Non-self-touching paths in plane graphs

A path in a graph G is called non-self-touching if two vertices are neighbours in the path if and only if they are neighbours in the graph. We investigate the existence of doubly infinite non-self-touching paths in infinite plane graphs. The matching graph G_* of an infinite plane graph G is obtained by adding all diagonals to all faces, and it plays an important role in the theory of site percolation on G. The main result of this paper is a necessary and sufficient condition on G for the existence of a doubly infinite non-self-touching path in G_* that traverses some diagonal. This is a key step in proving, for quasi-transitive G, that the critical points of site percolation on G and G_* satisfy the strict inequality pc(G_*) < pc(G), and it complements the earlier result of Grimmett and Li (Percolation critical points of matching lattice pairs, arXiv:2205.02734), proved by different methods, concerning the case of transitive graphs. Furthermore it implies, for quasi-transitive graphs, that pᵤ(G) + pc(G) ≥ 1, with equality if and only if the graph G_Δ, obtained from G by emptying all separating triangles, is a triangulation. Here, pᵤ is the critical probability for the existence of a unique infinite open cluster.