Binary trees in critical clusters of regular planar triangulations

Definitions and mathematical statement

Fix $k\ge7$, and let $H_k$ be the one-skeleton of the regular triangular tessellation of the hyperbolic plane in which exactly $k$ triangles meet at every vertex. Thus $H_k$ is an infinite vertex-transitive $k$-regular planar triangulation. Independently declare each edge open with probability $p\in[0,1]$.

Let $T$ be the rooted infinite binary tree, whose root has two children and whose other vertices have one parent and two children. For a fixed vertex $o\in H_k$, let $A_o$ be the event that there exists an injective map $f:V(T)\to V(H_k)$ with $f(\text{root})=o$ which sends every edge of $T$ to an open edge. Additional ambient edges between image vertices are allowed. Put \[ b_k(p)=\mathbb P_p(A_o),\qquad p_{\mathrm{bin}}(H_k)=\inf\{p:b_k(p)>0\}. \] Prove that $b_k(p_{\mathrm{bin}}(H_k))>0$.

Equivalently, at the threshold for containing a full binary-tree subgraph, some infinite open cluster already contains such a tree with positive probability. Transitivity and countability relate this global event to the rooted formulation; by ergodicity its existence then has probability one. This threshold is distinct from the ordinary threshold for existence of any infinite path.

The embedding uses actual edges, not paths of unbounded length, so subdivisions are not silently permitted. The hyperbolic range $k\ge7$ is essential for this formulation: the Euclidean $k=6$ triangular lattice has polynomial growth and cannot contain an unsubdivided infinite binary tree even at $p=1$.

Short English statement

Does the appearance of a full binary tree in percolation on a regular hyperbolic triangulation occur with a positive probability at its own threshold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000035 (AMR-099-0035), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Coarse Geometry and Randomness (Saint-Flour notes), Open problem 8.8, PDF page 58. Original source indexed by AMR-099-0035. https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=58.
• S3 I. Benjamini, Coarse Geometry and Randomness, primary Saint-Flour notes; the numbered question and its surrounding definitions. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.