug
  • Isotropic measures and stronger forms of the reverse isoperimetric inequality
  • Trans. Amer. Math. Soc. 369(10), 6987–7019 (2017)
  • 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)
  • Hug, Daniel and Kiderlen, Markus and Svane, Anne Marie
  • Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples
  • Discrete Comput. Geom. 57(3), 545–570 (2017)
  • Klatt, Michael A. and Last, Günter and Mecke, Klaus and Redenbach, Claudia and Schaller, Fabian M. and Schröder-Turk, Gerd E.
  • Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors
  • pages 385–421 in: Lecture Notes in Mathematics, Vol. 2177: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (editor(s): Vedel Jensen, Eva B. and Kiderlen, Markus), Springer International Publishing, Cham 2017
  • Wilts, Bodo D. and Apeleo Zubiri, Benjamin and Klatt, Michael A. and Butz, Benjamin and Fischer, Michael G. and Kelly, Stephen T. and Spiecker, Erdmann and Steiner, Ullrich and Schröder-Turk, Gerd E.
  • Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development
  • Sci. Adv. 3(4), e1603119 (2017)
  • Hug, Daniel and Reichenbacher, Andreas
  • Geometric inequalities, stability results and Kendall's problem in spherical space
  • arXiv:1709.06522 [math] (2017)
    note: arXiv: 1709.06522
  • Kabluchko, Zakhar and Last, Günter and Zaporozhets, Dmitry
  • Inclusion–Exclusion Principles for Convex Hulls and the Euler Relation
  • Discrete Comput. Geom. 58(2), 417–434 (2017)
  • Mughal, Adil and Libertiny, Tomas and Schröder-Turk, GE
  • How bees and foams respond to curved confinement: Level set boundary representations in the Surface Evolver
  • Colloids and Surfaces A: Physicochemical and Engineering Aspects 534, 94–104 (2017)
  • Mughal, Adil and Cox, SJ and Schröder-Turk, GE
  • Curvature driven motion of a bubble in a toroidal Hele-Shaw cell
  • Interface focus 7(4), 20160106 (2017)
  • Schmiedeberg, M. and Achim, C. V. and Hielscher, J. and Kapfer, S. C. and Löwen, H.
  • Dislocation-free growth of quasicrystals from two seeds due to additional phasonic degrees of freedom
  • Phys. Rev. E 96, 012602 (2017)
  • Hielscher, J. and Martinsons, M. and Schmiedeberg, M. and Kapfer, S. C.
  • Detection of phonon and phason modes in intrinsic colloidal quasicrystals by reconstructing their structure in hyperspace
  • Journal of Physics: Condensed Matter 29(9), 094002 (2017)
  • Schaller, Fabian M. and Punzmann, Horst and Schröder-Turk, Gerd E. and Saadatfar, Mohammad
  • Towards minimal models for realistic granular materials: Tomographic analysis of bidispersed assemblies of ellipsoids
  • EPJ Web Conf. 140, 06030 (2017)
  • Weis, Simon and Schönhöfer, Philipp W. A. and Schaller, Fabian M. and Schröter, Matthias and Schröder-Turk, Gerd E.
  • Pomelo, a tool for computing Generic Set Voronoi Diagrams of Aspherical Particles of Arbitrary Shape
  • EPJ Web Conf. 140, 06007 (2017)
  • M. Marechal AND S. Dussi AND M. Dijkstra
  • Density functional theory and simulations of colloidal triangular prisms
  • The Journal of Chemical Physics 146(12), 124905 (2017)
  • Sch{ö}nh{ö}fer, Philipp W. A. and Ellison, Laurence J. and Marechal, Matthieu and Cleaver, Douglas J. and Schr{ö}der-Turk, Gerd E.
  • Purely entropic self-assembly of the bicontinuous Ia3̅d gyroid phase in equilibrium hard-pear systems
  • Interface Focus 7(4) (2017)
  • Schella, André and Weis, Simon and Schröter, Matthias
  • Charging changes contact composition in binary sphere packings
  • Phys. Rev. E 95, 062903 (2017)
  • Weis, Simon and Schröter, Matthias
  • Analyzing X-ray tomographies of granular packings
  • Review of Scientific Instruments 88(5), 051809 (2017)


