### 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(2^{2n/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 - p_{c}(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 language | English (US) |
---|---|

Pages (from-to) | 395-410 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*26*(4), 395-410. https://doi.org/10.1007/s00493-006-0022-1

**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.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 26, no. 4, pp. 395-410. https://doi.org/10.1007/s00493-006-0022-1

}

TY - JOUR

T1 - Random subgraphs of finite graphs

T2 - III. The phase transition for the n-cube

AU - Borgs, Christian

AU - Chayes, Jennifer T.

AU - Van Der Hofstad, Remco

AU - Slade, Gordon

AU - Spencer, Joel

PY - 2006/8

Y1 - 2006/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33749317177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749317177&partnerID=8YFLogxK

U2 - 10.1007/s00493-006-0022-1

DO - 10.1007/s00493-006-0022-1

M3 - Article

AN - SCOPUS:33749317177

VL - 26

SP - 395

EP - 410

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -