Multiplicative connection bounds above criticality

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 a connected locally finite nonamenable hyperbolic graph. Hyperbolic means its graph-metric geodesic triangles are uniformly thin. Nonamenable means its Cheeger constant $\inf_{0<|A|<\infty}|\partial_E A|/|A|$ is positive, where $\partial_E A$ is the set of edges with one endpoint in $A$. In Bernoulli bond percolation, each edge is independently open with probability $p$. Write $x\leftrightarrow y$ for an open connection and $p_c$ for the threshold for infinite clusters. A geodesic is a shortest unit-edge path.

Statement recorded in the supplied source.

For which $p\in[0,1]$ is there $C=C(G,p)<\infty$ such that \[\mathbb P_p(x\leftrightarrow y)\le C\mathbb P_p(x\leftrightarrow z)\mathbb P_p(z\leftrightarrow y)\] for all vertices $x,y$ and every $z$ on a geodesic from $x$ to $y$? In particular, decide this at $p_c$ and for some $p>p_c$. [S2]

Short English statement

When does percolation connectivity on a nonamenable hyperbolic graph satisfy a uniform multiplicative upper bound through points on geodesics?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000040 (AMR-099-0040). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour lecture notes, version dated 30 October 2013; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 9.2, Open Problem 9.14 and immediately preceding paragraph, p. 66. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.