Isoperimetric bounds for critical probabilities of 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.For a nonempty finite vertex set $A$, let $\partial A=\{v\notin A:v\text{ has a neighbor in }A\}$ be its external vertex boundary, and set \[h(G)=\inf_{0<|A|<\infty}\frac{|\partial A|}{|A|},\qquad \operatorname{Dim}(G)=\sup\left\{d>0:\inf_{0<|A|<\infty}\frac{|\partial A|}{|A|^{(d-1)/d}}>0\right\}.\]

Statement recorded in the supplied source.

Prove that each of the following hypotheses suffices for $p_c(G)\le1/2$: (i) $\operatorname{Dim}(G)\ge2$; (ii) $\operatorname{Dim}(G)>1$; (iii) there is a function $f:\mathbb N\to(0,\infty)$ with $f(n)\to\infty$ such that $|\partial A|\ge f(|A|)\log|A|$ for every nonempty finite $A$. Moreover, prove that $h(G)>0$ implies $p_c(G)<1/2$. [S2]

Short English statement

Establish half-density site-percolation bounds for disc triangulations from isoperimetric dimension or boundary growth, with a strict bound under positive Cheeger constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000084 (AMR-099-0084). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini and Oded Schramm, Percolation beyond $\mathbb Z^d$, many questions and a few answers, Electronic Communications in Probability 1 (1996), 71--82, Section 2, boundary definition, p. 73; Section 3, Conjecture 3, p. 74. https://emis.de/ft/41630.