在密集区域中随机均匀超图的直径
On the diameter of random uniform hypergraphs in dense regime
AI总结：
本文研究了密集区域中随机均匀超图的直径，证明了其直径在两个点上集中，并利用 Stein-Chen 方法和耦合技术得出了相关结论。
摘要：
对于固定的自然数 $t\geq 2$，我们考虑 $t$-均匀随机超图 $\mathscr{H} (n,t,p)$，其在 $n$ 个顶点 $[n]=\{1,\ldots, n\}$ 上，每个 $t$-子集以概率 $p$ 被包含为超边，并且独立地。我们证明了在密集区域中，$\mathscr{H} (n,t,p)$ 的直径仅在两个点上集中。更具体地说，假设 $\mathcal H$ 是一个在 $n$ 个顶点上的超图，其直径记为 $diam(\mathcal H)$。我们证明，对于固定的 $t,c,d$ 常数，如果 $n$ 和 $p$（取决于 $t,c,d,n$）满足\n $$ \frac{ (t-1)^ {d} N^{d} p^{d}} {n}= \log \left( \frac{n^2}{c} \right), \text{ 其中 } N={n-1\choose t-1}, $$ $c$ 是一个正数常数，$d\geq2$ 是一个自然数，那么 $$ \lim_{n \to \infty} \mathbb{P} \left( diam( \mathcal{H}) = d \right) = e^{- \frac{c}{2}} \text{ 且 } \lim_{n \to \infty} \mathbb{P} \left( diam(\mathcal{H}) = d+1 \right) = 1- e^{- \frac{c}{2}}. $$ 特别地，当 $t = 2$ 时，对应于 Erdős-Rényi 图的直径，如 Bollobás 在 \cite[Theorem~6]{bollobas1981diameter} 中所建立的。Bollobás 的结果是通过动量方法证明的，这种方法在我们的情况下难以应用，因为模型的复杂性。在本文中，我们利用 Stein-Chen 方法和耦合技术来证明我们的结果。这种方法可能可以用于解决各种问题，特别是直径问题，在更复杂的网络中。
英文摘要：
For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently. We show that the diameter of $\mathscr{H} (n,t,p)$ is concentrated only at two points in the dense regime. More precisely, suppose $diam(\mathcal H)$ denotes the diameter of a hypergraph $\mathcal H$ on $n$ vertices. We show that, for fixed $t,c,d$ constants, if $n$ and $p$ (depends on $t,c,d,n$) satisfy $$ \frac{ (t-1)^ {d} N^{d} p^{d}} {n}= \log \left( \frac{n^2}{c} \right), \mbox{ where } N={n-1\choose t-1}, $$ $c$ is a positive constant and $d\geq2$ is a natural number, then $$ \lim_{n \to \infty} \mathbb{P} \left( diam( \mathcal{H}) = d \right) = e^{- \frac{c}{2}} \text{ and } \lim_{n \to \infty} \mathbb{P} \left( diam(\mathcal{H}) = d+1 \right) = 1- e^{- \frac{c}{2}}. $$ In particular, the case where $t = 2$ corresponds to the diameter of the Erdős-Rényi graph, as established by Bollobás in \cite[Theorem~6]{bollobas1981diameter}. Bollob\' as's result was proven using the moments method, which is challenging to apply in our context due to the complexity of the model. In this paper, we utilize the Stein-Chen method along with coupling techniques to prove our result. This approach can potentially be used to solve various problems, in particular diameter problems, in more complex networks.
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