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

*(English)*Zbl 1079.05087Summary: In a previous paper [Random Struct. Algorithms 27, 137–184 (2005; Zbl 1076.05071)] 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 \(\{0,1,\dots,r-1\}^n\), 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.

The latter is readily verified for several graphs with vertex set \(\{0,1,\dots,r-1\}^n\), 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.

##### MSC:

05C80 | Random graphs (graph-theoretic aspects) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

##### Keywords:

random graph; percolation triangle condition; bond percolation; percolation phase transition; lace expansion; random walks##### References:

[1] | Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear Phys. B [ FS ] 485 551–582. · Zbl 0925.82112 |

[2] | Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489–526. · Zbl 0618.60098 |

[3] | Aizenman, M. and Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 107–143. · Zbl 0586.60096 |

[4] | Alon, N. and Spencer, J. H. (2000). The Probabilistic Method , 2nd ed. Wiley, New York. · Zbl 0996.05001 |

[5] | Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520–1536. JSTOR: · Zbl 0747.60093 |

[6] | Bollobás, B. (2001). Random Graphs , 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 0979.05003 |

[7] | Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005). Random subgraphs of finite graphs: I. The scaling window under the triangle condition. Random Structures Algorithms 27 137–184. · Zbl 1076.05071 |

[8] | Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G. and Spencer, J. (2005). Random subgraphs of finite graphs: III. The phase transition for the \(n\)-cube. Combinatorica . · Zbl 1121.05108 |

[9] | Brydges, D. C. and Spencer, T. (1985). Self-avoiding walk in 5 or more dimensions. Comm. Math. Phys. 97 125–148. · Zbl 0575.60099 |

[10] | Grimmett, G. (1999). Percolation , 2nd ed. Springer, Berlin. · Zbl 0926.60004 |

[11] | Hara, T. (2005). Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. · Zbl 1142.82006 |

[12] | Hara, T., van der Hofstad, R. and Slade, G. (2003). Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 349–408. · Zbl 1044.82006 |

[13] | Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333–391. · Zbl 0698.60100 |

[14] | Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. In Probability and Phase Transition (G. R. Grimmett, ed.) 87–122. Kluwer, Dordrecht. · Zbl 0831.60107 |

[15] | Hara, T. and Slade, G. (1995). The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4 197–215. · Zbl 0838.60087 |

[16] | Hara, T. and Slade, G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 1244–1293. · Zbl 0977.82022 |

[17] | van der Hofstad, R. and Slade, G. (2002). A generalised inductive approach to the lace expansion. Probab. Theory Related Fields 122 389–430. · Zbl 1002.60095 |

[18] | van der Hofstad, R. and Slade, G. (2003). The lace expansion on a tree with application to networks of self-avoiding walks. Adv. in Appl. Math. 30 471–528. · Zbl 1027.60098 |

[19] | van der Hofstad, R. and Slade, G. (2005). Expansion in \(n^-1\) for percolation critical values on the \(n\)-cube and \(\mathbb Z^n\): The first three terms. Combin. Probab. Comput. · Zbl 1102.60090 |

[20] | van der Hofstad, R. and Slade, G. (2005). Asymptotic expansions in \(n^-1\) for percolation critical values on the \(n\)-cube and \(\mathbb Z^n\). Random Structures Algorithms . · Zbl 1077.60077 |

[21] | Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs . Wiley, New York. · Zbl 0968.05003 |

[22] | Madras, N. and Slade, G. (1993). The Self-Avoiding Walk . Birkhäuser, Boston. · Zbl 0780.60103 |

[23] | Menshikov, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 856–859. · Zbl 0615.60096 |

[24] | Nguyen, B. G. (1987). Gap exponents for percolation processes with triangle condition. J. Statist. Phys. 49 235–243. · Zbl 0962.82521 |

[25] | Slade, G. (1987). The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. 110 661–683. · Zbl 0628.60073 |

[26] | Slade, G. (1999). Lattice trees, percolation and super-Brownian motion. In Perplexing Problems in Probability : Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 35–51. Birkhäuser, Basel. · Zbl 0942.60072 |

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