On a form of coordinate percolation

  • Authors:

  • Elizabeth r. Moseman;Peter Winkler

  • Affiliations:

  • Department of mathematical sciences, usma, west point ny 10996, usa (e-mail: lizz.moseman@usma.edu);Department of mathematics, dartmouth college, hanover nh 03755-3551, usa (e-mail: peter.winkler@dartmouth.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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Abstract

Let ai,bi, i = 0, 1, 2,… be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) ε ℕ2 be open if ai + bj ≤ t, and closed otherwise. We obtain a simple, exact expression for the probability Θ(t) that there is an infinite path (oriented or not) of open sites, containing the origin. Θ(t) is continuous and has continuous first derivative except at the critical point (t=1), near which it has critical exponent (3 - √5 /2.