  • D.J. Daley, Sven Ebert, and Günter Last
  • Two lilypond systems of finite line-segments
  • Probab. Math. Statist., p. 221–246 (2016)
  • Daniel Hug and Rolf Schneider
  • Random conical tessellations
  • Discrete Comput. Geom. 52, 395–426 (2016)
  • Hug, Daniel and Reitzner, Matthias
  • Introduction to stochastic geometry
  • pages 145–184 in: Bocconi Springer Ser., Vol. 7: Stochastic analysis for Poisson point processes (editor(s): Peccati, Giovanni and Reitzner, Matthias), Bocconi Univ. Press, [place of publication not identified], 2016
  • Klatt, Michael A.
  • Morphometry of random spatial structures in physics
  • FAU University Press, Erlangen 2016
    note: PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg
  • Reichenbacher, Andreas
  • Random tessellations: Stereological formulae in Euclidean space and Kendall's Problem in spherical space
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2016
  • Ziesche, Sebastian
  • Perkolation auf zufälligen Mosaiken und im Boole'schen Modell
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2016
  • Klatt, Michael A. and Torquato, Salvatore
  • 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)
  • Mitschke, Holger and Schury, Fabian and Mecke, Klaus and Wein, Fabian and Stingl, Michael and Schröder-Turk, Gerd E.
  • 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)
  • Ziesche, Sebastian
  • First passage percolation in Euclidean space and on random tessellations
  • Preprint (2016)
    note: arXiv: 1611.02005
  • Ziesche, Sebastian
  • Bernoulli Percolation on random Tessellations
  • Preprint (2016)
    note: arXiv: 1609.04707
  • Fodor, Ferenc and Hug, Daniel and Ziebarth, Ines
  • The volume of random polytopes circumscribed around a convex body
  • Mathematika 62(1), 283–306 (2016)
  • Schulte, Matthias and Thäle, Christoph
  • Poisson Point Process Convergence and Extreme Values in Stochastic Geometry
  • pages 255–294 in: Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry (editor(s): Peccati, Giovanni and Reitzner, Matthias), Springer International Publishing, Cham 2016
  • Schulte, Matthias
  • Normal Approximation of Poisson Functionals in Kolmogorov Distance
  • J. Theoret. Probab. 29(1), 96–117 (2016)
  • Decreusefond, Laurent and Schulte, Matthias and Thäle, Christoph
  • Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry
  • Ann. Probab. 44(3), 2147–2197 (2016)
  • Schneider, Rolf
  • Second moments related to Poisson hyperplane tessellations
  • J. Math. Anal. Appl. 434(2), 1365–1375 (2016)
  • Schaller, Fabian M. and Weigel, Robert F. B. and Kapfer, Sebastian C.
  • Densest Local Structures of Uniaxial Ellipsoids
  • Phys. Rev. X 6, 041032 (2016)
  • Topic, Nikola and Schaller, Fabian M. and Schröder-Turk, Gerd E. and Pöschel, Thorsten
  • The microscopic structure of mono-disperse granular heaps and sediments of particles on inclined surfaces
  • Soft Matter 12, 3184-3188 (2016)


  • 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)
  • Gerd E Schröder-Turk
  • The Hard Core of Soft Matter: Kugelsysteme in der weichen Materie
  • Mitteilungen der Deutschen Mathematiker-Vereinigung 23(4), 208-217 (2015)
  • Susan Nachtrab and Matthias J F Hoffmann and Sebastian C Kapfer and Gerd E Schröder-Turk and Klaus Mecke
  • Beyond the percolation universality class: the vertex split model for tetravalent lattices
  • New Journal of Physics 17(4), 043061 (2015)


  • 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)
  • Schulte, Matthias and Thäle, Christoph
  • Distances Between Poisson k-Flats
  • Methodol. Comput. Appl. Probab. 16(2), 311–329 (2014)
  • Mughal, Adil and Weaire, Denis
  • Theory of cylindrical dense packings of disks
  • Physical Review E 89(4), 042307 (2014)
  • M. Marechal and S. Korden and K. Mecke
  • Deriving fundamental measure theory from the virial series: Consistency with the zero-dimensional limit
  • Phys. Rev. E 90, 042131 (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
  • CSGB Research Report 2013-05 (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)
  • Schneider, Rolf
  • Extremal Properties of Random Mosaics
  • pages 301–330 in: Bolyai Soc. Math. Stud. 24: Geometry — Intuitive, Discrete, and Convex: A Tribute to László Fejes Tóth (editor(s): Bárány, Imre and Böröczky, Károly J. and Tóth, Gábor Fejes and Pach, János), Springer Berlin Heidelberg, Budapest 2013
  • Mitschke, H. and Schröder-Turk, G. E. and Mecke, K. and Fowler, P. W. and Guest, S. D.
  • Symmetry detection of auxetic behaviour in 2D frameworks
  • EPL (Europhysics Letters) 102(6), 66005 (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)
  • Mitschke, Holger and Robins, Vanessa and Mecke, Klaus and Schroder-Turk, G. E.
  • Finite auxetic deformations of plane tessellations
  • Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences 469, 20120465–20120465 (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)