Critical site percolation and cutsets

In 2003, Kahn conjectured a characterization of the critical percolation probability pc in terms of vertex cut sets ([9]). Later, Lyons and Peres (2016) conjectured a similar characterization of pc, but in terms of edge cut sets ([14]). Both conjectures were subsequently proven by Tang ([16]) for bond percolation and site percolation on bounded-degree graphs. Tang further conjectured that Kahn’s vertex-cut characterization for pcsite and the Lyons-Peres edge-cut characterization for pcsite would hold for site percolation on any infinite, connected, locally finite graph. In this paper, we establish Kahn’s vertex-cut characterization for pcsite by adapting arguments from [8, 5]. Additionally, we disprove the Lyons-Peres edge-cut characterization for pcsite by constructing a counterexample.