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Dependent percolation in two dimensions
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  • Published: August 2000

Dependent percolation in two dimensions

  • P.N. Balister1,
  • B. Bollobás2 &
  • A.M. Stacey3 

Probability Theory and Related Fields volume 117, pages 495–513 (2000)Cite this article

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Abstract.

For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x i , y j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ2 closed if x i = y j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i 0, j 0)(i 1, j 1) · · · such that 0 = i 0≤i 1≤· · ·, 0 = j 0≤j 1≤· · ·, and each site (i n , j n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x i ) and (y j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.

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Authors and Affiliations

  1. Department of Mathematics, University of Memphis, Memphis, TN 38152, USA. e-mail: balistep@msci.memphis.edu, , , , , , US

    P.N. Balister

  2. Department of Mathematics, University of Memphis, Memphis, TN 38152, USA, , , , , , US

    B. Bollobás

  3. Department of Pure Mathematics and Statistics, Peterhouse, University of Cambridge, Cambridge CB2 1RD, England. email: A.M.Stacey@dpmms.cam.ac.uk, , , , , , GB

    A.M. Stacey

Authors
  1. P.N. Balister
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  2. B. Bollobás
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  3. A.M. Stacey
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Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000

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Balister, P., Bollobás, B. & Stacey, A. Dependent percolation in two dimensions. Probab Theory Relat Fields 117, 495–513 (2000). https://doi.org/10.1007/PL00008732

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  • Issue Date: August 2000

  • DOI: https://doi.org/10.1007/PL00008732

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  • Mathematics Subject Classification (1991): 60K35
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