大大小小的分支都恰如其分 II：常数度正则图上的超临界渗流
Components, large and small, are as they should be II: supercritical percolation on regular graphs of constant degree
AI总结：
本文证明在满足扩张性与局部稀疏条件的常数度正则图上，超临界边渗流存在大小约为 $yn$ 的巨型分支，其余分支仅为对数阶。
摘要：
设 $d\ge 3$ 为固定整数。令 $y:=y(p)$ 表示无限 $d$-正则树的根在 $p$-边渗流后属于某个无限簇的概率。我们证明，对于任意常数 $b,\alpha>0$ 和 $1<\lambda<d-1$，都存在常数 $c,C>0$，使以下结论成立。设 $G$ 是一个具有 $n$ 个顶点的 $d$-正则图，并满足：对于每个满足 $|U|\le \frac{n}{2}$ 的 $U\subseteq V(G)$，都有 $e(U,U^c)\ge b|U|$；且对于每个满足 $|U|\le \log^C n$ 的 $U\subseteq V(G)$，都有 $e(U)\le(1+c)|U|$。令 $p=\frac{\lambda}{d-1}$。那么，当 $n$ 趋于无穷时，以趋于一的概率，$G$ 的随机子图 $G_p$ 中最大分支 $L_1$ 满足 $\left|1-\frac{|L_1|}{yn}\right|\le\alpha$，并且 $G_p$ 中所有其他分支的阶均为 $O\left(\frac{\lambda\log n}{(\lambda-1)^2}\right)$。这推广（并改进）了关于随机 $d$-正则图的结果。
英文摘要：
Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,α>0$ and $1<λ< d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^c)\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log^Cn$, $e(U)\le (1+c)|U|$. Let $p=\fracλ{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_1$ in the random subgraph $G_p$ of $G$ satisfies $\left|1-\frac{|L_1|}{yn}\right|\le α$, and all the other components in $G_p$ are of order $O\left(\frac{λ\log n}{(λ-1)^2}\right)$. This generalises (and improves upon) results for random $d$-regular graphs.
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