Given p ∈ (0, 1), we let be the random subgraph of the d-dimensional hypercube Q^d where edges are present independently with probability p. It is well known that, as d → ∞, if then with high probability Q_p is connected; and if then with high probability Q_p consists of one giant component together with many smaller components which form the “fragment”. Here we fix , and investigate the fragment, and how it sits inside the hypercube. For example, we give asymptotic estimates for the mean numbers of components in the fragment of each size, and describe their asymptotic distributions, much extending earlier work of Weber.

中文翻译：

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超立方体的稠密随机子图的分量结构

给定p  ∈ (0, 1)，我们让d维超立方体Q ^d的随机子图，其中边以概率p独立存在。众所周知，当d  →  ∞ 时，如果有高概率Q _p是连通的；如果那么很有可能Q _p由一个巨大的组件和许多更小的组件组​​成，这些组件形成了“碎片”。在这里我们修复，并调查片段，以及它如何位于超立方体中。例如，我们给出了每个尺寸片段中成分平均数的渐近估计，并描述了它们的渐近分布，这大大扩展了 Weber 的早期工作。