SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2033-2062, January 2020.
One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph $\mathcal{G}(n,p)$, the threshold for the appearance of the unique largest component (also known as the giant component) is $p_g = n^{-1}$. More precisely, when $p$ changes from $(1-\varepsilon)p_g$ (subcritical case) to $p_g$ and then to $(1+\varepsilon)p_g$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest component increases smoothly from $O(\varepsilon^{-2}\log(\varepsilon^3 n))$ to $\Theta(n^{2/3})$ and then to $(1 \pm o(1)) 2 \varepsilon n$. Furthermore, in the supercritical case, with high probability the largest components except the giant component are trees of order $O(\varepsilon^{-2}\log(\varepsilon^3 n))$, exhibiting a structural symmetry between the subcritical random graph and the graph obtained from the supercritical random graph by deleting its giant component. As a natural generalization of random graphs and connectedness, we consider the binomial random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$ (where each $k$-tuple of vertices is present as a hyperedge with probability $p$ independently) and the following notion of high-order connectedness. Given an integer $1 \leq j \leq k-1$, two sets of $j$ vertices are called $j$-connected if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least $j$ vertices. A $j$-connected component is a maximal collection of pairwise $j$-connected $j$-tuples of vertices. Recently, the threshold for the appearance of the giant $j$-connected component in $\mathcal{H}^k(n,p)$ and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest $j$-connected components, with the help of a certain class of “hypertrees” and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant $j$-connected component.

中文翻译：

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次临界随机超图，高阶分量和超树

SIAM离散数学杂志，第34卷，第4期，第2033-2062页，2020年1月。
随机图理论的中心主题之一是按最大分量的顺序处理相变。在二项式随机图$ \ mathcal {G}（n，p）$中，唯一最大分量（也称为巨型分量）出现的阈值为$ p_g = n ^ {-1} $。更准确地说，当$ p $从$（1- \ varepsilon）p_g $（亚临界情况）变为$ p_g $，然后在$ \ varepsilon> 0 $时变为$（1+ \ varepsilon）p_g $（超临界情况）时，最大成分的顺序很有可能从$ O（\ varepsilon ^ {-2} \ log（\ varepsilon ^ 3 n））$平稳增加到$ \ Theta（n ^ {2/3}）$，然后到$（1 \ pm o（1））2 \ varepsilon n $。此外，在超临界情况下，除了巨型组件之外，最大的组件是$ O（\ varepsilon ^ {-2} \ log（\ varepsilon ^ 3 n））$阶的树，在亚临界随机图和通过删除超巨型随机图从超临界随机图获得的图之间呈现出结构对称性。作为随机图和连通性的自然概括，我们考虑二项式随机$ k $统一超图$ \ mathcal {H} ^ k（n，p）$（其中每个$ k $元组的顶点都作为超边存在）概率分别为$ p $）和以下高阶连通性的概念。给定一个整数$ 1 \ leq j \ leq k-1 $，如果两套$ j $顶点之间有一条超边，那么任何两个连续的超边在至少$ j $处相交，则称为$ j $ -connected。顶点。$ j $连接的组件是成对的$ j $连接的$ j $-元组的最大集合。最近，确定$ \ mathcal {H} ^ k（n，p）$中与$ j $相连的巨型成分出现的阈值及其顺序。在本文中，我们将仔细研究亚临界随机超图。我们借助特定类别的“超树”和相关对象，确定最大的$ j $连接的组件的结构，顺序和大小。在我们的证明中，我们结合了各种概率和枚举技术，例如生成函数和与分支过程的耦合。我们的研究将为删除次临界随机超图和从超临界随机超图获得的超图之间的对称性铺平道路，方法是删除其巨型的$ j $连接分量。$ j $连接的最大组件的顺序和大小，借助特定类别的“超树”和相关对象。在我们的证明中，我们结合了各种概率和枚举技术，例如生成函数和与分支过程的耦合。我们的研究将为删除次临界随机超图和从超临界随机超图获得的超图之间的对称性铺平道路，方法是删除其巨型的$ j $连接分量。$ j $连接的最大组件的顺序和大小，借助特定类别的“超树”和相关对象。在我们的证明中，我们结合了各种概率和枚举技术，例如生成函数和与分支过程的耦合。我们的研究将为删除次临界随机超图和从超临界随机超图获得的超图之间的对称性铺平道路，方法是删除其巨型的$ j $连接分量。