### Abstract

The two-dimensional Hamming graph H(2,n) consists of the n^{2} vertices (i,j), 1 ≤ i,j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, in such a way that the average degree satisfies 2(n - 1)p = 1 + ε. Previous work [8] has shown that in the barely supercritical region n^{-2/3} ln^{1/3} n « ε » 1, the largest component satisfies a law of large numbers with mean 2εn. Here we show that the second largest component has, with high probability, size bounded by 2^{8}ε^{-2} log(n^{2}ε^{3}), so that the dominant component has emerged. This result also suggests that a discrete duality principle holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime.

Original language | English (US) |
---|---|

Pages (from-to) | 80-89 |

Number of pages | 10 |

Journal | Random Structures and Algorithms |

Volume | 36 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2010 |

### Fingerprint

### Keywords

- Percolation
- Phase transition
- Random graphs
- Scaling window

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*36*(1), 80-89. https://doi.org/10.1002/rsa.20288

**The second largest component in the supercritical 2D Hamming graph.** / van der Hofstad, Remco; Luczak, Malwina J.; Spencer, Joel.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 36, no. 1, pp. 80-89. https://doi.org/10.1002/rsa.20288

}

TY - JOUR

T1 - The second largest component in the supercritical 2D Hamming graph

AU - van der Hofstad, Remco

AU - Luczak, Malwina J.

AU - Spencer, Joel

PY - 2010/1

Y1 - 2010/1

N2 - The two-dimensional Hamming graph H(2,n) consists of the n2 vertices (i,j), 1 ≤ i,j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, in such a way that the average degree satisfies 2(n - 1)p = 1 + ε. Previous work [8] has shown that in the barely supercritical region n-2/3 ln1/3 n « ε » 1, the largest component satisfies a law of large numbers with mean 2εn. Here we show that the second largest component has, with high probability, size bounded by 28ε-2 log(n2ε3), so that the dominant component has emerged. This result also suggests that a discrete duality principle holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime.

AB - The two-dimensional Hamming graph H(2,n) consists of the n2 vertices (i,j), 1 ≤ i,j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, in such a way that the average degree satisfies 2(n - 1)p = 1 + ε. Previous work [8] has shown that in the barely supercritical region n-2/3 ln1/3 n « ε » 1, the largest component satisfies a law of large numbers with mean 2εn. Here we show that the second largest component has, with high probability, size bounded by 28ε-2 log(n2ε3), so that the dominant component has emerged. This result also suggests that a discrete duality principle holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime.

KW - Percolation

KW - Phase transition

KW - Random graphs

KW - Scaling window

UR - http://www.scopus.com/inward/record.url?scp=70749125909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70749125909&partnerID=8YFLogxK

U2 - 10.1002/rsa.20288

DO - 10.1002/rsa.20288

M3 - Article

VL - 36

SP - 80

EP - 89

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -