Half-density percolation on transient disk triangulations

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be the connected one-skeleton of a locally finite triangulation of an open topological disc, with a uniform bound on vertex degrees. In Bernoulli site percolation, vertices are independently open with probability $p$, and clusters are the connected components induced by the open vertices. Write $\mathbb P_p$ for this product law and \[p_c(G)=\inf\{p:\mathbb P_p(\text{some open cluster is infinite})>0\}.\] Simple random walk moves from its current vertex to a uniformly chosen neighbor. Recurrence means return to its starting vertex infinitely often almost surely; transience means finitely many returns almost surely.

Statement recorded in the supplied source.

If $G$ is transient, prove $\mathbb P_{1/2}(\text{some open cluster is infinite})=1$. [S2]

Short English statement

Must half-density site percolation on every transient bounded-degree disc triangulation contain an infinite cluster?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000050 (AMR-099-0050). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Percolation and Coarse Conformal Uniformization, June 2015; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 2, Conjecture 2.1, p. 2. https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf.