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

  • Authors:

  • Christian Borgs;Jennifer T. Chayes;Remco van der Hofstad;Gordon Slade;Joel Spencer

  • Affiliations:

  • Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA;Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA;Department of Mathematics and Computer Science, Eindhoven University of Technology, 513, One Microsoft Way, 5600 MB, Eindhoven, The Netherlands;Department of Mathematics, University of British Columbia, 513, One Microsoft Way, 5600 MB, Vancouver, BC V6T 1Z2, Canada;Courant Institute of Mathematical Sciences, New York University, 513, 251 Mercer St., 5600 MB, New York, NY 10012, USA

  • Venue:
  • Combinatorica
  • Year:
  • 2006

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Abstract

We study random subgraphs of the n-cube {0,1}n, where nearest-neighbor edges are occupied with probability p. Let pc(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(p−pc(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2−n/3) is of size Θ(22n/3), below this scaling window it is at most 2(log 2)nε−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.