Planar half-density percolation has no unique infinite cluster

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 $G=(V,E)$ be a connected locally finite planar graph. In independent site percolation with parameter $p$, each vertex is open with probability $p$, independently of all others. An open cluster is a connected component of the subgraph induced by open vertices. Write $N_\infty$ for the number of infinite open clusters and $\mathbb P_p$ for this product probability law.

Statement recorded in the supplied source.

At $p=1/2$, does \[\mathbb P_{1/2}(N_\infty\ge1)=1\quad\Longrightarrow\quad\mathbb P_{1/2}(N_\infty=\infty)=1\] hold for every such planar graph? This is the site-percolation question, with no transitivity or minimum-degree hypothesis. [S2]

Short English statement

Does almost-sure site percolation at probability one half on a planar graph force infinitely many infinite open clusters?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000087 (AMR-099-0087). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 I. Benjamini and O. Schramm, Percolation beyond $\mathbb Z^d$, many questions and a few answers, Electronic Communications in Probability 1 (1996), pp. 71--82, Conjecture 8, p. 80, with the conventions of Section 2. https://emis.de/ft/41630.