Skip to main content
SpringerLink
Log in

Alignment Percolation

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

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, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Hilário, M., Ungaretti, D.: A note on the phase transition for independent alignment percolation. https://arxiv.org/pdf/2007.00539.pdf

  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)

    Article  ADS  MathSciNet  Google Scholar 

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

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

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Authors and Affiliations

  1. School of Mathematics & Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Nicholas R. Beaton, Geoffrey R. Grimmett & Mark Holmes

  2. Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB2 0WB, UK

    Geoffrey R. Grimmett

Authors
  1. Nicholas R. Beaton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geoffrey R. Grimmett
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mark Holmes
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Check for updates. 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). https://doi.org/10.1007/s11040-021-09373-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-021-09373-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation