Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Percolation of poisson sticks on the plane
Download PDF
Download PDF
  • Published: December 1991

Percolation of poisson sticks on the plane

  • Rahul Roy1 

Probability Theory and Related Fields volume 89, pages 503–517 (1991)Cite this article

  • 127 Accesses

  • 15 Citations

  • Metrics details

Summary

We consider a percolation model on the plane which consists of 1-dimensional sticks placed at points of a Poisson process onR 2 each stick having a random, but bounded length and a random direction. The critical probabilities are defined with respect to the occupied clusters and vacant clusters and they are shown to be equal. The equality is shown through a ‘pivotal cell’ argument, using a version of the Russo-Seymour-Welsh theorem which we obtain for this model.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Domany, E., Kinzel, W.: Equivalence of cellular automata to Ising models and directed percolation. Phys. Rev. Lett.53, 311–314 (1984)

    Google Scholar 

  2. Hall, P.: On continuum percolation. Ann. Probab.13, 1250–1266 (1985)

    Google Scholar 

  3. Kemperman, J.H.B.: On the FKG Inequality for measures on a partially orderes space. Proc. K. Ned. Akad. Wet. Ser. A, Math. Sci.80, 313–331 (1977)

    Google Scholar 

  4. Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982

    Google Scholar 

  5. Menshikov, M.V.: Coincidence of critical points in percolation problems. Sov. Math., Dokl24, 856–859 (1986)

    Google Scholar 

  6. Roy, R.: Equality of the critical densities and dual critical densities for continuum percolation inR 2. Ann. Probab.18, 1563–1575 (1990)

    Google Scholar 

  7. Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229–237 (1981)

    Google Scholar 

  8. Zuev, S.A., Sidorenko, A.F.: Continuous models of percolation theory I, II. Theor. Math. Phys.62, 76–86 and 253–262 (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, 110016, New Delhi, India

    Rahul Roy

Authors
  1. Rahul Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Roy, R. Percolation of poisson sticks on the plane. Probab. Th. Rel. Fields 89, 503–517 (1991). https://doi.org/10.1007/BF01199791

Download citation

  • Received: 21 March 1990

  • Revised: 06 May 1991

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01199791

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Poisson Process
  • Mathematical Biology
  • Percolation Model
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature