### 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 |
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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