Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}n, where nearest-neighbor edges are occupied with probability p. Let p c (n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2n/3, where λ is a small positive constant. Let ε=n(pp c (n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2n/3) is of size Θ(22n/3), below this scaling window it is at most 2(log 2)−2, and above this scaling window it is at most O(ε2n). In this paper, we prove that for \( p - p_{c} {\left( n \right)} \geqslant e^{{cn^{{1/3}} }} \) the size of the largest component is at least Θ(ε2n), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.

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Correspondence to Christian Borgs.

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Borgs, C., Chayes, J.T., van der Hofstad, R. et al. Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube. Combinatorica 26, 395–410 (2006). https://doi.org/10.1007/s00493-006-0022-1

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Mathematics Subject Classification (2000):

  • 05C80
  • 82B43