Alignment Percolation


The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in \(d \geqslant 2\) dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on \(\mathbb {Z}^{d}\) with parameter p ∈ (0,1]. For each occupied site v, and for each of the 2d possible coordinate directions, declare the entire line segment from v to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the ‘one-choice model’, each occupied site declares one of its 2d incident segments to be blue. In the ‘independent model’, the states of different line segments are independent.

This is a preview of subscription content, access via your institution.


  1. 1.

    Aizenman, M., Grimmett, G.R.: Strict monotonicity of critical points in percolation and ferromagnetic models. J. Statist. Phys. 63, 817–835 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Athreya, K.B., Ney, P.E.: Branching Processes Grundlehren Der Mathematischen Wissenschaften, vol. 196. Springer-Verlag, Heidelberg (1972)

    Google Scholar 

  3. 3.

    Barsky, D.J., Grimmett, G.R., Newman, C.M.: Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90, 111–148 (1991)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys. 121, 501–505 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Burton, R.M., Keane, M.: Topological metric properties of infinite clusters in stationary two-dimensional site percolation. Israel J. Math. 76, 299–316 (1991)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chayes, L., Schonmann, R.H.: Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Probab. 10, 1182–1196 (2000)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Daley, D.J., Ebert, S., Last, G.: Two lilypond systems of finite line-segments. Probab. Math. Stat. 36, 221–246 (2016)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gandolfi, A., Keane, M.S., Newman, C.M.: Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Relat. Fields 92, 511–527 (1992)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Georgii, H.-O., Häggström, O., Maes, C.: The Random Geometry of Equilibrium Phases, Phase Transitions and Critical Phenomena, vol. 18, pp 1–142. Academic Press, London (2000)

    Google Scholar 

  10. 10.

    Grimmett, G.R.: Percolation, Second Ed. Grundlehren Der Mathematischen Wissenschaften, vol. 321. Springer-Verlag, Berlin (1999)

    Google Scholar 

  11. 11.

    Grimmett, G.R., Marstrand, J.M.: The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430, 439–457 (1990)

    ADS  MathSciNet  MATH  Google Scholar 

  12. 12.

    Häggström, O., Meester, R.: Nearest neighbor and hard sphere models in continuum percolation. Random Struct. Algor. 9, 295–315 (1996)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hammersley, J.M.: Percolation processes: Lower bounds for the critical probability. Ann. Math. Statist. 28, 790–795 (1957)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hilário, M., Ungaretti, D.: A note on the phase transition for independent alignment percolation.

  15. 15.

    Hirsch, C.: On the absence of percolation in a line-segment based lilypond model. Ann. Inst. Henri Poincaré, Probab. Statist. 52, 127–145 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Hirsch, C., Holmes, M., Kleptsyn, V.: Absence of WARM percolation in the very strong reinforcement regime. Annals of Applied Probability - to appear

  17. 17.

    Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25, 71–95 (1997)

    MathSciNet  Article  Google Scholar 

Download references


NRB is supported by grant DE170100186 from the Australian Research Council. MH is supported by Future Fellowship FT160100166 from the Australian Research Council. The authors are grateful to a colleague and two referees for their comments.

Author information



Corresponding author

Correspondence to Geoffrey R. Grimmett.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Beaton, N.R., Grimmett, G.R. & Holmes, M. Alignment Percolation. Math Phys Anal Geom 24, 3 (2021).

Download citation


  • Percolation
  • One-choice model
  • Independent model

Mathematics Subject Classification (2010)

  • 60K35