计数最小顶点割集及顶点传递图上点渗流在1处的间隙
Counting Minimal Vertex Cutsets and a Gap at 1 for Site Percolation on Vertex-Transitive Graphs
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中文总结 AI 辅助
本文研究顶点传递图上的点渗流，通过随机游走逃逸概率和等周型不等式给出最小顶点割集数量的指数上界，并证明临界概率在1处存在普适间隙。
中文摘要 AI 辅助
我们考虑一般图上的点渗流，并给出一个关于具有依赖于度的电导的某随机游走逃逸概率的充分条件，使得将给定顶点与无穷远分离的大小为$n$的最小顶点割集的数量在$n$上以指数方式有上界。该结果在稍强的假设下推广了Easo、Severo和Tassion的一个类似定理。此外，我们给出了一个基于等周维数或更具体地说是等周型不等式的替代充分条件。进一步地，我们的定理足以推广Panagiotis和Severo的结果，并表明存在一个普适正常数$\varepsilon_1$，使得在每个无限、连通、局部有限的顶点传递图上的点渗流满足$p_c = 1$或$p_c \leq 1-\varepsilon_1$。
英文摘要
We consider site percolation on general graphs and give a sufficient condition on the escape probability of a certain random walk with degree-dependent conductances, such that the number of minimal vertex cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This result extends an analogous theorem of Easo, Severo and Tassion under slightly stronger assumptions. Moreover, we give an alternative sufficient condition in terms of the isoperimetric dimension or, more specifically, an isoperimetric-type inequality. Furthermore, our theorem is sufficient to extend the results of Panagiotis and Severo, and show that there exists a universal positive constant $\varepsilon_1$ such that site percolation on every infinite, connected, locally finite, vertex-transitive graph satisfies $p_c = 1$ or $p_c \leq 1-\varepsilon_1$.
