Benjamini–Schramm Conjecture 7 for general planar graphs

Prove that every planar graph of minimal degree at least seven satisfies $p_c<1/2$ and $p_u\geq 1-p_c$, where $p_c$ is the percolation threshold and $p_u$ is the uniqueness threshold for Bernoulli site percolation.

Background

The paper relates its percolation theorem to Conjecture 7 of Benjamini and Schramm. The conjecture concerns universal bounds for Bernoulli site percolation on planar graphs of sufficiently large minimum degree. The paper records that the two inequalities had been established only in restricted settings involving embeddings without accumulation points, leaving the general formulation as a stated conjectural problem.

References

Theorem~\ref{thm:perco} is also linked toConjecture~$7$ that states that, for any planar graph of minimal degree at least~$7$, $p_c<1/2$ and~$p_u\geq 1-p_c$, where~$p_c$ and~$p_u$ are defined by

— Planar percolation and the loop O(n) model  (2508.20917 - Glazman et al., 28 Aug 2025) in Section 1, subsection “Site percolation models on planar graphs”