### Abstract

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let p_{c}^{(d)} be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, p_{c}^{(d)} = 1/m + 2d^{2} - 1/2(d -1)^{2} 1/m^{2} + O (m^{-3})+O (m^{-1}V^{-1/3}), which extends the asymptotics found in [10] by one order. The term O(m^{-1}V^{-1/3}) is the width of the critical window. For d=4,5,6 we have & m^{-3} = O(m^{-1}V^{-1/3}), and so the above formula represents the full asymptotic expansion of p_{c}^{(d)}. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös-Rényi random graph.

Original language | English |
---|---|

Pages (from-to) | 68-100 |

Journal | Combinatorics, Probability and Computing |

Volume | 29 |

Issue number | 1 |

Early online date | 5 Aug 2019 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

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### Keywords

- 2010 MSC Codes:
- 82B43
- Primary 60K35
- Secondary 60K37

### Cite this

*Combinatorics, Probability and Computing*,

*29*(1), 68-100. https://doi.org/10.1017/S0963548319000208

}

*Combinatorics, Probability and Computing*, vol. 29, no. 1, pp. 68-100. https://doi.org/10.1017/S0963548319000208

**Expansion of percolation critical points for hamming graphs.** / Federico, Lorenzo (Corresponding author); van der Hofstad, Remco W.; den Hollander, Frank; Hulshof, Tim.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Expansion of percolation critical points for hamming graphs

AU - Federico, Lorenzo

AU - van der Hofstad, Remco W.

AU - den Hollander, Frank

AU - Hulshof, Tim

PY - 2020/1/1

Y1 - 2020/1/1

N2 - The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let pc(d) be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, pc(d) = 1/m + 2d2 - 1/2(d -1)2 1/m2 + O (m-3)+O (m-1V-1/3), which extends the asymptotics found in [10] by one order. The term O(m-1V-1/3) is the width of the critical window. For d=4,5,6 we have & m-3 = O(m-1V-1/3), and so the above formula represents the full asymptotic expansion of pc(d). In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös-Rényi random graph.

AB - The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let pc(d) be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, pc(d) = 1/m + 2d2 - 1/2(d -1)2 1/m2 + O (m-3)+O (m-1V-1/3), which extends the asymptotics found in [10] by one order. The term O(m-1V-1/3) is the width of the critical window. For d=4,5,6 we have & m-3 = O(m-1V-1/3), and so the above formula represents the full asymptotic expansion of pc(d). In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös-Rényi random graph.

KW - 2010 MSC Codes:

KW - 82B43

KW - Primary 60K35

KW - Secondary 60K37

UR - http://www.scopus.com/inward/record.url?scp=85070364860&partnerID=8YFLogxK

U2 - 10.1017/S0963548319000208

DO - 10.1017/S0963548319000208

M3 - Article

AN - SCOPUS:85070364860

VL - 29

SP - 68

EP - 100

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

IS - 1

ER -