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


5. Percolation

Percolation is the occurrence of an unbounded connected component in a random graph or in a random subset of Euclidean space. Accurate approximations of percolation thresholds and the study of the behaviour of the system near this critical threshold are central topics of percolation theory.
This project deals with percolation problems, where the underlying system depends on a random spatial structure such as a random tessellation or a Boolean model. Examples that are investigated within this project are the Boolean model, the lilypond model, face percolation on stationary tessellations in Euclidean space and loop percolation. Special emphasis is given to the relationship between percolation properties and Minkowski functionals.

Project Members

Cooperating partners



  • Ziesche, Sebastian
  • Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^d$
  • Ann. Inst. H. Poincaré Probab. Statist. 54(2), 866–878 (2018)
  • Last, Günter and Nestmann, Franz and Schulte, Matthias
  • The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation
  • Preprint (2018)
    note: arXiv: 1808.01203


  • Klatt, Michael A and Schröder-Turk, Gerd E and Mecke, Klaus
  • Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals
  • J. Stat. Mech. Theor. Exp. 2017(2), 023302 (2017)
  • Last, Günter and Penrose, Mathew D. and Zuyev, Sergei
  • On the capacity functional of the infinite cluster of a Boolean model
  • Ann. Appl. Probab. 27, 1678–1701 (2017)
  • Last, Günter and Ziesche, Sebastian
  • On the Ornstein–Zernike equation for stationary cluster processes and the random connection model
  • Adv. Appl. Probab. 49(4), 1260–1287 (2017)


  • Klatt, Michael A.
  • Morphometry of random spatial structures in physics
  • FAU University Press, Erlangen 2016
    note: PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg
  • Ziesche, Sebastian
  • Perkolation auf zufälligen Mosaiken und im Boole'schen Modell
  • PhD thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe 2016
  • Markus Spanner, Felix Höfling, Sebastian C. Kapfer, Klaus R. Mecke, Gerd E. Schröder-Turk and Thomas Franosch
  • Splitting of the Universality Class of Anomalous Transport in Crowded Media
  • Phys. Rev. Lett. 116, 060601 (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


  • Susan Nachtrab, Matthias J.F. Hoffmann, Sebastian C Kapfer, 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)
  • 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)


  • Günter Last and Eva Ochsenreither
  • Percolation on stationary tessellations: models, mean values and second order structure
  • J. Appl. Probab. 51A, 311–332 (2014)


  • Günter Last and Mathew D. Penrose
  • Percolation and limit theory for the Poisson lilypond model
  • Random Structures & Algorithms 42, 226–249 (2013)


  • Christian Scholz, Frank Wirner, Jan Götze, Ulrich Rüde, Gerd E. Schröder-Turk, Klaus Mecke, and Clemens Bechinger
  • Permeability of porous materials determinded from the Euler characteristic
  • Physical Review Letters 109, 264504 (2012)


  • Susan Nachtrab, Sebastian C. Kapfer, Christoph H. Arns, Mahyar Madadi, Klaus Mecke, and Gerd E. Schröder-Turk
  • Morphology and linear-elastic moduli of random network solids
  • Advanced Material 23(22–23), 2633–2637 (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)
  • Markus Spanner, Felix Höfling, Gerd E. Schröder-Turk, Klaus Mecke, and Thomas Franosch
  • Anomalous transport of a tracer on percolating clusters
  • Journal of Physiscs: Condensed Matter 23, 234120 (2011)