Counting Minimal Vertex Cutsets and a Gap at 1 for Site Percolation on Vertex-Transitive Graphs

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 ${\epsilon }_1$ such that site percolation on every infinite, connected, locally finite, vertex-transitive graph satisfies $p_c=1$ or $p_c\le 1-{\epsilon }_1$.