A phase transition for the metric distortion of percolation on the hypercube

  • Authors:

  • Omer Angel;Itai Benjamini

  • Affiliations:

  • University of Toronto, Department of Mathematics, M5S 2E4, Toronto, ON, Canada;Weizmann Institute of Science, Department of Mathematics, 76100, Rehovot, ON, Israel

  • Venue:
  • Combinatorica
  • Year:
  • 2007

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Abstract

Let H n be the hypercube {0, 1} n , and denote by H n,p Bernoulli bond percolation on H n , with parameter p = n −α . It is shown that at α = 1/2 there is a phase transition for the metric distortion between H n and H n,p . For α H n,p is likely to be quasi-isometric to H n with constant distortion (depending only on α). For 1/2 α n. We argue that the phase 1/2 α