Critical site percolation and cutsets
Zhongyang Li
TL;DR
This work addresses the problem of characterizing the critical site percolation threshold $p_c^{site}(G)$ on infinite graphs. It develops a differential-inequalities framework and introduces a site-percolation specific $\phi_p^v(S)$ to extend Kahn's vertex-cut characterization to all infinite, connected, locally finite graphs, removing the bounded-degree restriction. The key results show $p_c^{site}(G)=p'_{cut,V}(G)$ for every such graph, and provide a counterexample demonstrating that the Lyons–Peres edge-cut characterization for $p_c^{site}$ does not hold in general. Overall, the paper strengthens the vertex-cut perspective for site percolation and clarifies limitations of edge-cut characterizations, with implications for non-transitive and broadly general graphs.
Abstract
In 2003, Kahn conjectured a characterization of the critical percolation probability $p_c$ in terms of vertex cut sets (\cite{JK03}). Later, Lyons and Peres (2016) conjectured a similar characterization of $p_c$ , but in terms of edge cut sets (\cite{LP16}). Both conjectures were subsequently proven by Tang (\cite{pt23}) for bond percolation and site percolation on bounded-degree graphs. Tang further conjectured that Kahn's vertex-cut characterization for $p_c^{site}$ and the Lyons-Peres edge-cut characterization for $p_c^{site}$ would hold for site percolation on any infinite, connected, locally finite graph. In this paper, we establish Kahn's vertex-cut characterization for $p_c^{site}$ by adapting arguments from \cite{jmh57a, DCT15}. Additionally, we disprove the Lyons-Peres edge-cut characterization for $p_c^{site}$ by constructing a counterexample.