### Abstract

In a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper we use a new and simplified approach to the lace expansion to prove quite generally that, for finite graphs that are tori, the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set {0, 1,..., r - 1} ^{n}, including the Hamming cube on an alphabet of r letters (the n-cube, for r = 2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n > 6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

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

Pages (from-to) | 1886-1944 |

Number of pages | 59 |

Journal | Annals of Probability |

Volume | 33 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2005 |

### Fingerprint

### Keywords

- Lace expansion
- Percolation
- Phase transition
- Random graph
- Triangle condition

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*33*(5), 1886-1944. https://doi.org/10.1214/009117905000000260

**Random subgraphs of finite graphs. II. The lace expansion and the triangle condition.** / Borgs, Christian; Chayes, Jennifer T.; Van Der Hofstad, Remco; Slade, Gordon; Spencer, Joel.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 33, no. 5, pp. 1886-1944. https://doi.org/10.1214/009117905000000260

}

TY - JOUR

T1 - Random subgraphs of finite graphs. II. The lace expansion and the triangle condition

AU - Borgs, Christian

AU - Chayes, Jennifer T.

AU - Van Der Hofstad, Remco

AU - Slade, Gordon

AU - Spencer, Joel

PY - 2005/9

Y1 - 2005/9

N2 - In a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper we use a new and simplified approach to the lace expansion to prove quite generally that, for finite graphs that are tori, the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set {0, 1,..., r - 1} n, including the Hamming cube on an alphabet of r letters (the n-cube, for r = 2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n > 6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

AB - In a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper we use a new and simplified approach to the lace expansion to prove quite generally that, for finite graphs that are tori, the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set {0, 1,..., r - 1} n, including the Hamming cube on an alphabet of r letters (the n-cube, for r = 2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n > 6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

KW - Lace expansion

KW - Percolation

KW - Phase transition

KW - Random graph

KW - Triangle condition

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U2 - 10.1214/009117905000000260

DO - 10.1214/009117905000000260

M3 - Article

VL - 33

SP - 1886

EP - 1944

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 5

ER -