论文标题
关键渗透在扩展图序列上的局部性
Locality of critical percolation on expanding graph sequences
论文作者
论文摘要
我们研究有限图上关键渗透的局部性：让$ g_n $是一系列有限图，在本地趋于本地收敛到（随机，根，根）的无限图$ g $。考虑Bernoulli Edge Percolation：在$ G $上出现无限组件的关键概率是否与出现在$ G_N $上的线性大小组件出现的关键概率相吻合？在这篇简短的文章中，我们给出了一个积极的答案，只要图形$ g_n $满足扩展条件，限制图$ g $具有有限的预期根学位。 Benjamini，Nachmias和Peres（2011）的主要结果，首先提出了这个问题，该结果表明结果假设$ g_n $满足统一的界限和统一的扩展条件，并收敛到确定性的限制$ g $。后来的Sarkar（2021）的工作扩展了结果，以允许随机限制$ G $，但仍需要统一的限制和统一的扩展，以$ g_n $。我们的结果将$ g_n $的学位替换为（温和）要求$ g $必须具有有限的预期根学位的要求。我们的证明是对先前结果的修改，使用修剪过程和控制无界度的第二刻方法方法。
We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probability for the emergence of an infinite component on $G$ coincide with the critical probability for the emergence of a linear-sized component on $G_n$? In this short article we give a positive answer provided the graphs $G_n$ satisfy an expansion condition, and the limiting graph $G$ has finite expected root degree. The main result of Benjamini, Nachmias, and Peres (2011), where this question was first formulated, showed the result assuming the $G_n$ satisfy a uniform degree bound and uniform expansion condition, and converge to a deterministic limit $G$. Later work of Sarkar (2021) extended the result to allow for a random limit $G$, but still required a uniform degree bound and uniform expansion for $G_n$. Our result replaces the degree bound on $G_n$ with the (milder) requirement that $G$ must have finite expected root degree. Our proof is a modification of the previous results, using a pruning procedure and the second moment method to control unbounded degrees.
