Random subgraphs of finite graphs: III. The phase transition for the n-cube

Christian Borgs, Jennifer T. Chayes, Remco Van Der Hofstad, Gordon Slade, Joel Spencer

Research output: Contribution to journalArticle

Abstract

We study random subgraphs of the n-cube {01} 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 λ2 n/3 where λ is a small positive constant. Let ε=n(p-p c (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(ε2 n ). In this paper we prove that for p - pc(n)≥ e-cn1/3 the size of the largest component is at least Θ(ε2 n ) 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.

Original languageEnglish (US)
Pages (from-to)395-410
Number of pages16
JournalCombinatorica
Volume26
Issue number4
DOIs
StatePublished - Aug 2006

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N-cube
Finite Graph
Subgraph
Phase Transition
Phase transitions
Scaling
Geometry
Nearest Neighbor
Upper bound
Vertex of a graph

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics

Cite this

Random subgraphs of finite graphs : III. The phase transition for the n-cube. / Borgs, Christian; Chayes, Jennifer T.; Van Der Hofstad, Remco; Slade, Gordon; Spencer, Joel.

In: Combinatorica, Vol. 26, No. 4, 08.2006, p. 395-410.

Research output: Contribution to journalArticle

Borgs, Christian ; Chayes, Jennifer T. ; Van Der Hofstad, Remco ; Slade, Gordon ; Spencer, Joel. / Random subgraphs of finite graphs : III. The phase transition for the n-cube. In: Combinatorica. 2006 ; Vol. 26, No. 4. pp. 395-410.
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