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

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 < 1/2, the giant component of H n,p is likely to be quasi-isometric to H n with constant distortion (depending only on α). For 1/2 < α < 1 the minimal distortion tends to infinity as a power of n. We argue that the phase 1/2 < α < 1 is an analogue of the non-uniqueness phase appearing in percolation on non-amenable graphs.

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Correspondence to Omer Angel.

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Angel, O., Benjamini, I. A phase transition for the metric distortion of percolation on the hypercube. Combinatorica 27, 645–658 (2007). https://doi.org/10.1007/s00493-007-2241-5

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Mathematics Subject Classification (2000)

  • 05D40
  • 68R10
  • 60D05