Let be the Erdős–Rényi graph with connection probability as N → ∞ for a fixed t ∈ (0, ∞). We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of , registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order N), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at t = 1 as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus–Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition.

中文翻译：

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稀疏 Erdős-Rényi 图的所有簇大小的大偏差原理

让是ERDOS-莱利图形与连接概率作为ñ  →交通 ∞对于固定吨 ∈（0，  ∞）。我们推导出了一个大偏差原理，用于根据微观尺寸（即有限阶）、宏观尺寸（即N阶）和介观尺寸（在之间）。速率函数明确地描述了微观和宏观分量以及介观尺寸分量中顶点的分数。此外，它清楚地捕捉到了t处众所周知的相变 = 1 作为综合图的一部分。证明依赖于基本组合学和已知的估计和子图连接概率的渐近性。我们还为乘法聚结剂的强相关模型、具有单分散初始条件的 Marcus-Lushnikov 凝固模型及其凝胶相变得出结论。