Critical site percolation and cutsets

TL;DR

This paper investigates characterizations of the critical percolation probability in infinite graphs, confirming Kahn's vertex-cut conjecture for site percolation and disproving Lyons-Peres's edge-cut conjecture through counterexamples.

Contribution

It proves Kahn's vertex-cut characterization for site percolation on any infinite, connected, locally finite graph and provides a counterexample to the Lyons-Peres edge-cut characterization.

Findings

Confirmed Kahn's vertex-cut characterization for site percolation.

Disproved Lyons-Peres edge-cut characterization with a counterexample.

Extended understanding of percolation thresholds in infinite 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.