The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ 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 $d \in \mathbb{N}$ fixed and $n \to \infty$, $$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$ 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 $d=2,3,4$. 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.

中文翻译：

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汉明图渗透临界点的扩展

汉明图H(d,n) 是的笛卡尔积d完整的图表n顶点。让${m=d(n-1)}$成为学位和$V = n^d$是的顶点数H(d,n）。让$p_c^{(d)}$成为债券渗透的临界点H(d,n）。我们证明，对于$d \in \mathbb{N}$固定和$n \to \infty$,$$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over { {m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$它将 [10] 中发现的渐近性扩展了一个阶。术语$O(m^{-1}V^{-1/3})$是关键窗口的宽度。为了$d=4,5,6$我们有$m^{-3} = O(m^{-1}V^{-1/3})$，所以上式表示完全渐近展开$p_c^{(d)}$. 在[16]中，我们表明该公式是研究临界键渗透的关键成分H(d,n） 为了$d=2,3,4$. 该证明对上限使用了花边扩展，对下限使用了分支随机游走的新颖比较。下界的证明也为亚临界 Erdös-Rényi 随机图的敏感性产生了一个改进的渐近线。