大小组件均符合预期I：度增长正则图上的超临界渗流
Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree
AI总结：
本文针对度增长正则图给出超临界渗流出现类G(n,p)相变的充分条件，刻画了巨分量与其余分量的规模，证明了结果的紧性，为多类已知图的渗流结论提供统一证明。
摘要：
我们针对度d随规模增长的正则图G给出了充分条件，可保证当p·d≈1时，其随机子图G_p会出现与G(n,p)相似的相变。这些条件覆盖了若干已被充分研究的图类，包括完全图K_n、二进制超立方体Q^d、d正则扩张子以及随机d正则图上的渗流场景，尤其可为这些及更多同类场景提供统一证明。假设G是含n个顶点的d正则图，其中d=ω(1)，令ε>0为小常数，取p=(1+ε)/d，记y(ε)为后代分布服从Po(1+ε)的高尔顿-沃森树的存活概率。我们证明，若G满足极弱的边扩张要求，且对G中小集合的扩张性有较好的控制，则渗流随机子图G_p通常包含一个唯一巨分量，其渐近规模为y(ε)n，其余所有分量的规模均为O(log n/ε²)量级。我们还证明该结果是紧的：若将G中小集合扩张性的控制条件略微放宽，则存在含n个顶点的d正则图G，其第二大分量的典型规模为Ω(d log(n/d))=ω(log n)。本文是两篇系列论文的首篇，后续工作将研究恒定度正则图上的超临界渗流，并在该场景下建立类似的充分（本质紧）条件。
英文摘要：
We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied graphs, such as (percolation on) the complete graph $K_n$, the binary hypercube $Q^d$, $d$-regular expanders, and random $d$-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that $G$ is a $d$-regular graph on $n$ vertices, with $d=ω(1)$. Let $ε>0$ be a small constant, and let $p=\frac{1+ε}{d}$. Let $y(ε)$ be the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. We show that if $G$ satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in $G$, then typically the percolated random subgraph $G_p$ contains a unique giant component of asymptotic order $y(ε)n$, and all the other components in $G_p$ are of order $O(\log n/ε^2)$. We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in $G$, then there are $d$-regular graphs $G$ on $n$ vertices, where typically the second largest component is of order $Ω(d\log (n/d))=ω(\log n)$. This is the first of a two-part sequence of papers. In the subsequent work, we consider supercritical percolation on regular graphs of constant degree, and establish similar sufficient (and essentially tight) conditions in that setting.
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