AI 中文总结
该研究针对Easo-Severo-Tassion割集定理的位点渗流类似物构造了反例，指出将原边割集定理朴素推广到位点渗流和顶点割集的结论不成立，反例基于修改有根二叉树并替换边为特定小装置构建。
AI 中文摘要
Easo、Severo和Tassion[EST25]证明了关于无限图中最小边割集的两个定理：均匀临界渗流参数的非平凡性等价于将顶点与无穷远分离的大小为n的最小边割集数量的指数增长，且均匀暂态是另一个充分假设。我们表明，将这些定理（的原始贡献）朴素地推广到位点渗流和顶点割集是错误的；我们的反例涉及修改一棵有根二叉树，并将边替换为小装置，这些小装置在构造割集时允许对两条局部路径中的一条选择在何处阻断，同时保持每个顶点均匀靠近超临界、均匀暂态的捷径骨架。
英文摘要
Easo, Severo and Tassion [EST25] proved two theorems about minimal edge cutsets in infinite graphs: that non-triviality of the uniform critical percolation parameter is equivalent to exponential growth of the number of minimal edge cutsets of size n separating a vertex from infinity, and that uniform transience is an alternative sufficient assumption. We show that the naive extensions of (the original contribution to) these theorems to site percolation and vertex cutsets are false; our counterexample involves modifying a rooted binary tree and replacing edges with gadgets that allow many choices for where to block (in constructing a cutset) one of two local routes, while keeping every vertex uniformly close to a supercritical, uniformly transient shortcut skeleton.
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